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+The Project Gutenberg EBook of Encyclopaedia Britannica, 11th Edition,
+Volume 11, Slice 6, by Various
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+Title: Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6
+       "Geodesy" to "Geometry"
+
+Author: Various
+
+Release Date: September 17, 2011 [EBook #37461]
+
+Language: English
+
+Character set encoding: ISO-8859-1
+
+*** START OF THIS PROJECT GUTENBERG EBOOK ENCYCLOPAEDIA BRITANNICA ***
+
+
+
+
+Produced by Marius Masi, Don Kretz and the Online
+Distributed Proofreading Team at http://www.pgdp.net
+
+
+
+
+
+
+
+
+
+Transcriber's notes:
+
+(1) Numbers following letters (without space) like C2 were originally
+      printed in subscript. Letter subscripts are preceded by an
+      underscore, like C_n.
+
+(2) Characters following a carat (^) were printed in superscript.
+
+(3) Side-notes were relocated to function as titles of their respective
+      paragraphs.
+
+(4) Macrons and breves above letters and dots below letters were not
+      inserted.
+
+(5) [root] stands for the root symbol; [alpha], [beta], etc. for greek
+      letters.
+
+(6) The following typographical errors have been corrected:
+
+    ARTICLE GEOFFREY: "... his history in chiefly one of quarrels, with
+      the see of Canterbury, with the chancellor William Longchamp, with
+      his half-brothers Richard and John, and especially with his canons
+      at York." 'William' amended from 'Willian'.
+
+    ARTICLE GEOLOGY: "... and at the same time greater appreciation has
+      been shown of the signification and strength of the geological
+      proofs of the high antiquity of our planet." 'strength' amended
+      from 'stength'.
+
+    ARTICLE GEOLOGY: "... it can be demonstrated that sometimes an inch
+      or two of sediment might, on certain horizons, represent the
+      deposit of an enormously longer period than a hundred or a thousand
+      times the same amount of sediment on other horizons." 'might'
+      amended from 'much'.
+
+    ARTICLE GEOLOGY: "But no such extensive disturbance of the normal
+      conditions of the distribution of life can take place without
+      carrying with it many secondary effects, and setting in motion a
+      wide cycle of change and of reaction in the animal and vegetable
+      kingdoms." 'kingdoms' amended from 'kindgoms'.
+
+    ARTICLE GEOMETRY: "The bases and altitudes of equal solid
+      parallelepipeds are reciprocally proportional; and if the bases and
+      altitudes be reciprocally proportional, the solid parallelepipeds
+      are equal." 'are' amended from 'and'.
+
+    ARTICLE GEOMETRY: "An alternative method of testing a relation is
+      illustrated in the following example:--If A, B, C, D, E,
+      F be six collinear points, then" 'following example:--' amended
+      from 'example: following'.
+
+    ARTICLE GEOMETRY: "3. In an hyperbolic involution any two conjugate
+      points are harmonic conjugates with regard to the two foci." 'an'
+      amended from 'a'.
+
+    ARTICLE GEOMETRY: "If two lines, given by their projections,
+      intersect, the intersection of their planes and the intersection of
+      their elevations must lie in a line perpendicular to the axis,
+      because they must be the projections of the point common to the two
+      lines." 'planes' amended from 'plans'.
+
+    ARTICLE GEOMETRY: "Where this is the case, if [alpha] be the measure
+      of curvature, the linear element can be put into the form" 'if'
+      amended from 'it'.
+
+    ARTICLE GEOMETRY: "The development of the consequences of these
+      metrical definitions is the subject of non-Euclidean geometry."
+      'subject' amended from 'subjct'.
+
+
+
+
+          ENCYCLOPAEDIA BRITANNICA
+
+  A DICTIONARY OF ARTS, SCIENCES, LITERATURE
+           AND GENERAL INFORMATION
+
+              ELEVENTH EDITION
+
+
+            VOLUME XI, SLICE VI
+
+            GEODESY to GEOMETRY
+
+
+
+
+ARTICLES IN THIS SLICE:
+
+
+  GEODESY                          GEOFFROY, ÉTIENNE FRANĒOIS
+  GEOFFREY (Martel)                GEOFFROY, JULIEN LOUIS
+  GEOFFREY (Plantagenet)           GEOFFROY SAINT-HILAIRE, ÉTIENNE
+  GEOFFREY (duke of Brittany)      GEOFFROY SAINT-HILAIRE, ISIDORE
+  GEOFFREY (archbishop of York)    GEOGRAPHY
+  GEOFFREY DE MONTBRAY             GEOID
+  GEOFFREY OF MONMOUTH             GEOK-TEPE
+  GEOFFREY OF PARIS                GEOLOGY
+  GEOFFREY THE BAKER               GEOMETRICAL CONTINUITY
+  GEOFFRIN, MARIE THÉRČSE RODET    GEOMETRY
+
+
+
+
+GEODESY (from the Gr. [Greek: gź], the earth, and [Greek: daiein], to
+divide), the science of surveying (q.v.) extended to large tracts of
+country, having in view not only the production of a system of maps of
+very great accuracy, but the determination of the curvature of the
+surface of the earth, and eventually of the figure and dimensions of the
+earth. This last, indeed, may be the sole object in view, as was the
+case in the operations conducted in Peru and in Lapland by the
+celebrated French astronomers P. Bouguer, C.M. de la Condamine, P.L.M.
+de Maupertuis, A.C. Clairault and others; and the measurement of the
+meridian arc of France by P.F.A. Méchain and J.B.J. Delambre had for
+its end the determination of the true length of the "metre" which was to
+be the legal standard of length of France (see EARTH, FIGURE OF THE).
+
+The basis of every extensive survey is an accurate triangulation, and
+the operations of geodesy consist in the measurement, by theodolites, of
+the angles of the triangles; the measurement of one or more sides of
+these triangles on the ground; the determination by astronomical
+observations of the azimuth of the whole network of triangles; the
+determination of the actual position of the same on the surface of the
+earth by observations, first for latitude at some of the stations, and
+secondly for longitude; the determination of altitude for all stations.
+
+For the computation, the points of the actual surface of the earth are
+imagined as projected along their plumb lines on the mathematical
+figure, which is given by the stationary sea-level, and the extension of
+the sea through the continents by a system of imaginary canals. For many
+purposes the mathematical surface is assumed to be a plane; in other
+cases a sphere of radius 6371 kilometres (20,900,000 ft.). In the case
+of extensive operations the surface must be considered as a compressed
+ellipsoid of rotation, whose minor axis coincides with the earth's axis,
+and whose compression, flattening, or ellipticity is about 1/298.
+
+
+_Measurement of Base Lines._
+
+  To determine by actual measurement on the ground the length of a side
+  of one of the triangles ("base line"), wherefrom to infer the lengths
+  of all the other sides in the triangulation, is not the least
+  difficult operation of a trigonometrical survey. When the problem is
+  stated thus--To determine the number of times that a certain standard
+  or unit of length is contained between two finely marked points on the
+  surface of the earth at a distance of some miles asunder, so that the
+  error of the result may be pronounced to lie between certain very
+  narrow limits,--then the question demands very serious consideration.
+  The representation of the unit of length by means of the distance
+  between two fine lines on the surface of a bar of metal at a certain
+  temperature is never itself free from uncertainty and probable error,
+  owing to the difficulty of knowing at any moment the precise
+  temperature of the bar; and the transference of this unit, or a
+  multiple of it, to a measuring bar will be affected not only with
+  errors of observation, but with errors arising from uncertainty of
+  temperature of both bars. If the measuring bar be not
+  self-compensating for temperature, its expansion must be determined by
+  very careful experiments. The thermometers required for this purpose
+  must be very carefully studied, and their errors of division and index
+  error determined.
+
+  In order to avoid the difficulty in exactly determining the
+  temperature of a bar by the mercury thermometer, F.W. Bessel
+  introduced in 1834 near Königsberg a compound bar which constituted a
+  metallic thermometer.[1] A zinc bar is laid on an iron bar two toises
+  long, both bars being perfectly planed and in free contact, the zinc
+  bar being slightly shorter and the two bars rigidly united at one end.
+  As the temperature varies, the difference of the lengths of the bars,
+  as perceived by the other end, also varies, and affords a quantitative
+  correction for temperature variations, which is applied to reduce the
+  length to standard temperature. During the measurement of the base
+  line the bars were not allowed to come into contact, the interval
+  being measured by the insertion of glass wedges. The results of the
+  comparisons of four measuring rods with one another and with the
+  standards were elaborately computed by the method of least-squares.
+  The probable error of the measured length of 935 toises (about 6000
+  ft.) has been estimated as 1/863500 or 1.2 µ (µ denoting a millionth).
+  With this apparatus fourteen base lines were measured in Prussia and
+  some neighbouring states; in these cases a somewhat higher degree of
+  accuracy was obtained.
+
+  The principal triangulation of Great Britain and Ireland has seven
+  base lines: five have been measured by steel chains, and two, more
+  exactly, by the compensation bars of General T.F. Colby, an apparatus
+  introduced in 1827-1828 at Lough Foyle in Ireland. Ten base lines were
+  measured in India in 1831-1869 by the same apparatus. This is a system
+  of six compound-bars self-correcting for temperature. The bars may be
+  thus described: Two bars, one of brass and the other of iron, are laid
+  in parallelism side by side, firmly united at their centres, from
+  which they may freely expand or contract; at the standard temperature
+  they are of the same length. Let AB be one bar, A'B' the other; draw
+  lines through the corresponding extremities AA' (to P) and BB' (to Q),
+  and make A'P = B'Q, AA' being equal to BB'. If the ratio A'P/AP equals
+  the ratio of the coefficients of expansion of the bars A'B' and AB,
+  then, obviously, the distance PQ is constant (or nearly so). In the
+  actual instrument P and Q are finely engraved dots 10 ft. apart. In
+  practice the bars, when aligned, are not in contact, an interval of 6
+  in. being allowed between each bar and its neighbour. This distance is
+  accurately measured by an ingenious micrometrical arrangement
+  constructed on exactly the same principle as the bars themselves.
+
+  The last base line measured in India had a length of 8913 ft. In
+  consequence of some suspicion as to the accuracy of the compensation
+  apparatus, the measurement was repeated four times, the operations
+  being conducted so as to determine the actual values of the probable
+  errors of the apparatus. The direction of the line (which is at Cape
+  Comorin) is north and south. In two of the measurements the brass
+  component was to the west, in the others to the east; the differences
+  between the individual measurements and the mean of the four were
+  +0.0017, -0.0049, -0.0015, +0.0045 ft. These differences are very
+  small; an elaborate investigation of all sources of error shows that
+  the probable error of a base line in India is on the average ±2.8 µ.
+  These compensation bars were also used by Sir Thomas Maclear in the
+  measurement of the base line in his extension of Lacaille's arc at the
+  Cape. The account of this operation will be found in a volume entitled
+  _Verification and Extension of Lacaille's Arc of Meridian at the Cape
+  of Good Hope_, by Sir Thomas Maclear, published in 1866. A
+  rediscussion has been given by Sir David Gill in his _Report on the
+  Geodetic Survey of South Africa, &c., 1896_.
+
+  A very simple base apparatus was employed by W. Struve in his
+  triangulations in Russia from 1817 to 1855. This consisted of four
+  wrought-iron bars, each two toises (rather more than 13 ft.) long; one
+  end of each bar is terminated in a small steel cylinder presenting a
+  slightly convex surface for contact, the other end carries a contact
+  lever rigidly connected with the bar. The shorter arm of the lever
+  terminates below in a polished hemisphere, the upper and longer arm
+  traversing a vertical divided arc. In measuring, the plane end of one
+  bar is brought into contact with the short arm of the contact lever
+  (pushed forward by a weak spring) of the next bar. Each bar has two
+  thermometers, and a level for determining the inclination of the bar
+  in measuring. The manner of transferring the end of a bar to the
+  ground is simply this: under the end of the bar a stake is driven very
+  firmly into the ground, carrying on its upper surface a disk, capable
+  of movement in the direction of the measured line by means of
+  slow-motion screws. A fine mark on this disk is brought vertically
+  under the end of the bar by means of a theodolite which is planted at
+  a distance of 25 ft. from the stake in a direction perpendicular to
+  the base. Struve investigated for each base the probable errors of the
+  measurement arising from each of these seven causes: Alignment,
+  inclination, comparisons with standards, readings of index, personal
+  errors, uncertainties of temperature, and the probable errors of
+  adopted rates of expansion. He found that ±0.8 µ was the mean of the
+  probable errors of the seven bases measured by him. The
+  Austro-Hungarian apparatus is similar; the distance of the rods is
+  measured by a slider, which rests on one of the ends of each rod.
+  Twenty-two base lines were measured in 1840-1899.
+
+  General Carlos Ibańez employed in 1858-1879, for the measurement of
+  nine base lines in Spain, two apparatus similar to the apparatus
+  previously employed by Porro in Italy; one is complicated, the other
+  simplified. The first, an apparatus of the brothers Brunner of Paris,
+  was a thermometric combination of two bars, one of platinum and one of
+  brass, in length 4 metres, furnished with three levels and four
+  thermometers. Suppose A, B, C three micrometer microscopes very firmly
+  supported at intervals of 4 metres with their axes vertical, and
+  aligned in the plane of the base line by means of a transit
+  instrument, their micrometer screws being in the line of measurement.
+  The measuring bar is brought under say A and B, and those micrometers
+  read; the bar is then shifted and brought under B and C. By repetition
+  of this process, the reading of a micrometer indicating the end of
+  each position of the bar, the measurement is made.
+
+  Quite similar apparatus (among others) has been employed by the French
+  and Germans. Since, however, it only permitted a distance of about 300
+  m. to be measured daily, Ibańez introduced a simplification; the
+  measuring rod being made simply of steel, and provided with inlaid
+  mercury thermometers. This apparatus was used in Switzerland for the
+  measurement of three base lines. The accuracy is shown by the
+  estimated probable errors: ±0.2 µ to ±0.8 µ. The distance measured
+  daily amounts at least to 800 m.
+
+  A greater daily distance can be measured with the same accuracy by
+  means of Bessel's apparatus; this permits the ready measurement of
+  2000 m. daily. For this, however, it is important to notice that a
+  large staff and favourable ground are necessary. An important
+  improvement was introduced by Edward Jäderin of Stockholm, who
+  measures with stretched wires of about 24 metres long; these wires are
+  about 1.65 mm. in diameter, and when in use are stretched by an
+  accurate spring balance with a tension of 10 kg.[2] The nature of the
+  ground has a very trifling effect on this method. The difficulty of
+  temperature determinations is removed by employing wires made of
+  invar, an alloy of steel (64%) and nickel (36%) which has practically
+  no linear expansion for small thermal changes at ordinary
+  temperatures; this alloy was discovered in 1896 by Benōit and
+  Guillaume of the International Bureau of Weights and Measures at
+  Breteuil. Apparently the future of base-line measurements rests with
+  the invar wires of the Jäderin apparatus; next comes Porro's apparatus
+  with invar bars 4 to 5 metres long.
+
+  Results have been obtained in the United States, of great importance
+  in view of their accuracy, rapidity of determination and economy. For
+  the measurement of the arc of meridian in longitude 98° E., in 1900,
+  nine base lines of a total length of 69.2 km. were measured in six
+  months. The total cost of one base was $1231. At the beginning and at
+  the end of the field-season a distance of exactly 100 m. was measured
+  with R.S. Woodward's "5-m. ice-bar" (invented in 1891); by means of
+  the remeasurement of this length the standardization of the apparatus
+  was done under the same conditions as existed in the case of the base
+  measurements. For the measurements there were employed two steel tapes
+  of 100 m. long, provided with supports at distances of 25 m., two of
+  50 m., and the duplex apparatus of Eimbeck, consisting of four 5-m.
+  rods. Each base was divided into sections of about 1000 m.; one of
+  these, the "test kilometre," was measured with all the five apparatus,
+  the others only with two apparatus, mostly tapes. The probable error
+  was about ±0.8 µ, and the day's work a distance of about 2000 m. Each
+  of the four rods of the duplex apparatus consists of two bars of brass
+  and steel. Mercury thermometers are inserted in both bars; these serve
+  for the measurement of the length of the base lines by each of the
+  bars, as they are brought into their consecutive positions, the
+  contact being made by an elastic-sliding contact. The length of the
+  base lines may be calculated for each bar only, and also by the
+  supposition that both bars have the same temperature. The apparatus
+  thus affords three sets of results, which mutually control themselves,
+  and the contact adjustments permit rapid work. The same device has
+  been applied to the older bimetallic-compensating apparatus of
+  Bache-Würdemann (six bases, 1847-1857) and of Schott. There was also
+  employed a single rod bimetallic apparatus on F. Porro's principle,
+  constructed by the brothers Repsold for some base lines. Excellent
+  results have been more recently obtained with invar tapes.
+
+  The following results show the lengths of the same German base lines
+  as measured by different apparatus:
+
+                                                 metres.
+    Base at Berlin   1864 Apparatus of Bessel   2336·3920
+        "     "      1880     "        Brunner      ·3924
+    Base at Strehlen 1854     "        Bessel   2762·5824
+        "     "      1879     "        Brunner      ·5852
+    Old base at Bonn 1847     "        Bessel   2133·9095
+        "     "      1892     "          "          ·9097
+    New base at Bonn 1892     "          "      2512·9612
+        "     "      1892     "        Brunner      ·9696
+
+  It is necessary that the altitude above the level of the sea of every
+  part of a base line be ascertained by spirit levelling, in order that
+  the measured length may be reduced to what it would have been had the
+  measurement been made on the surface of the sea, produced in
+  imagination. Thus if l be the length of a measuring bar, h its height
+  at any given position in the measurement, r the radius of the earth,
+  then the length radially projected on to the level of the sea is l(1 -
+  h/r). In the Salisbury Plain base line the reduction to the level of
+  the sea is -0.6294 ft.
+
+  The total number of base lines measured in Europe up to the present
+  time is about one hundred and ten, nineteen of which do not exceed in
+  length 2500 metres, or about 1½ miles, and three--one in France, the
+  others in Bavaria--exceed 19,000 metres. The question has been
+  frequently discussed whether or not the advantage of a long base is
+  sufficiently great to warrant the expenditure of time that it
+  requires, or whether as much precision is not obtainable in the end by
+  careful triangulation from a short base. But the answer cannot be
+  given generally; it must depend on the circumstances of each
+  particular case. With Jäderin's apparatus, provided with invar wires,
+  bases of 20 to 30 km. long are obtained without difficulty.
+
+  [Illustration: FIG. 1.]
+
+  In working away from a base line ab, stations c, d, e, f are carefully
+  selected so as to obtain from well-shaped triangles gradually
+  increasing sides. Before, however, finally leaving the base line, it
+  is usual to verify it by triangulation thus: during the measurement
+  two or more points, as p, q (fig. 1), are marked in the base in
+  positions such that the lengths of the different segments of the line
+  are known; then, taking suitable external stations, as h, k, the
+  angles of the triangles bhp, phq, hqk, kqa are measured. From these
+  angles can be computed the ratios of the segments, which must agree,
+  if all operations are correctly performed, with the ratios resulting
+  from the measures. Leaving the base line, the sides increase up to
+  10, 30 or 50 miles occasionally, but seldom reaching 100 miles. The
+  triangulation points may either be natural objects presenting
+  themselves in suitable positions, such as church towers; or they may
+  be objects specially constructed in stone or wood on mountain tops or
+  other prominent ground. In every case it is necessary that the precise
+  centre of the station be marked by some permanent mark. In India no
+  expense is spared in making permanent the principal trigonometrical
+  stations--costly towers in masonry being erected. It is essential that
+  every trigonometrical station shall present a fine object for
+  observation from surrounding stations.
+
+
+  _Horizontal Angles._
+
+  In placing the theodolite over a station to be observed from, the
+  first point to be attended to is that it shall rest upon a perfectly
+  solid foundation. The method of obtaining this desideratum must depend
+  entirely on the nature of the ground; the instrument must if possible
+  be supported on rock, or if that be impossible a solid foundation must
+  be obtained by digging. When the theodolite is required to be raised
+  above the surface of the ground in order to command particular points,
+  it is necessary to build two scaffolds,--the outer one to carry the
+  observatory, the inner one to carry the instrument,--and these two
+  edifices must have no point of contact. Many cases of high scaffolding
+  have occurred on the English Ordnance Survey, as for instance at
+  Thaxted church, where the tower, 80 ft. high, is surmounted by a spire
+  of 90 ft. The scaffold for the observatory was carried from the base
+  to the top of the spire; that for the instrument was raised from a
+  point of the spire 140 ft. above the ground, having its bearing upon
+  timbers passing through the spire at that height. Thus the instrument,
+  at a height of 178 ft. above the ground, was insulated, and not
+  affected by the action of the wind on the observatory.
+
+  At every station it is necessary to examine and correct the
+  adjustments of the theodolite, which are these: the line of
+  collimation of the telescope must be perpendicular to its axis of
+  rotation; this axis perpendicular to the vertical axis of the
+  instrument; and the latter perpendicular to the plane of the horizon.
+  The micrometer microscopes must also measure correct quantities on the
+  divided circle or circles. The method of observing is this. Let A, B,
+  C ... be the stations to be observed taken in order of azimuth; the
+  telescope is first directed to A and the cross-hairs of the telescope
+  made to bisect the object presented by A, then the microscopes or
+  verniers of the horizontal circle (also of the vertical circle if
+  necessary) are read and recorded. The telescope is then turned to B,
+  which is observed in the same manner; then C and the other stations.
+  Coming round by continuous motion to A, it is again observed, and the
+  agreement of this second reading with the first is some test of the
+  stability of the instrument. In taking this round of angles--or "arc,"
+  as it is called on the Ordnance Survey--it is desirable that the
+  interval of time between the first and second observations of A should
+  be as small as may be consistent with due care. Before taking the next
+  arc the horizontal circle is moved through 20° or 30°; thus a
+  different set of divisions of the circle is used in each arc, which
+  tends to eliminate the errors of division.
+
+  It is very desirable that all arcs at a station should contain one
+  point in common, to which all angular measurements are thus
+  referred,--the observations on each arc commencing and ending with
+  this point, which is on the Ordnance Survey called the "referring
+  object." It is usual for this purpose to select, from among the points
+  which have to be observed, that one which affords the best object for
+  precise observation. For mountain tops a "referring object" is
+  constructed of two rectangular plates of metal in the same vertical
+  plane, their edges parallel and placed at such a distance apart that
+  the light of the sky seen through appears as a vertical line about 10"
+  in width. The best distance for this object is from 1 to 2 miles.
+
+  This method seems at first sight very advantageous; but if, however,
+  it be desired to attain the highest accuracy, it is better, as shown
+  by General Schreiber of Berlin in 1878, to measure only single angles,
+  and as many of these as possible between the directions to be
+  determined. Division-errors are thus more perfectly eliminated, and
+  errors due to the variation in the stability, &c., of the instruments
+  are diminished. This method is rapidly gaining precedence.
+
+  The theodolites used in geodesy vary in pattern and in size--the
+  horizontal circles ranging from 10 in. to 36 in. in diameter. In
+  Ramsden's 36-in. theodolite the telescope has a focal length of 36 in.
+  and an aperture of 2.5 in., the ordinarily used magnifying power being
+  54; this last, however, can of course be changed at the requirements
+  of the observer or of the weather. The probable error of a single
+  observation of a fine object with this theodolite is about 0".2. Fig.
+  2 represents an altazimuth theodolite of an improved pattern used on
+  the Ordnance Survey. The horizontal circle of 14-in. diameter is read
+  by three micrometer microscopes; the vertical circle has a diameter of
+  12 in., and is read by two microscopes. In the great trigonometrical
+  survey of India the theodolites used in the more important parts of
+  the work have been of 2 and 3 ft. diameter--the circle read by five
+  equidistant microscopes. Every angle is measured twice in each
+  position of the zero of the horizontal circle, of which there are
+  generally ten; the entire number of measures of an angle is never
+  less than 20. An examination of 1407 angles showed that the probable
+  error of an observed angle is on the average ± 0".28.
+
+  For the observations of very distant stations it is usual to employ a
+  heliotrope (from the Gr. [Greek: hźlios], sun; [Greek: tropos], a
+  turn), invented by Gauss at Göttingen in 1821. In its simplest form
+  this is a plane mirror, 4, 6, or 8 in. in diameter, capable of
+  rotation round a horizontal and a vertical axis. This mirror is placed
+  at the station to be observed, and in fine weather it is kept so
+  directed that the rays of the sun reflected by it strike the distant
+  observing telescope. To the observer the heliotrope presents the
+  appearance of a star of the first or second magnitude, and is
+  generally a pleasant object for observing.
+
+  Observations at night, with the aid of light-signals, have been
+  repeatedly made, and with good results, particularly in France by
+  General Franēois Perrier, and more recently in the United States by
+  the Coast and Geodetic Survey; the signal employed being an acetylene
+  bicycle-lamp, with a lens 5 in. in diameter. Particularly noteworthy
+  are the trigonometrical connexions of Spain and Algeria, which were
+  carried out in 1879 by Generals Ibańez and Perrier (over a distance of
+  270 km.), of Sicily and Malta in 1900, and of the islands of Elba and
+  Sardinia in 1902 by Dr Guarducci (over distances up to 230 km.); in
+  these cases artificial light was employed: in the first case electric
+  light and in the two others acetylene lamps.
+
+  [Illustration: FIG. 2.--Altazimuth Theodolite.]
+
+
+  _Astronomical Observations._
+
+  The direction of the meridian is determined either by a theodolite or
+  a portable transit instrument. In the former case the operation
+  consists in observing the angle between a terrestrial
+  object--generally a mark specially erected and capable of illumination
+  at night--and a close circumpolar star at its greatest eastern or
+  western azimuth, or, at any rate, when very near that position. If the
+  observation be made t minutes of time before or after the time of
+  greatest azimuth, the azimuth then will differ from its maximum value
+  by (450t)² sin 1" sin 2[delta]/ sin z, in seconds of angle, omitting
+  smaller terms, [delta] being the star's declination and z its zenith
+  distance. The collimation and level errors are very carefully
+  determined before and after these observations, and it is usual to
+  arrange the observations by the reversal of the telescope so that
+  collimation error shall disappear. If b, c be the level and
+  collimation errors, the correction to the circle reading is b cot z ±
+  c cosec z, b being positive when the west end of the axis is high. It
+  is clear that any uncertainty as to the real state of the level will
+  produce a corresponding uncertainty in the resulting value of the
+  azimuth,--an uncertainty which increases with the latitude and is very
+  large in high latitudes. This may be partly remedied by observing in
+  connexion with the star its reflection in mercury. In determining the
+  value of "one division" of a level tube, it is necessary to bear in
+  mind that in some the value varies considerably with the temperature.
+  By experiments on the level of Ramsden's 3-foot theodolite, it was
+  found that though at the ordinary temperature of 66° the value of a
+  division was about one second, yet at 32° it was about five seconds.
+
+  In a very excellent portable transit used on the Ordnance Survey, the
+  uprights carrying the telescope are constructed of mahogany, each
+  upright being built of several pieces glued and screwed together; the
+  base, which is a solid and heavy plate of iron, carries a reversing
+  apparatus for lifting the telescope out of its bearings, reversing it
+  and letting it down again. Thus is avoided the change of temperature
+  which the telescope would incur by being lifted by the hands of the
+  observer. Another form of transit is the German diagonal form, in
+  which the rays of light after passing through the object-glass are
+  turned by a total reflection prism through one of the transverse arms
+  of the telescope, at the extremity of which arm is the eye-piece. The
+  unused half of the ordinary telescope being cut away is replaced by a
+  counterpoise. In this instrument there is the advantage that the
+  observer without moving the position of his eye commands the whole
+  meridian, and that the level may remain on the pivots whatever be the
+  elevation of the telescope. But there is the disadvantage that the
+  flexure of the transverse axis causes a variable collimation error
+  depending on the zenith distance of the star to which it is directed;
+  and moreover it has been found that in some cases the personal error
+  of an observer is not the same in the two positions of the telescope.
+
+  To determine the direction of the meridian, it is well to erect two
+  marks at nearly equal angular distances on either side of the north
+  meridian line, so that the pole star crosses the vertical of each mark
+  a short time before and after attaining its greatest eastern and
+  western azimuths.
+
+  If now the instrument, perfectly levelled, is adjusted to have its
+  centre wire on one of the marks, then when elevated to the star, the
+  star will traverse the wire, and its exact position in the field at
+  any moment can be measured by the micrometer wire. Alternate
+  observations of the star and the terrestrial mark, combined with
+  careful level readings and reversals of the instrument, will enable
+  one, even with only one mark, to determine the direction of the
+  meridian in the course of an hour with a probable error of less than a
+  second. The second mark enables one to complete the station more
+  rapidly and gives a check upon the work. As an instance, at Findlay
+  Seat, in latitude 57° 35', the resulting azimuths of the two marks
+  were 177° 45' 37".29 ± 0".20 and 182° 17' 15".61 ± 0".13, while the
+  angle between the two marks directly measured by a theodolite was
+  found to be 4° 31' 37".43 ± 0".23.
+
+  [Illustration: FIG. 3.]
+
+  We now come to the consideration of the determination of time with the
+  transit instrument. Let fig. 3 represent the sphere stereographically
+  projected on the plane of the horizon,--ns being the meridian, we the
+  prime vertical, Z, P the zenith and the pole. Let p be the point in
+  which the production of the axis of the instrument meets the celestial
+  sphere, S the position of a star when observed on a wire whose
+  distance from the collimation centre is c. Let a be the azimuthal
+  deviation, namely, the angle wZp, b the level error so that Zp = 90° -
+  b. Let also the hour angle corresponding to p be 90° - n, and the
+  declination of the same = m, the star's declination being [delta], and
+  the latitude [phi]. Then to find the hour angle ZPS = [tau] of the
+  star when observed, in the triangles pPS, pPZ we have, since pPS = 90
+  + [tau] - n,
+
+         -Sin c = sin m sin [delta] + cos m cos [delta] sin (n - [tau]),
+          Sin m = sin b sin [phi]   - cos b cos [phi] sin a,
+    Cos m sin n = sin b cos [phi]   + cos b sin [phi] sin a.
+
+  And these equations solve the problem, however large be the errors of
+  the instrument. Supposing, as usual, a, b, m, n to be small, we have
+  at once [tau] = n + c sec [delta] + m tan [delta], which is the
+  correction to the observed time of transit. Or, eliminating m and n by
+  means of the second and third equations, and putting z for the zenith
+  distance of the star, t for the observed time of transit, the
+  corrected time is t + (a sin z + b cos z + c) / cos [delta]. Another
+  very convenient form for stars near the zenith is [tau] = b sec [phi]
+  + c sec [delta] + m (tan [delta] - tan [phi]).
+
+  Suppose that in commencing to observe at a station the error of the
+  chronometer is not known; then having secured for the instrument a
+  very solid foundation, removed as far as possible level and
+  collimation errors, and placed it by estimation nearly in the
+  meridian, let two stars differing considerably in declination be
+  observed--the instrument not being reversed between them. From these
+  two stars, neither of which should be a close circumpolar star, a good
+  approximation to the chronometer error can be obtained; thus let
+  [epsilon]1, [epsilon]2, be the apparent clock errors given by these
+  stars if [delta]1, [delta]2 be their declinations the real error is
+
+    [epsilon] = [epsilon]1 + ([epsilon]1 - [epsilon]2)
+      (tan [phi] - tan [delta]1) / (tan [delta]1 - tan [delta]2).
+
+  Of course this is still only approximate, but it will enable the
+  observer (who by the help of a table of natural tangents can compute
+  [epsilon] in a few minutes) to find the meridian by placing at the
+  proper time, which he now knows approximately, the centre wire of his
+  instrument on the first star that passes--not near the zenith.
+
+  The transit instrument is always reversed at least once in the course
+  of an evening's observing, the level being frequently read and
+  recorded. It is necessary in most instruments to add a correction for
+  the difference in size of the pivots.
+
+  The transit instrument is also used in the prime vertical for the
+  determination of latitudes. In the preceding figure let q be the point
+  in which the northern extremity of the axis of the instrument produced
+  meets the celestial sphere. Let nZq be the azimuthal deviation = a,
+  and b being the level error, Zq = 90° - b; let also nPq = [tau] and Pq
+  = [psi]. Let S' be the position of a star when observed on a wire
+  whose distance from the collimation centre is c, positive when to the
+  south, and let h be the observed hour angle of the star, viz. ZPS'.
+  Then the triangles qPS', gPZ give
+
+    -Sin c = sin [delta] cos [psi] - cos [delta] sin [psi] cos (h + [tau]),
+     Cos [psi] = sin b sin [phi] + cos b cos [phi] cos a,
+     Sin [psi] sin [tau] = cos b sin a.
+
+  Now when a and b are very small, we see from the last two equations
+  that [psi] = [phi] - b, a = [tau] sin [psi], and if we calculate
+  [phi]' by the formula cot [phi]' = cot [delta] cos h, the first
+  equation leads us to this result--
+
+    [phi] = [phi]' + (a sin z + b cos z + c)/cos z,
+
+  the correction for instrumental error being very similar to that
+  applied to the observed time of transit in the case of meridian
+  observations. When a is not very small and z is small, the formulae
+  required are more complicated.
+
+  [Illustration: FIG. 4.--Zenith Telescope constructed for the
+  International Stations at Mizusawa, Carloforte, Gaithersburg and
+  Ukiah, by Hermann Wanschaff, Berlin.]
+
+  The method of determining latitude by transits in the prime vertical
+  has the disadvantage of being a somewhat slow process, and of
+  requiring a very precise knowledge of the time, a disadvantage from
+  which the zenith telescope is free. In principle this instrument is
+  based on the proposition that when the meridian zenith distances of
+  two stars at their upper culminations--one being to the north and the
+  other to the south of the zenith--are equal, the latitude is the mean
+  of their declinations; or, if the zenith distance of a star
+  culminating to the south of the zenith be Z, its declination being
+  [delta], and that of another culminating to the north with zenith
+  distance Z' and declination [delta]', then clearly the latitude is
+  ½([delta] + [delta]') + ½(Z - Z'). Now the zenith telescope does away
+  with the divided circle, and substitutes the measurement
+  micrometrically of the quantity Z' - Z.
+
+  In fig. 4 is shown a zenith telescope by H. Wanschaff of Berlin, which
+  is the type used (according to the Central Bureau at Potsdam) since
+  about 1890 for the determination of the variations of latitude due to
+  different, but as yet imperfectly understood, influences. The
+  instrument is supported on a strong tripod, fitted with levelling
+  screws; to this tripod is fixed the azimuth circle and a long vertical
+  steel axis. Fitting on this axis is a hollow axis which carries on its
+  upper end a short transverse horizontal axis with a level. This latter
+  carries the telescope, which, supported at the centre of its length,
+  is free to rotate in a vertical plane. The telescope is thus mounted
+  eccentrically with respect to the vertical axis around which it
+  revolves. Two extremely sensitive levels are attached to the
+  telescope, which latter carries a micrometer in its eye-piece, with a
+  screw of long range for measuring differences of zenith distance. Two
+  levels are employed for controlling and increasing the accuracy. For
+  this instrument stars are selected in pairs, passing north and south
+  of the zenith, culminating within a few minutes of time and within
+  about twenty minutes (angular) of zenith distance of each other. When
+  a pair of stars is to be observed, the telescope is set to the mean of
+  the zenith distances and in the plane of the meridian. The first star
+  on passing the central meridional wire is bisected by the micrometer;
+  then the telescope is rotated very carefully through 180° round the
+  vertical axis, and the second star on passing through the field is
+  bisected by the micrometer on the centre wire. The micrometer has thus
+  measured the difference of the zenith distances, and the calculation
+  to get the latitude is most simple. Of course it is necessary to read
+  the level, and the observations are not necessarily confined to the
+  centre wire. In fact if n, s be the north and south readings of the
+  level for the south star, n', s' the same for the north star, l the
+  value of one division of the level, m the value of one division of the
+  micrometer, r, r' the refraction corrections, µ, µ' the micrometer
+  readings of the south and north star, the micrometer being supposed to
+  read from the zenith, then, supposing the observation made on the
+  centre wire,--
+
+    [phi] = ½([delta] + [delta]') + ½(µ - µ')m + ¼(n + n' - s - s')l +
+      ½(r - r').
+
+  It is of course of the highest importance that the value m of the
+  screw be well determined. This is done most effectually by observing
+  the vertical movement of a close circumpolar star when at its greatest
+  azimuth.
+
+  In a single night with this instrument a very accurate result, say
+  with a probable error of about 0".2, could be obtained for latitude
+  from, say, twenty pair of stars; but when the latitude is required to
+  be obtained with the highest possible precision, two nights at least
+  are necessary. The weak point of the zenith telescope lies in the
+  circumstance that its requirements prevent the selection of stars
+  whose positions are well fixed; very frequently it is necessary to
+  have the declinations of the stars selected for this instrument
+  specially observed at fixed observatories. The zenith telescope is
+  made in various sizes from 30 to 54 in. in focal length; a 30-in.
+  telescope is sufficient for the highest purposes and is very portable.
+  The net observation probable-error for one pair of stars is only
+  ±0".1.
+
+  The zenith telescope is a particularly pleasant instrument to work
+  with, and an observer has been known (a sergeant of Royal Engineers,
+  on one occasion) to take every star in his list during eleven hours on
+  a stretch, namely, from 6 o'clock P.M. until 5 A.M., and this on a
+  very cold November night on one of the highest points of the
+  Grampians. Observers accustomed to geodetic operations attain
+  considerable powers of endurance. Shortly after the commencement of
+  the observations on one of the hills in the Isle of Skye a storm
+  carried away the wooden houses of the men and left the observatory
+  roofless. Three observatory roofs were subsequently demolished, and
+  for some time the observatory was used without a roof, being filled
+  with snow every night and emptied every morning. Quite different,
+  however, was the experience of the same party when on the top of Ben
+  Nevis, 4406 ft. high. For about a fortnight the state of the
+  atmosphere was unusually calm, so much so, that a lighted candle could
+  often be carried between the tents of the men and the observatory,
+  whilst at the foot of the hill the weather was wild and stormy.
+
+  The determination of the difference of longitude between two stations
+  A and B resolves itself into the determination of the local time at
+  each of the stations, and the comparison by signals of the clocks at A
+  and B. Whenever telegraphic lines are available these comparisons are
+  made by telegraphy. A small and delicately-made apparatus introduced
+  into the mechanism of an astronomical clock or chronometer breaks or
+  closes by the action of the clock an electric circuit every second. In
+  order to record the minutes as well as seconds, one second in each
+  minute, namely that numbered 0 or 60, is omitted. The seconds are
+  recorded on a chronograph, which consists of a cylinder revolving
+  uniformly at the rate of one revolution per minute covered with white
+  paper, on which a pen having a slow movement in the direction of the
+  axis of the cylinder describes a continuous spiral. This pen is
+  deflected through the agency of an electromagnet every second, and
+  thus the seconds of the clock are recorded on the chronograph by
+  offsets from the spiral curve. An observer having his hand on a
+  contact key in the same circuit can record in the same manner his
+  observed times of transits of stars. The method of determination of
+  difference of longitude is, therefore, virtually as follows. After the
+  necessary observations for instrumental corrections, which are
+  recorded only at the station of observation, the clock at A is put in
+  connexion with the circuit so as to write on both chronographs,
+  namely, that at A and that at B. Then the clock at B is made to write
+  on both chronographs. It is clear that by this double operation one
+  can eliminate the effect of the small interval of time consumed in the
+  transmission of signals, for the difference of longitude obtained from
+  the one chronograph will be in excess by as much as that obtained from
+  the other will be in defect. The determination of the personal errors
+  of the observers in this delicate operation is a matter of the
+  greatest importance, as therein lies probably the chief source of
+  residual error.
+
+  These errors can nevertheless be almost entirely avoided by using the
+  impersonal micrometer of Dr Repsold (Hamburg, 1889). In this device
+  there is a movable micrometer wire which is brought by hand into
+  coincidence with the star and moved along with it; at fixed points
+  there are electrical contacts, which replace the fixed wires.
+  Experiments at the Geodetic Institute and Central Bureau at Potsdam in
+  1891 gave the following personal equations in the case of four
+  observers:--
+
+          Older Procedure.   New Procedure.
+
+    A-B      -0^s.108          -0^s.004
+    A-G      -0^s.314          -0^s.035
+    A-S      -0^s.184          -0^s.027
+    B-G      -0^s.225          +0^s.013
+    B-S      -0^s.086          -0^s.023
+    G-S      +0^s.109          -0^s.006
+
+  These results show that in the later method the personal equation is
+  small and not so variable; and consequently the repetition of
+  longitude determinations with exchanged observers and apparatus
+  entirely eliminates the constant errors, the probable error of such
+  determinations on ten nights being scarcely ±0^s.01.
+
+
+  _Calculation of Triangulation._
+
+  The surface of Great Britain and Ireland is uniformly covered by
+  triangulation, of which the sides are of various lengths from 10 to
+  111 miles. The largest triangle has one angle at Snowdon in Wales,
+  another on Slieve Donard in Ireland, and a third at Scaw Fell in
+  Cumberland; each side is over a hundred miles and the spherical excess
+  is 64". The more ordinary method of triangulation is, however, that of
+  chains of triangles, in the direction of the meridian and
+  perpendicular thereto. The principal triangulations of France, Spain,
+  Austria and India are so arranged. Oblique chains of triangles are
+  formed in Italy, Sweden and Norway, also in Germany and Russia, and in
+  the United States. Chains are composed sometimes merely of consecutive
+  plain triangles; sometimes, and more frequently in India, of
+  combinations of triangles forming consecutive polygonal figures. In
+  this method of triangulating, the sides of the triangles are generally
+  from 20 to 30 miles in length--seldom exceeding 40.
+
+  The inevitable errors of observation, which are inseparable from all
+  angular as well as other measurements, introduce a great difficulty
+  into the calculation of the sides of a triangulation. Starting from a
+  given base in order to get a required distance, it may generally be
+  obtained in several different ways--that is, by using different sets
+  of triangles. The results will certainly differ one from another, and
+  probably no two will agree. The experience of the computer will then
+  come to his aid, and enable him to say which is the most trustworthy
+  result; but no experience or ability will carry him through a large
+  network of triangles with anything like assurance. The only way to
+  obtain trustworthy results is to employ the method of least squares.
+  We cannot here give any illustration of this method as applied to
+  general triangulation, for it is most laborious, even for the simplest
+  cases.
+
+  Three stations, projected on the surface of the sea, give a spherical
+  or spheroidal triangle according to the adoption of the sphere or the
+  ellipsoid as the form of the surface. A spheroidal triangle differs
+  from a spherical triangle, not only in that the curvatures of the
+  sides are different one from another, but more especially in this
+  that, while in the spherical triangle the normals to the surface at
+  the angular points meet at the centre of the sphere, in the spheroidal
+  triangle the normals at the angles A, B, C meet the axis of revolution
+  of the spheroid in three different points, which we may designate
+  [alpha], ß, [gamma] respectively. Now the angle A of the triangle as
+  measured by a theodolite is the inclination of the planes BA[alpha]
+  and CA[alpha], and the angle at B is that contained by the planes ABß
+  and CBß. But the planes AB[alpha] and ABß containing the line AB in
+  common cut the surface in two distinct plane curves. In order,
+  therefore, that a spheroidal triangle may be exactly defined, it is
+  necessary that the nature of the lines joining the three vertices be
+  stated. In a mathematical point of view the most natural definition is
+  that the sides be geodetic or shortest lines. C.C.G. Andrae, of
+  Copenhagen, has also shown that other lines give a less convenient
+  computation.
+
+  K.F. Gauss, in his treatise, _Disquisitiones generales circa
+  superficies curvas_, entered fully into the subject of geodetic (or
+  geodesic) triangles, and investigated expressions for the angles of a
+  geodetic triangle whose sides are given, not certainly finite
+  expressions, but approximations inclusive of small quantities of the
+  fourth order, the side of the triangle or its ratio to the radius of
+  the nearly spherical surface being a small quantity of the first
+  order. The terms of the fourth order, as given by Gauss for any
+  surface in general, are very complicated even when the surface is a
+  spheroid. If we retain small quantities of the second order only, and
+  put [A], [B], [C] for the angles of the geodetic triangle, while A, B,
+  C are those of a plane triangle having sides equal respectively to
+  those of the geodetic triangle, then, [sigma] being the area of the
+  plane triangle and [a], [b], [c] the measures of curvature at the
+  angular points,
+
+    [A] = A + [sigma](2[a] + [b] + [c])/12,
+    [B] = B + [sigma]([a] + 2[b] + [c])/12,
+    [C] = C + [sigma]([a] + [b] + 2[c])/12.
+
+  For the sphere [a] = [b] = [r], and making this simplification, we
+  obtain the theorem previously given by A.M. Legendre. With the terms
+  of the fourth order, we have (after Andrae):
+
+              [epsilon]   [sigma]   /m² - a²    [a] - k \
+    [A] - A = --------- + -------k ( -------k + -------  ),
+                  3          3      \  20          4k   /
+
+              [epsilon]   [sigma]   /m² - b²    [b] - k \
+    [B] - B = --------- + -------k ( -------k + -------- ),
+                  3          3      \  20          4k   /
+
+              [epsilon]   [sigma]   /m² - c²    [c] - k \
+    [C] - C = --------- + -------k ( -------k + -------- ),
+                  3          3      \  20          4k   /
+
+  in which [epsilon] = [sigma] k {1 + (m²k / 8)}, 3m² = a² + b² + c², 3k
+  = [a] + [b] + [c]. For the ellipsoid of rotation the measure of
+  curvature is equal to 1 / [rho]n, [rho] and n being the radii of
+  curvature of the meridian and perpendicular.
+
+  It is rarely that the terms of the fourth order are required. As a
+  rule spheroidal triangles are calculated as spherical (after
+  Legendre), i.e. like plane triangles with a decrease of each angle of
+  about [epsilon] / 3; [epsilon] must, however, be calculated for each
+  triangle separately with its mean measure of curvature k.
+
+  The geodetic line being the shortest that can be drawn on any surface
+  between two given points, we may be conducted to its most important
+  characteristics by the following considerations: let p, q be adjacent
+  points on a curved surface; through s the middle point of the chord pq
+  imagine a plane drawn perpendicular to pq, and let S be any point in
+  the intersection of this plane with the surface; then pS + Sq is
+  evidently least when sS is a minimum, which is when sS is a normal to
+  the surface; hence it follows that of all plane curves on the surface
+  joining p, q, when those points are indefinitely near to one another,
+  that is the shortest which is made by the normal plane. That is to
+  say, the osculating plane at any point of a geodetic line contains the
+  normal to the surface at that point. Imagine now three points in
+  space, A, B, C, such that AB = BC = c; let the direction cosines of AB
+  be l, m, n, those of BC l', m', n', then x, y, z being the
+  co-ordinates of B, those of A and C will be respectively--
+
+    x - cl : y - cm : z - cn
+    x + cl': y + cm': z + cn'.
+
+  Hence the co-ordinates of the middle point M of AC are x + ½c(l' - l),
+  y + ½c(m' - m), z + ½c(n' - n), and the direction cosines of BM are
+  therefore proportional to l' - l : m' - m : n' - n. If the angle made
+  by BC with AB be indefinitely small, the direction cosines of BM are
+  as [delta]l : [delta]m : [delta]n. Now if AB, BC be two contiguous
+  elements of a geodetic, then BM must be a normal to the surface, and
+  since [delta]l, [delta]m, [delta]n are in this case represented by
+  [delta](dx/ds), [delta](dy/ds), [delta](dz/ds), and if the equation of
+  the surface be u = 0, we have
+
+    d²x   / du   d²y   / du   d²z   / du
+    ---  /  -- = ---  /  -- = ---  /  --,
+    ds² /   dx   ds² /   dy   ds² /   dz
+
+  which, however, are equivalent to only one equation. In the case of
+  the spheroid this equation becomes
+
+      d²x     d²y
+    y --- - x --- = 0,
+      ds²     ds²
+
+  which integrated gives ydx - xdy = Cds. This again may be put in the
+  form r sin a = C, where a is the azimuth of the geodetic at any
+  point--the angle between its direction and that of the meridian--and r
+  the distance of the point from the axis of revolution.
+
+  From this it may be shown that the azimuth at A of the geodetic
+  joining AB is not the same as the astronomical azimuth at A of B or
+  that determined by the vertical plane A[alpha]B. Generally speaking,
+  the geodetic lies between the two plane section curves joining A and B
+  which are formed by the two vertical planes, supposing these points
+  not far apart. If, however, A and B are nearly in the same latitude,
+  the geodetic may cross (between A and B) that plane curve which lies
+  nearest the adjacent pole of the spheroid. The condition of crossing
+  is this. Suppose that for a moment we drop the consideration of the
+  earth's non-sphericity, and draw a perpendicular from the pole C on
+  AB, meeting it in S between A and B. Then A being that point which is
+  nearest the pole, the geodetic will cross the plane curve if AS be
+  between ¼AB and 3/8 AB. If AS lie between this last value and ½AB, the
+  geodetic will lie wholly to the north of both plane curves, that is,
+  supposing both points to be in the northern hemisphere.
+
+  The difference of the azimuths of the vertical section AB and of the
+  geodetic AB, i.e. the astronomical and geodetic azimuths, is very
+  small for all observable distances, being approximately:--
+
+  Geod. azimuth = Astr. azimuth -(1/12) [e²/(1 - e²)] (s²/[rho]n)
+  (cos²[phi] sin 2[alpha] + (s/4a)|sin 2[phi] sin [alpha]), in which: e
+  and a are the numerical eccentricity and semi-major axis respectively
+  of the meridian ellipse, [phi] and [alpha] are the latitude and
+  azimuth at A, s = AB, and [rho] and n are the radii of curvature of
+  the meridian and perpendicular at A. For s = 100 kilometres, only the
+  first term is of moment; its value is 0".028 cos² [phi] sin 2[alpha],
+  and it lies well within the errors of observation. If we imagine the
+  geodetic AB, it will generally trisect the angles between the vertical
+  sections at A and B, so that the geodetic at A is near the vertical
+  section AB, and at B near the section BA.[3] The greatest distance of
+  the vertical sections one from another is e²s³ cos² [phi]0 sin
+  2[alpha]0/16a², in which [phi]0 and [alpha]0 are the mean latitude and
+  azimuth respectively of the middle point of AB. For the value s = 64
+  kilometres, the maximum distance is 3 mm.
+
+  An idea of the course of a longer geodetic line may be gathered from
+  the following example. Let the line be that joining Cadiz and St
+  Petersburg, whose approximate positions are--
+
+        Cadiz.         St Petersburg.
+    Lat. 36° 22' N.      59° 56' N.
+    Long. 6° 18' W.      30° 17' E.
+
+  If G be the point on the geodetic corresponding to F on that one of
+  the plane curves which contains the normal at Cadiz (by
+  "corresponding" we mean that F and G are on a meridian) then G is to
+  the north of F; at a quarter of the whole distance from Cadiz GF is
+  458 ft., at half the distance it is 637 ft., and at three-quarters it
+  is 473 ft. The azimuth of the geodetic at Cadiz differs 20" from that
+  of the vertical plane, which is the astronomical azimuth.
+
+  The azimuth of a geodetic line cannot be observed, so that the line
+  does not enter of necessity into practical geodesy, although many
+  formulae connected with its use are of great simplicity and elegance.
+  The geodetic line has always held a more important place in the
+  science of geodesy among the mathematicians of France, Germany and
+  Russia than has been assigned to it in the operations of the English
+  and Indian triangulations. Although the observed angles of a
+  triangulation are not geodetic angles, yet in the calculation of the
+  distance and reciprocal bearings of two points which are far apart,
+  and are connected by a long chain of triangles, we may fall upon the
+  geodetic line in this manner:--
+
+  If A, Z be the points, then to start the calculation from A, we obtain
+  by some preliminary calculation the approximate azimuth of Z, or the
+  angle made by the direction of Z with the side AB or AC of the first
+  triangle. Let P1 be the point where this line intersects BC; then, to
+  find P2, where the line cuts the next triangle side CD, we make the
+  angle BP1P2 such that BP1P2 + BP1A = 180°. This fixes P2, and P3 is
+  fixed by a repetition of the same process; so for P4, P5 .... Now it
+  is clear that the points P1, P2, P3 so computed are those which would
+  be actually fixed by an observer with a theodolite, proceeding in the
+  following manner. Having set the instrument up at A, and turned the
+  telescope in the direction of the computed bearing, an assistant
+  places a mark P1 on the line BC, adjusting it till bisected by the
+  cross-hairs of the telescope at A. The theodolite is then placed over
+  P1, and the telescope turned to A; the horizontal circle is then moved
+  through 180°. The assistant then places a mark P2 on the line CD, so
+  as to be bisected by the telescope, which is then moved to P2, and in
+  the same manner P3 is fixed. Now it is clear that the series of points
+  P1, P2, P3 approaches to the geodetic line, for the plane of any two
+  consecutive elements P_(n-1) P_n, P_n P_(n+1) contains the normal at
+  P_n.
+
+  If the objection be raised that not the geodetic azimuths but the
+  astronomical azimuths are observed, it is necessary to consider that
+  the observed vertical sections do not correspond to points on the
+  sea-level but to elevated points. Since the normals of the ellipsoid
+  of rotation do not in general intersect, there consequently arises an
+  influence of the height on the azimuth. In the case of the measurement
+  of the azimuth from A to B, the instrument is set to a point A' over
+  the surface of the ellipsoid (the sea-level), and it is then adjusted
+  to a point B', also over the surface, say at a height h'. The vertical
+  plane containing A' and B' also contains A but not B: it must
+  therefore be rotated through a small azimuth in order to contain B.
+  The correction amounts approximately to -e²h' cos²[phi] sin
+  2[alpha]/2a; in the case of h' = 1000 m., its value is 0".108
+  cos²[phi] sin 2[alpha].
+
+  This correction is therefore of greater importance in the case of
+  observed azimuths and horizontal angles than in the previously
+  considered case of the astronomical and the geodetic azimuths. The
+  observed azimuths and horizontal angles must therefore also be
+  corrected in the case, where it is required to dispense with geodetic
+  lines.
+
+  When the angles of a triangulation have been adjusted by the method of
+  least squares, and the sides are calculated, the next process is to
+  calculate the latitudes and longitudes of all the stations starting
+  from one given point. The calculated latitudes, longitudes and
+  azimuths, which are designated geodetic latitudes, longitudes and
+  azimuths, are not to be confounded with the observed latitudes,
+  longitudes and azimuths, for these last are subject to somewhat large
+  errors. Supposing the latitudes of a number of stations in the
+  triangulation to be observed, practically the mean of these determines
+  the position in latitude of the network, taken as a whole. So the
+  orientation or general azimuth of the whole is inferred from all the
+  azimuth observations. The triangulation is then supposed to be
+  projected on a spheroid of given elements, representing as nearly as
+  one knows the real figure of the earth. Then, taking the latitude of
+  one point and the direction of the meridian there as given--obtained,
+  namely, from the astronomical observations there--one can compute the
+  latitudes of all the other points with any degree of precision that
+  may be considered desirable. It is necessary to employ for this
+  purpose formulae which will give results true even for the longest
+  distances to the second place of decimals of seconds, otherwise there
+  will arise an accumulation of errors from imperfect calculation which
+  should always be avoided. For very long distances, eight places of
+  decimals should be employed in logarithmic calculations; if seven
+  places only are available very great care will be required to keep the
+  last place true. Now let [phi], [phi]' be the latitudes of two
+  stations A and B; [alpha], [alpha]^* their mutual azimuths counted
+  from north by east continuously from 0° to 360°; [omega] their
+  difference of longitude measured from west to east; and s the distance
+  AB.
+
+  First compute a latitude [phi]1 by means of the formula [phi]1 = [phi]
+  + (s cos [alpha]) / [rho], where [rho] is the radius of curvature of
+  the meridian at the latitude [phi]; this will require but four places
+  of logarithms. Then, in the first two of the following, five places
+  are sufficient--
+
+                   s²                                   s²
+    [epsilon] = ------- sin [alpha] cos a,   [eta] = ------- sin²[alpha] tan[phi]1,
+                2[rho]n                              2[rho]n
+
+                      s
+    [phi]' - [phi] = ---- cos ([alpha] - 2/3[epsilon]) - [eta],
+                     rho0
+
+              s sin (alpha - 1/3[epsilon])
+    [omega] = ----------------------------,
+               n cos ([phi]' + 1/3[eta])
+
+    [alpha]^* - [alpha] = [omega] sin ([phi]' + 2/3[eta]) - [epsilon] + 180°.
+
+  Here n is the normal or radius of curvature perpendicular to the
+  meridian; both n and [rho] correspond to latitude [phi]1, and [rho]0
+  to latitude ½([phi] + [phi]'). For calculations of latitude and
+  longitude, tables of the logarithmic values of [rho] sin 1", n sin 1",
+  and 2 n [rho] sin 1" are necessary. The following table contains these
+  logarithms for every ten minutes of latitude from 52° to 53° computed
+  with the elements a = 20926060 and a : b = 295 : 294 :--
+
+    +------+------------------+--------------+--------------------+
+    |      |          1       |        1     |           1        |
+    | Lat. | Log.------------.| Log.--------.| Log.--------------.|
+    |      |     [rho] sin 1" |     n sin 1" |     2[rho]n sin 1" |
+    +------+------------------+--------------+--------------------+
+    | °  ' |                  |              |                    |
+    |52  0 |    7.9939434     |  7.9928231   |      0.37131       |
+    |   10 |         9309     |       8190   |           29       |
+    |   20 |         9185     |       8148   |           28       |
+    |   30 |         9060     |       8107   |           26       |
+    |   40 |         8936     |       8065   |           24       |
+    |   50 |         8812     |       8024   |           23       |
+    |53  0 |         8688     |       7982   |           22       |
+    +------+------------------+--------------+--------------------+
+
+  The logarithm in the last column is that required also for the
+  calculation of spherical excesses, the spherical excess of a triangle
+  being expressed by a b sin (C/2[rho]n) sin 1".
+
+  It is frequently necessary to obtain the co-ordinates of one point
+  with reference to another point; that is, let a perpendicular arc be
+  drawn from B to the meridian of A meeting it in P, then, [alpha] being
+  the azimuth of B at A, the co-ordinates of B with reference to A are
+
+    AP = s cos ([alpha] - 2/3[epsilon]), BP = s sin ([alpha] -
+      1/3[epsilon]),
+
+  where [epsilon] is the spherical excess of APB, viz. s² sin [alpha]
+  cos [alpha] multiplied by the quantity whose logarithm is in the
+  fourth column of the above table.
+
+  If it be necessary to determine the geographical latitude and
+  longitude as well as the azimuths to a greater degree of accuracy than
+  is given by the above formulae, we make use of the following formula:
+  given the latitude [phi] of A, and the azimuth [alpha] and the
+  distance s of B, to determine the latitude [phi]' and longitude
+  [omega] of B, and the back azimuth [alpha]'. Here it is understood
+  that [alpha]' is symmetrical to [alpha], so that [alpha]^* + [alpha]'
+  = 360°.
+
+  Let
+
+    [theta] = s [Delta] / a, where [Delta] = (1 - e² sin² [phi])^½
+
+  and
+
+           e² [theta]²
+    [xi] = -----------  cos² [phi] sin 2[alpha],
+           (4 (1 - e²)
+
+            e² [theta]³
+    [xi]' = ----------- cos² [phi] cos² [alpha];
+            (6 (1 - e²)
+
+  [xi], [xi]' are always very minute quantities even for the longest
+  distances; then, putting [kappa] = 90° - [phi],
+
+       [alpha]' + [xi] - [omega]   sin ½([kappa] - [theta] - [xi]')     [alpha]
+    tan------------------------- = -------------------------------- cot -------
+                   2               sin ½([kappa] + [theta] + [xi]')        2
+
+       [alpha]' + [xi] + [omega]   cos ½([kappa] - [theta] - [xi]')     [alpha]
+    tan------------------------- = -------------------------------- cot -------
+                   2               cos ½([kappa] + [theta] + [xi]')        2
+
+                        s sin ½([alpha]' + [xi] - [alpha])    /    [theta]²     [alpha]' - [alpha]\
+    [phi]' - [phi] = --------------------------------------- ( 1 + --------cos² ------------------ );
+                     [rho]0 sin ½([alpha]' + [xi] + [alpha])  \       12                 2        /
+
+  here [rho]0 is the radius of curvature of the meridian for the mean
+  latitude ½([phi] + [phi]'). These formulae are approximate only, but
+  they are sufficiently precise even for very long distances.
+
+  For lines of any length the formulae of F.W. Bessel (_Astr. Nach._,
+  1823, iv. 241) are suitable.
+
+  If the two points A and B be defined by their geographical
+  co-ordinates, we can accurately calculate the corresponding
+  astronomical azimuths, i.e. those of the vertical section, and then
+  proceed, in the case of not too great distances, to determine the
+  length and the azimuth of the shortest lines. For _any_ distances
+  recourse must again be made to Bessel's formula.[4]
+
+  Let [alpha], [alpha]' be the mutual azimuths of two points A, B on a
+  spheroid, k the chord line joining them, µ, µ' the angles made by the
+  chord with the normals at A and B, [phi], [phi]', [omega] their
+  latitudes and difference of longitude, and (x² + y²)/a² + z² b² = 1
+  the equation of the surface; then if the plane xz passes through A the
+  co-ordinates of A and B will be
+
+    x = (a/[Delta]) cos [phi], x' = (a/[Delta]') cos [phi]' cos [omega],
+
+    y = 0                      y' = (a/[Delta]') cos [phi]' sin [omega],
+
+    z = (a/[Delta]) (1 - e²) sin [phi], z' = (a/[Delta]') (1 - e²) sin [phi]',
+
+  where [Delta] = (1 - e² sin² [phi])^½, [Delta]' = (1 - e² sin²
+  [phi]')^½, and e is the eccentricity. Let f, g, h be the direction
+  cosines of the normal to that plane which contains the normal at A and
+  the point B, and whose inclinations to the meridian plane of A is =
+  [alpha]; let also l, m, n and l', m', n' be the direction cosines of
+  the normal at A, and of the tangent to the surface at A which lies in
+  the plane passing through B, then since the first line is
+  perpendicular to each of the other two and to the chord k, whose
+  direction cosines are proportional to x' - x, y' - y, z' - z, we have
+  these three equations
+
+    f(x' - x) + gy' + h(z' - z) = 0
+
+                fl + gm + hn    = 0
+
+                fl' + gm' + hn' = 0.
+
+  Eliminate f, g, h from these equations, and substitute
+
+    l = cos [phi]   l' = - sin [phi] cos [alpha]
+
+    m = 0           m' = sin [alpha]
+
+    n = sin [phi]   n' = cos [phi] cos [alpha],
+
+  and we get
+
+    (x' - x) sin [phi] + y' cot [alpha] - (z' - z) cos [phi] = 0.
+
+  The substitution of the values of x, z, x', y', z' in this equation
+  will give immediately the value of cot [alpha]; and if we put [zeta],
+  [zeta]' for the corresponding azimuths on a sphere, or on the
+  supposition e = 0, the following relations exist
+
+                                     cos [phi] Q
+    cot [alpha] - cot [zeta] = e² ------------------
+                                  cos [phi]' [Delta]
+
+                                       cos [phi]' Q
+    cot [alpha]' - cot [zeta]' = e² ------------------
+                                    cos [phi] [Delta]'
+
+    [Delta]' sin [phi] - [Delta] sin [phi]' = Q sin [omega].
+
+  If from B we let fall a perpendicular on the meridian plane of A, and
+  from A let fall a perpendicular on the meridian plane of B, then the
+  following equations become geometrically evident:
+
+    k sin µ sin [alpha] = (a/[Delta]') cos [phi]' sin [omega]
+
+    k sin µ' sin [alpha]' = (a/[Delta]) cos [phi] sin [omega].
+
+  Now in any surface u = 0 we have
+
+    k² = (x' - x)² + (y' - y)² + (z' - z)²
+               _                                      _
+              |         du            du            du |  /   / du²   du²   du² \ ½
+    -cos µ =  |(x' - x) -- + (y' - y) -- + (z' - z) -- | / k (  --- + --- + ---  )
+              |_        dx            dy            dz_|/     \ dx²   dy²   dz² /
+               _                                         _
+              |         du             du             du  |  /   / du²    du²    du²  \ ½
+    -cos µ' = |(x' - x) --- + (y' - y) --- + (z' - z) --- | / k (  ---- + ---- + ----  ).
+              |_        dx'            dy'            dz'_|/     \ dx'²   dy'²   dz'² /
+
+  In the present case, if we put
+
+        xx'   zz'
+    1 - --- - --- = U,
+        a²    b²
+
+  then
+
+    k²            /z' - z \ ²
+    -- = 2U - e² ( ------  )
+    a²            \   b   /
+
+    cos µ = (a/k) [Delta]U; cos µ' = (a/k) [Delta]'U.
+
+  Let u be such an angle that
+
+    (1 - e²)^½ sin [phi] = [Delta] sin u
+
+               cos [phi] = [Delta] cos u,
+
+  then on expressing x, x', z, z' in terms of u and u',
+
+    U = 1 - cos u cos u' cos [omega] - sin u sin u';
+
+  also, if v be the third side of a spherical triangle, of which two
+  sides are ½[pi] - u and ½[pi] - u' and the included angle [omega],
+  using a subsidiary angle [psi] such that
+
+    sin [psi] sin ½v = e sin ½(u' - u) cos ½(u' + u),
+
+  we obtain finally the following equations:--
+
+                      k = 2a cos [psi] sin ½v
+
+                  cos µ = [Delta] sec [psi] sin ½v
+
+                 cos µ' = [Delta]' sec [psi] sin ½v
+
+      sin µ sin [alpha] = (a/k) cos u' sin [omega]
+
+    sin µ' sin [alpha]' = (a/k) cos u sin [omega].
+
+  These determine rigorously the distance, and the mutual zenith
+  distances and azimuths, of any two points on a spheroid whose
+  latitudes and difference of longitude are given.
+
+  By a series of reductions from the equations containing [zeta],
+  [zeta]' it may be shown that
+
+    [alpha] + [alpha]' = [zeta] + [zeta]' + ¼e^4[omega]([phi]' - [phi])²
+      cos^4 [phi]0 sin [phi]0 + ...,
+
+  where [phi]0 is the mean of [phi] and [phi]', and the higher powers of
+  e are neglected. A short computation will show that the small quantity
+  on the right-hand side of this equation cannot amount even to the
+  thousandth part of a second for k < 0.1a, which is, practically
+  speaking, zero; consequently the sum of the azimuths [alpha] +
+  [alpha]' on the spheroid is equal to the sum of the spherical
+  azimuths, whence follows this very important theorem (known as Dalby's
+  theorem). If [phi], [phi]' be the latitudes of two points on the
+  surface of a spheroid, [omega] their difference of longitude, [alpha],
+  [alpha]' their reciprocal azimuths,
+
+    tan ½[omega] = cot ½([alpha] + [alpha]') {cos ½([phi]' - [phi])/
+      sin ½([phi]' + [phi])}.
+
+  The computation of the geodetic from the astronomical azimuths has
+  been given above. From k we can now compute the length s of the
+  vertical section, and from this the shortest length. The difference of
+  length of the geodetic line and either of the plane curves is
+
+    e^4 s^5 cos^4 [phi]0 sin² 2[alpha]0/360 a^4.
+
+  At least this is an approximate expression. Supposing s = 0.1a, this
+  quantity would be less than one-hundredth of a millimetre. The line s
+  is now to be calculated as a circular arc with a mean radius r along
+  AB. If [phi]0 = ½([phi] + [phi]'), [alpha]0 = ½(180° + [alpha] -
+  [alpha]'), [Delta]0 = (1 - e² sin² [phi]0)^½, then 1/r = [Delta]0/a [1
+  + e²/(1 - e²) (cos² [phi]0 cos² [alpha]0)], and approximately sin
+  (s/2r) = k/2r. These formulae give, in the case of k = 0.1a, values
+  certain to eight logarithmic decimal places. An excellent series of
+  formulae for the solution of the problem, to determine the azimuths,
+  chord and distance along the surface from the geographical
+  co-ordinates, was given in 1882 by Ch. M. Schols (_Archives
+  Néerlandaises_, vol. xvii.).
+
+
+  _Irregularities of the Earth's Surface._
+
+  In considering the effect of unequal distribution of matter in the
+  earth's crust on the form of the surface, we may simplify the matter
+  by disregarding the considerations of rotation and eccentricity. In
+  the first place, supposing the earth a sphere covered with a film of
+  water, let the density [rho] be a function of the distance from the
+  centre so that surfaces of equal density are concentric spheres. Let
+  now a disturbance of the arrangement of matter take place, so that the
+  density is no longer to be expressed by [rho], a function of r only,
+  but is expressed by [rho] + [rho]', where [rho]' is a function of
+  three co-ordinates [theta], [phi], r. Then [rho]' is the density of
+  what may be designated disturbing matter; it is positive in some
+  places and negative in others, and the whole quantity of matter whose
+  density is [rho]' is zero. The previously spherical surface of the sea
+  of radius a now takes a new form. Let P be a point on the disturbed
+  surface, P' the corresponding point vertically below it on the
+  undisturbed surface, PP' = N. The knowledge of N over the whole
+  surface gives us the form of the disturbed or actual surface of the
+  sea; it is an equipotential surface, and if V be the potential at P of
+  the disturbing matter [rho]', M the mass of the earth (the
+  attraction-constant is assumed equal to unity)
+
+      M             M    M
+    ----- + V = C = -- - -- N + V.
+    a + N           a    a²
+
+  As far as we know, N is always a very small quantity, and we have with
+  sufficient approximation N = 3V/4[pi][delta]a, where [delta] is the
+  mean density of the earth. Thus we have the disturbance in elevation
+  of the sea-level expressed in terms of the potential of the disturbing
+  matter. If at any point P the value of N remain constant when we pass
+  to any adjacent point, then the actual surface is there parallel to
+  the ideal spherical surface; as a rule, however, the normal at P is
+  inclined to that at P', and astronomical observations have shown that
+  this inclination, the deflection or deviation, amounting ordinarily to
+  one or two seconds, may in some cases exceed 10", or, as at the foot
+  of the Himalayas, even 60". By the expression "mathematical figure of
+  the earth" we mean the surface of the sea produced in imagination so
+  as to percolate the continents. We see then that the effect of the
+  uneven distribution of matter in the crust of the earth is to produce
+  small elevations and depressions on the mathematical surface which
+  would be otherwise spheroidal. No geodesist can proceed far in his
+  work without encountering the irregularities of the mathematical
+  surface, and it is necessary that he should know how they affect his
+  astronomical observations. The whole of this subject is dealt with in
+  his usual elegant manner by Bessel in the _Astronomische Nachrichten_,
+  Nos. 329, 330, 331, in a paper entitled "Ueber den Einfluss der
+  Unregelmässigkeiten der Figur der Erde auf geodätische Arbeiten, &c."
+  But without entering into further details it is not difficult to see
+  how local attraction at any station affects the determinations of
+  latitude, longitude and azimuth there.
+
+  Let there be at the station an attraction to the north-east throwing
+  the zenith to the south-west, so that it takes in the celestial sphere
+  a position Z', its undisturbed position being Z. Let the rectangular
+  components of the displacement ZZ' be [xi] measured southwards and
+  [eta] measured westwards. Now the great circle joining Z' with the
+  pole of the heavens P makes there an angle with the meridian PZ =
+  [eta] cosec PZ' = [eta] sec [phi], where [phi] is the latitude of the
+  station. Also this great circle meets the horizon in a point whose
+  distance from the great circle PZ is [eta] sec [phi] sin [phi] = [eta]
+  tan [phi]. That is, a meridian mark, fixed by observations of the pole
+  star, will be placed that amount to the east of north. Hence the
+  observed latitude requires the correction [xi]; the observed longitude
+  a correction [eta] sec [phi]; and any observed azimuth a correction
+  [eta] tan [phi]. Here it is supposed that azimuths are measured from
+  north by east, and longitudes eastwards. The horizontal angles are
+  also influenced by the deflections of the plumb-line, in fact, just as
+  if the direction of the vertical axis of the theodolite varied by the
+  same amount. This influence, however, is slight, so long as the sights
+  point almost horizontally at the objects, which is always the case in
+  the observation of distant points.
+
+  The expression given for N enables one to form an approximate estimate
+  of the effect of a compact mountain in raising the sea-level. Take,
+  for instance, Ben Nevis, which contains about a couple of cubic miles;
+  a simple calculation shows that the elevation produced would only
+  amount to about 3 in. In the case of a mountain mass like the
+  Himalayas, stretching over some 1500 miles of country with a breadth
+  of 300 and an average height of 3 miles, although it is difficult or
+  impossible to find an expression for V, yet we may ascertain that an
+  elevation amounting to several hundred feet may exist near their base.
+  The geodetical operations, however, rather negative this idea, for it
+  was shown by Colonel Clarke (_Phil. Mag._, 1878) that the form of the
+  sea-level along the Indian arc departs but slightly from that of the
+  mean figure of the earth. If this be so, the action of the Himalayas
+  must be counteracted by subterranean tenuity.
+
+  Suppose now that A, B, C, ... are the stations of a network of
+  triangulation projected on or lying on a spheroid of semiaxis major
+  and eccentricity a, e, this spheroid having its axis parallel to the
+  axis of rotation of the earth, and its surface coinciding with the
+  mathematical surface of the earth at A. Then basing the calculations
+  on the observed elements at A, the calculated latitudes, longitudes
+  and directions of the meridian at the other points will be the true
+  latitudes, &c., of the points as projected on the spheroid. On
+  comparing these geodetic elements with the corresponding astronomical
+  determinations, there will appear a system of differences which
+  represent the inclinations, at the various points, of the actual
+  irregular surface to the surface of the spheroid of reference. These
+  differences will suggest two things,--first, that we may improve the
+  agreement of the two surfaces, by not restricting the spheroid of
+  reference by the condition of making its surface coincide with the
+  mathematical surface of the earth at A; and secondly, by altering the
+  form and dimensions of the spheroid. With respect to the first
+  circumstance, we may allow the spheroid two degrees of freedom, that
+  is, the normals of the surfaces at A may be allowed to separate a
+  small quantity, compounded of a meridional difference and a difference
+  perpendicular to the same. Let the spheroid be so placed that its
+  normal at A lies to the north of the normal to the earth's surface by
+  the small quantity [xi] and to the east by the quantity [eta]. Then in
+  starting the calculation of geodetic latitudes, longitudes and
+  azimuths from A, we must take, not the observed elements [phi],
+  [alpha], but for [phi], [phi] + [xi], and for [alpha], [alpha] + [eta]
+  tan [phi], and zero longitude must be replaced by [eta] sec [phi]. At
+  the same time suppose the elements of the spheroid to be altered from
+  a, e to a + da, e + de. Confining our attention at first to the two
+  points A, B, let ([phi]'), ([alpha]'), ([omega]) be the numerical
+  elements at B as obtained in the first calculation, viz. before the
+  shifting and alteration of the spheroid; they will now take the form
+
+    ([phi]') + f[xi] + g[eta] + hda + kde,
+
+    ([alpha]') + f'[xi] + g'[eta] + h'da + k'de,
+
+    [omega] + f"[xi] + g"[eta] + h"da + k"de,
+
+  where the coefficients f, g, ... &c. can be numerically calculated.
+  Now these elements, corresponding to the projection of B on the
+  spheroid of reference, must be equal severally to the astronomically
+  determined elements at B, corrected for the inclination of the
+  surfaces there. If [xi]', [eta]' be the components of the inclination
+  at that point, then we have
+
+    [xi]' = ([phi]') - [phi]' + f[xi] + g[eta] + hda + kde,
+
+    [eta]' tan [phi]' = ([alpha]') - [alpha]' + f'[xi] + g'[eta] + h'da + k'de,
+
+    [eta]' sec [phi]' = ([omega]) - [omega] + f"[xi] + g"[eta] + h"da + k"de,
+
+  where [phi]', [alpha]', [omega] are the observed elements at B. Here
+  it appears that the observation of longitude gives no additional
+  information, but is available as a check upon the azimuthal
+  observations.
+
+  If now there be a number of astronomical stations in the
+  triangulation, and we form equations such as the above for each point,
+  then we can from them determine those values of [xi], [eta], da, de,
+  which make the quantity [xi]² + [eta]² + [xi]'² + [eta]'² + ... a
+  minimum. Thus we obtain that spheroid which best represents the
+  surface covered by the triangulation.
+
+  In the _Account of the Principal Triangulation of Great Britain and
+  Ireland_ will be found the determination, from 75 equations, of the
+  spheroid best representing the surface of the British Isles. Its
+  elements are a = 20927005 ± 295 ft., b : a - b = 280 ± 8; and it is so
+  placed that at Greenwich Observatory [xi] = 1".864, [eta] = -0".546.
+
+  Taking Durham Observatory as the origin, and the tangent plane to the
+  surface (determined by [xi] = -0".664, [eta] = -4".117) as the plane
+  of x and y, the former measured northwards, and z measured vertically
+  downwards, the equation to the surface is
+
+    .99524953 x² + .99288005 y² + .99763052 z² - 0.00671003xz - 41655070z = 0.
+
+
+  _Altitudes._
+
+  The precise determination of the altitude of his station is a matter
+  of secondary importance to the geodesist; nevertheless it is usual to
+  observe the zenith distances of all trigonometrical points. Of great
+  importance is a knowledge of the height of the base for its reduction
+  to the sea-level. Again the height of a station does influence a
+  little the observation of terrestrial angles, for a vertical line at B
+  does not lie generally in the vertical plane of A (see above). The
+  height above the sea-level also influences the geographical latitude,
+  inasmuch as the centrifugal force is increased and the magnitude and
+  direction of the attraction of the earth are altered, and the effect
+  upon the latitude is a very small term expressed by the formula h (g'-
+  g) sin 2 [phi] / ag, where g, g' are the values of gravity at the
+  equator and at the pole. This is h sin 2 [phi] / 5820 seconds, h being
+  in metres, a quantity which may be neglected, since for ordinary
+  mountain heights it amounts to only a few hundredths of a second. We
+  can assume this amount as joined with the northern component of the
+  plumb-line perturbations.
+
+  The uncertainties of terrestrial refraction render it impossible to
+  determine accurately by vertical angles the heights of distant points.
+  Generally speaking, refraction is greatest at about daybreak; from
+  that time it diminishes, being at a minimum for a couple of hours
+  before and after mid-day; later in the afternoon it again increases.
+  This at least is the general march of the phenomenon, but it is by no
+  means regular. The vertical angles measured at the station on Hart
+  Fell showed on one occasion in the month of September a refraction of
+  double the average amount, lasting from 1 P.M. to 5 P.M. The mean
+  value of the coefficient of refraction k determined from a very large
+  number of observations of terrestrial zenith distances in Great
+  Britain is .0792 ± .0047; and if we separate those rays which for a
+  considerable portion of their length cross the sea from those which do
+  not, the former give k = .0813 and the latter k = .0753. These values
+  are determined from high stations and long distances; when the
+  distance is short, and the rays graze the ground, the amount of
+  refraction is extremely uncertain and variable. A case is noted in the
+  Indian survey where the zenith distance of a station 10.5 miles off
+  varied from a depression of 4' 52".6 at 4.30 P.M. to an elevation of
+  2' 24".0 at 10.50 P.M.
+
+  If h, h' be the heights above the level of the sea of two stations,
+  90° + [delta], 90° + [delta]' their mutual zenith distances ([delta]
+  being that observed at h), s their distance apart, the earth being
+  regarded as a sphere of radius = a, then, with sufficient precision,
+
+                    /   1 - 2k           \                   / 1 - 2k            \
+    h' - h = s tan ( s -------- - [delta] ), h - h' = s tan ( -------- - [delta]' ).
+                    \     2a             /                   \   2a              /
+
+  If from a station whose height is h the horizon of the sea be observed
+  to have a zenith distance 90° + [delta], then the above formula gives
+  for h the value
+
+        a  tan² [delta]
+    h = -- ------------.
+        2     1 - 2k
+
+  Suppose the depression [delta] to be n minutes, then h = 1.054n² if
+  the ray be for the greater part of its length crossing the sea; if
+  otherwise, h = 1.040n². To take an example: the mean of eight
+  observations of the zenith distance of the sea horizon at the top of
+  Ben Nevis is 91° 4' 48", or [delta] = 64.8; the ray is pretty equally
+  disposed over land and water, and hence h = 1.047n² = 4396 ft. The
+  actual height of the hill by spirit-levelling is 4406 ft., so that the
+  error of the height thus obtained is only 10 ft.
+
+  The determination of altitudes by means of spirit-levelling is
+  undoubtedly the most exact method, particularly in its present
+  development as precise-levelling, by which there have been determined
+  in all civilized countries close-meshed nets of elevated points
+  covering the entire land.     (A. R. C; F. R. H.)
+
+
+FOOTNOTES:
+
+  [1] An arrangement acting similarly had been previously introduced by
+    Borda.
+
+  [2] _Geodetic Survey of South Africa_, vol. iii. (1905), p. viii;
+    _Les Nouveaux Appareils pour la mesure rapide des bases géod._, par
+    J. René Benoīt et Ch. Éd. Guillaume (1906).
+
+  [3] See a paper "On the Course of Geodetic Lines on the Earth's
+    Surface" in the _Phil. Mag._ 1870; Helmert, _Theorien der höheren
+    Geodäsie_, 1. 321.
+
+  [4] Helmert, Theorien der höheren Geodäsie, 1. 232, 247.
+
+
+
+
+GEOFFREY, surnamed MARTEL (1006-1060), count of Anjou, son of the count
+Fulk Nerra (q.v.) and of the countess Hildegarde or Audegarde, was born
+on the 14th of October 1006. During his father's lifetime he was
+recognized as suzerain by Fulk l'Oison ("the Gosling"), count of
+Vendōme, the son of his half-sister Adela. Fulk having revolted, he
+confiscated the countship, which he did not restore till 1050. On the
+1st of January 1032 he married Agnes, widow of William the Great, duke
+of Aquitaine, and taking arms against William the Fat, eldest son and
+successor of William the Great, defeated him and took him prisoner at
+Mont-Couėr near Saint-Jouin-de-Marnes on the 20th of September 1033. He
+then tried to win recognition as dukes of Aquitaine for the sons of his
+wife Agnes by William the Great, who were still minors, but Fulk Nerra
+promptly took up arms to defend his suzerain William the Fat, from whom
+he held the Loudunois and Saintonge in fief against his son. In 1036
+Geoffrey Martel had to liberate William the Fat, on payment of a heavy
+ransom, but the latter having died in 1038, and the second son of
+William the Great, Odo, duke of Gascony, having fallen in his turn at
+the siege of Mauzé (10th of March 1039) Geoffrey made peace with his
+father in the autumn of 1039, and had his wife's two sons recognized as
+dukes. About this time, also, he had interfered in the affairs of Maine,
+though without much result, for having sided against Gervais, bishop of
+Le Mans, who was trying to make himself guardian of the young count of
+Maine, Hugh, he had been beaten and forced to make terms with Gervais in
+1038. In 1040 he succeeded his father in Anjou and was able to conquer
+Touraine (1044) and assert his authority over Maine (see ANJOU). About
+1050 he repudiated Agnes, his first wife, and married Grécie, the widow
+of Bellay, lord of Montreuil-Bellay (before August 1052), whom he
+subsequently left in order to marry Adela, daughter of a certain Count
+Odo. Later he returned to Grécie, but again left her to marry Adelaide
+the German. When, however, he died on the 14th of November 1060, at the
+monastery of St Nicholas at Angers, he left no children, and transmitted
+the countship to Geoffrey the Bearded, the eldest of his nephews (see
+ANJOU).
+
+  See Louis Halphen, _Le Comté d'Anjou au XI^e sičcle_ (Paris, 1906). A
+  summary biography is given by Célestin Port, _Dictionnaire historique,
+  géographique et biographique de Maine-et-Loire_ (3 vols.,
+  Paris-Angers, 1874-1878), vol. ii. pp. 252-253, and a sketch of the
+  wars by Kate Norgate, _England under the Angevin Kings_ (2 vols.,
+  London, 1887), vol. i. chs. iii. iv.     (L. H.*)
+
+
+
+
+GEOFFREY, surnamed PLANTAGENET [or PLANTEGENET] (1113-1151), count of
+Anjou, was the son of Count Fulk the Young and of Eremburge (or
+Arembourg of La Flčche); he was born on the 24th of August 1113. He is
+also called "le bel" or "the handsome," and received the surname of
+Plantagenet from the habit which he is said to have had of wearing in
+his cap a sprig of broom (_genźt_). In 1127 he was made a knight, and on
+the 2nd of June 1129 married Matilda, daughter of Henry I. of England,
+and widow of the emperor Henry V. Some months afterwards he succeeded to
+his father, who gave up the countship when he definitively went to the
+kingdom of Jerusalem. The years of his government were spent in subduing
+the Angevin barons and in conquering Normandy (see ANJOU). In 1151,
+while returning from the siege of Montreuil-Bellay, he took cold, in
+consequence of bathing in the Loir at Chāteau-du-Loir, and died on the
+7th of September. He was buried in the cathedral of Le Mans. By his wife
+Matilda he had three sons: Henry Plantagenet, born at Le Mans on Sunday,
+the 5th of March 1133; Geoffrey, born at Argentan on the 1st of June
+1134; and William Long-Sword, born on the 22nd of July 1136.
+
+  See Kate Norgate, _England under the Angevin Kings_ (2 vols., London,
+  1887), vol. i. chs. v.-viii.; Célestin Port, _Dictionnaire historique,
+  géographique et biographique de Maine-et-Loire_ (3 vols.,
+  Paris-Angers, 1874-1878), vol. ii. pp. 254-256. A history of Geoffrey
+  le Bel has yet to be written; there is a biography of him written in
+  the 12th century by Jean, a monk of Marmoutier, _Historia Gaufredi,
+  ducis Normannorum et comitis Andegavorum_, published by Marchegay et
+  Salmon; "Chroniques des comtes d'Anjou" (_Société de l'histoire de
+  France_, Paris, 1856), pp. 229-310.     (L. H.*)
+
+
+
+
+GEOFFREY (1158-1186), duke of Brittany, fourth son of the English king
+Henry II. and his wife Eleanor of Aquitaine, was born on the 23rd of
+September 1158. In 1167 Henry suggested a marriage between Geoffrey and
+Constance (d. 1201), daughter and heiress of Conan IV., duke of Brittany
+(d. 1171); and Conan not only assented, perhaps under compulsion, to
+this proposal, but surrendered the greater part of his unruly duchy to
+the English king. Having received the homage of the Breton nobles,
+Geoffrey joined his brothers, Henry and Richard, who, in alliance with
+Louis VII. of France, were in revolt against their father; but he made
+his peace in 1174, afterwards helping to restore order in Brittany and
+Normandy, and aiding the new French king, Philip Augustus, to crush some
+rebellious vassals. In July 1181 his marriage with Constance was
+celebrated, and practically the whole of his subsequent life was spent
+in warfare with his brother Richard. In 1183 he made peace with his
+father, who had come to Richard's assistance; but a fresh struggle soon
+broke out for the possession of Anjou, and Geoffrey was in Paris
+treating for aid with Philip Augustus, when he died on the 19th of
+August 1186. He left a daughter, Eleanor, and his wife bore a
+posthumous son, the unfortunate Arthur.
+
+
+
+
+GEOFFREY (c. 1152-1212), archbishop of York, was a bastard son of Henry
+II., king of England. He was distinguished from his legitimate
+half-brothers by his consistent attachment and fidelity to his father.
+He was made bishop of Lincoln at the age of twenty-one (1173); but
+though he enjoyed the temporalities he was never consecrated and
+resigned the see in 1183. He then became his father's chancellor,
+holding a large number of lucrative benefices in plurality. Richard
+nominated him archbishop of York in 1189, but he was not consecrated
+till 1191, or enthroned till 1194. Geoffrey, though of high character,
+was a man of uneven temper; his history in chiefly one of quarrels, with
+the see of Canterbury, with the chancellor William Longchamp, with his
+half-brothers Richard and John, and especially with his canons at York.
+This last dispute kept him in litigation before Richard and the pope for
+many years. He led the clergy in their refusal to be taxed by John and
+was forced to fly the kingdom in 1207. He died in Normandy on the 12th
+of December 1212.
+
+  See Giraldus Cambrensis, _Vita Galfridi_; Stubbs's prefaces to _Roger
+  de Hoveden_, vols. iii. and iv. (Rolls Series).     (H. W. C. D.)
+
+
+
+
+GEOFFREY DE MONTBRAY (d. 1093), bishop of Coutances (_Constantiensis_),
+a right-hand man of William the Conqueror, was a type of the great
+feudal prelate, warrior and administrator at need. He knew, says
+Orderic, more about marshalling mailed knights than edifying
+psalm-singing clerks. Obtaining, as a young man, in 1048, the see of
+Coutances, by his brother's influence (see MOWBRAY), he raised from his
+fellow nobles and from their Sicilian spoils funds for completing his
+cathedral, which was consecrated in 1056. With bishop Odo, a warrior
+like himself, he was on the battle-field of Hastings, exhorting the
+Normans to victory; and at William's coronation it was he who called on
+them to acclaim their duke as king. His reward in England was a mighty
+fief scattered over twelve counties. He accompanied William on his visit
+to Normandy (1067), but, returning, led a royal force to the relief of
+Montacute in September 1069. In 1075 he again took the field, leading
+with Bishop Odo a vast host against the rebel earl of Norfolk, whose
+stronghold at Norwich they besieged and captured.
+
+Meanwhile the Conqueror had invested him with important judicial
+functions. In 1072 he had presided over the great Kentish suit between
+the primate and Bishop Odo, and about the same time over those between
+the abbot of Ely and his despoilers, and between the bishop of Worcester
+and the abbot of Ely, and there is some reason to think that he acted as
+a Domesday commissioner (1086), and was placed about the same time in
+charge of Northumberland. The bishop, who attended the Conqueror's
+funeral, joined in the great rising against William Rufus next year
+(1088), making Bristol, with which (as Domesday shows) he was closely
+connected and where he had built a strong castle, his base of
+operations. He burned Bath and ravaged Somerset, but had submitted to
+the king before the end of the year. He appears to have been at Dover
+with William in January 1090, but, withdrawing to Normandy, died at
+Coutances three years later. In his fidelity to Duke Robert he seems to
+have there held out for him against his brother Henry, when the latter
+obtained the Cotentin.
+
+  See E.A. Freeman, _Norman Conquest_ and _William Rufus_; J.H. Round,
+  _Feudal England_; and, for original authorities, the works of Orderic
+  Vitalis and William of Poitiers, and of Florence of Worcester; the
+  Anglo-Saxon Chronicle; William of Malmesbury's _Gesta pontificum_, and
+  Lanfranc's works, ed. Giles; Domesday Book.     (J. H. R.)
+
+
+
+
+GEOFFREY OF MONMOUTH (d. 1154), bishop of St Asaph and writer on early
+British history, was born about the year 1100. Of his early life little
+is known, except that he received a liberal education under the eye of
+his paternal uncle, Uchtryd, who was at that time archdeacon, and
+subsequently bishop, of Llandaff. In 1129 Geoffrey appears at Oxford
+among the witnesses of an Oseney charter. He subscribes himself Geoffrey
+Arturus; from this we may perhaps infer that he had already begun his
+experiments in the manufacture of Celtic mythology. A first edition of
+his _Historia Britonum_ was in circulation by the year 1139, although
+the text which we possess appears to date from 1147. This famous work,
+which the author has the audacity to place on the same level with the
+histories of William of Malmesbury and Henry of Huntingdon, professes to
+be a translation from a Celtic source; "a very old book in the British
+tongue" which Walter, archdeacon of Oxford, had brought from Brittany.
+Walter the archdeacon is a historical personage; whether his book has
+any real existence may be fairly questioned. There is nothing in the
+matter or the style of the _Historia_ to preclude us from supposing that
+Geoffrey drew partly upon confused traditions, partly on his own powers
+of invention, and to a very slight degree upon the accepted authorities
+for early British history. His chronology is fantastic and incredible;
+William of Newburgh justly remarks that, if we accepted the events which
+Geoffrey relates, we should have to suppose that they had happened in
+another world. William of Newburgh wrote, however, in the reign of
+Richard I. when the reputation of Geoffrey's work was too well
+established to be shaken by such criticisms. The fearless romancer had
+achieved an immediate success. He was patronized by Robert, earl of
+Gloucester, and by two bishops of Lincoln; he obtained, about 1140, the
+archdeaconry of Llandaff "on account of his learning"; and in 1151 was
+promoted to the see of St Asaph.
+
+Before his death the _Historia Britonum_ had already become a model and
+a quarry for poets and chroniclers. The list of imitators begins with
+Geoffrey Gaimar, the author of the _Estorie des Engles_ (c. 1147), and
+Wace, whose _Roman de Brut_ (1155) is partly a translation and partly a
+free paraphrase of the _Historia_. In the next century the influence of
+Geoffrey is unmistakably attested by the _Brut_ of Layamon, and the
+rhyming English chronicle of Robert of Gloucester. Among later
+historians who were deceived by the _Historia Britonum_ it is only
+needful to mention Higdon, Hardyng, Fabyan (1512), Holinshed (1580) and
+John Milton. Still greater was the influence of Geoffrey upon those
+writers who, like Warner in _Albion's England_ (1586), and Drayton in
+_Polyolbion_ (1613), deliberately made their accounts of English history
+as poetical as possible. The stories which Geoffrey preserved or
+invented were not infrequently a source of inspiration to literary
+artists. The earliest English tragedy, _Gorboduc_ (1565), the _Mirror
+for Magistrates_ (1587), and Shakespeare's Lear, are instances in point.
+It was, however, the Arthurian legend which of all his fabrications
+attained the greatest vogue. In the work of expanding and elaborating
+this theme the successors of Geoffrey went as far beyond him as he had
+gone beyond Nennius; but he retains the credit due to the founder of a
+great school. Marie de France, who wrote at the court of Henry II., and
+Chrétien de Troyes, her French contemporary, were the earliest of the
+avowed romancers to take up the theme. The succeeding age saw the
+Arthurian story popularized, through translations of the French
+romances, as far afield as Germany and Scandinavia. It produced in
+England the _Roman du Saint Graal_ and the _Roman de Merlin_, both from
+the pen of Robert de Borron; the _Roman de Lancelot_; the _Roman de
+Tristan_, which is attributed to a fictitious Lucas de Gast. In the
+reign of Edward IV. Sir Thomas Malory paraphrased and arranged the best
+episodes of these romances in English prose. His _Morte d'Arthur_,
+printed by Caxton in 1485, epitomizes the rich mythology which
+Geoffrey's work had first called into life, and gave the Arthurian story
+a lasting place in the English imagination. The influence of the
+_Historia Britonum_ may be illustrated in another way, by enumerating
+the more familiar of the legends to which it first gave popularity. Of
+the twelve books into which it is divided only three (Bks. IX., X., XI.)
+are concerned with Arthur. Earlier in the work, however, we have the
+adventures of Brutus; of his follower Corineus, the vanquisher of the
+Cornish giant Goemagol (Gogmagog); of Locrinus and his daughter Sabre
+(immortalized in Milton's _Comus_); of Bladud the builder of Bath; of
+Lear and his daughters; of the three pairs of brothers, Ferrex and
+Porrex, Brennius and Belinus, Elidure and Peridure. The story of
+Vortigern and Rowena takes its final form in the _Historia Britonum_;
+and Merlin makes his first appearance in the prelude to the Arthur
+legend. Besides the _Historia Britonum_ Geoffrey is also credited with
+a _Life of Merlin_ composed in Latin verse. The authorship of this work
+has, however, been disputed, on the ground that the style is distinctly
+superior to that of the _Historia_. A minor composition, the _Prophecies
+of Merlin_, was written before 1136, and afterwards incorporated with
+the _Historia_, of which it forms the seventh book.
+
+  For a discussion of the manuscripts of Geoffrey's work, see Sir T.D.
+  Hardy's _Descriptive Catalogue_ (Rolls Series), i. pp. 341 ff. The
+  _Historia Britonum_ has been critically edited by San Marte (Halle,
+  1854). There is an English translation by J.A. Giles (London, 1842).
+  The _Vita Merlini_ has been edited by F. Michel and T. Wright (Paris,
+  1837). See also the _Dublin Univ. Magazine_ for April 1876, for an
+  article by T. Gilray on the literary influence of Geoffrey; G.
+  Heeger's _Trojanersage der Britten_ (1889); and La Borderie's _Études
+  historiques bretonnes_ (1883).     (H. W. C. D.)
+
+
+
+
+GEOFFREY OF PARIS (d. c. 1320), French chronicler, was probably the
+author of the _Chronique métrique de Philippe le Bel, or Chronique rimée
+de Geoffroi de Paris_. This work, which deals with the history of France
+from 1300 to 1316, contains 7918 verses, and is valuable as that of a
+writer who had a personal knowledge of many of the events which he
+relates. Various short historical poems have also been attributed to
+Geoffrey, but there is no certain information about either his life or
+his writings.
+
+  The _Chronique_ was published by J.A. Buchon in his _Collection des
+  chroniques_, tome ix. (Paris, 1827), and it has also been printed in
+  tome xxii. of the _Recueil des historiens des Gaules et de la France_
+  (Paris, 1865). See G. Paris, _Histoire de la littérature franēaise au
+  moyen āge_ (Paris, 1890); and A. Molinier, _Les Sources de l'histoire
+  de France_, tome iii. (Paris, 1903).
+
+
+
+
+GEOFFREY THE BAKER (d. c. 1360), English chronicler, is also called
+Walter of Swinbroke, and was probably a secular clerk at Swinbrook in
+Oxfordshire. He wrote a _Chronicon Angliae temporibus Edwardi II. et
+Edwardi III._, which deals with the history of England from 1303 to
+1356. From the beginning until about 1324 this work is based upon Adam
+Murimuth's _Continuatio chronicarum_, but after this date it is valuable
+and interesting, containing information not found elsewhere, and closing
+with a good account of the battle of Poitiers. The author obtained his
+knowledge about the last days of Edward II. from William Bisschop, a
+companion of the king's murderers, Thomas Gurney and John Maltravers.
+Geoffrey also wrote a _Chroniculum_ from the creation of the world until
+1336, the value of which is very slight. His writings have been edited
+with notes by Sir E.M. Thompson as the _Chronicon Galfridi le Baker de
+Swynebroke_ (Oxford, 1889). Some doubt exists concerning Geoffrey's
+share in the compilation of the _Vita et mors Edwardi II._, usually
+attributed to Sir Thomas de la More, or Moor, and printed by Camden in
+his _Anglica scripta_. It has been maintained by Camden and others that
+More wrote an account of Edward's reign in French, and that this was
+translated into Latin by Geoffrey and used by him in compiling his
+_Chronicon_. Recent scholarship, however, asserts that More was no
+writer, and that the _Vita et mors_ is an extract from Geoffrey's
+_Chronicon_, and was attributed to More, who was the author's patron. In
+the main this conclusion substantiates the verdict of Stubbs, who has
+published the _Vita et mors_ in his _Chronicles of the reigns of Edward
+I. and Edward II._ (London, 1883). The manuscripts of Geoffrey's works
+are in the Bodleian library at Oxford.
+
+
+
+
+GEOFFRIN, MARIE THÉRČSE RODET (1699-1777), a Frenchwoman who played an
+interesting part in French literary and artistic life, was born in Paris
+in 1699. She married, on the 19th of July 1713, Pierre Franēois
+Geoffrin, a rich manufacturer and lieutenant-colonel of the National
+Guard, who died in 1750. It was not till Mme Geoffrin was nearly fifty
+years of age that we begin to hear of her as a power in Parisian
+society. She had learned much from Mme de Tencin, and about 1748 began
+to gather round her a literary and artistic circle. She had every week
+two dinners, on Monday for artists, and on Wednesday for her friends the
+Encyclopaedists and other men of letters. She received many foreigners
+of distinction, Hume and Horace Walpole among others. Walpole spent much
+time in her society before he was finally attached to Mme du Deffand,
+and speaks of her in his letters as a model of common sense. She was
+indeed somewhat of a small tyrant in her circle. She had adopted the
+pose of an old woman earlier than necessary, and her coquetry, if such
+it can be called, took the form of being mother and mentor to her
+guests, many of whom were indebted to her generosity for substantial
+help. Although her aim appears to have been to have the _Encyclopédie_
+in conversation and action around her, she was extremely displeased with
+any of her friends who were so rash as to incur open disgrace. Marmontel
+lost her favour after the official censure of _Bélisaire_, and her
+advanced views did not prevent her from observing the forms of religion.
+A devoted Parisian, Mme Geoffrin rarely left the city, so that her
+journey to Poland in 1766 to visit the king, Stanislas Poniatowski, whom
+she had known in his early days in Paris, was a great event in her life.
+Her experiences induced a sensible gratitude that she had been born
+"_Franēaise_" and "_particuličre_." In her last illness her daughter,
+Thérčse, marquise de la Ferté Imbault, excluded her mother's old friends
+so that she might die as a good Christian, a proceeding wittily
+described by the old lady: "My daughter is like Godfrey de Bouillon, she
+wished to defend my tomb from the infidels." Mme Geoffrin died in Paris
+on the 6th of October 1777.
+
+  See _Correspondance inédite du roi Stanislas Auguste Poniatowski et de
+  Madame Geoffrin_, edited by the comte de Mou’ (1875); P. de Ségur, _Le
+  Royaume de la rue Saint-Honoré, Madame Geoffrin et sa fille_ (1897);
+  A. Tornezy, _Un Bureau d'esprit au XVIII^e sičcle: le salon de Madame
+  Geoffrin_ (1895); and Janet Aldis, _Madame Geoffrin, her Salon and her
+  Times, 1750-1777_ (1905).
+
+
+
+
+GEOFFROY, ÉTIENNE FRANĒOIS (1672-1731), French chemist, born in Paris on
+the 13th of February 1672, was first an apothecary and then practised
+medicine. After studying at Montpellier he accompanied Marshal Tallard
+on his embassy to London in 1698 and thence travelled to Holland and
+Italy. Returning to Paris he became professor of chemistry at the Jardin
+du Roi and of pharmacy and medicine at the Collčge de France, and dean
+of the faculty of medicine. He died in Paris on the 6th of January 1731.
+His name is best known in connexion with his tables of affinities
+(_tables des rapports_), which he presented to the French Academy in
+1718 and 1720. These were lists, prepared by collating observations on
+the actions of substances one upon another, showing the varying degrees
+of affinity exhibited by analogous bodies for different reagents, and
+they retained their vogue for the rest of the century, until displaced
+by the profounder conceptions introduced by C.L. Berthollet. Another of
+his papers dealt with the delusions of the philosopher's stone, but
+nevertheless he believed that iron could be artificially formed in the
+combustion of vegetable matter. His _Tractatus de materia medica_,
+published posthumously in 1741, was long celebrated.
+
+His brother CLAUDE JOSEPH, known as Geoffroy the younger (1685-1752),
+was also an apothecary and chemist who, having a considerable knowledge
+of botany, devoted himself especially to the study of the essential oils
+in plants.
+
+
+
+
+GEOFFROY, JULIEN LOUIS (1743-1814), French critic, was born at Rennes in
+1743. He studied in the school of his native town and at the Collčge
+Louis le Grand in Paris. He took orders and fulfilled for some time the
+humble functions of an usher, eventually becoming professor of rhetoric
+at the _Collčge Mazarin_. A bad tragedy, Caton, was accepted at the
+_Théātre Franēais_, but was never acted. On the death of Élie Fréron in
+1776 the other collaborators in the _Année littéraire_ asked Geoffroy to
+succeed him, and he conducted the journal until in 1792 it ceased to
+appear. Geoffroy was a bitter critic of Voltaire and his followers, and
+made for himself many enemies. An enthusiastic royalist, he published
+with Fréron's brother-in-law, the abbé Thomas Royou (1741-1792), a
+journal, _L'Ami du roi_ (1790-1792), which possibly did more harm than
+good to the king's cause by its ill-advised partisanship. During the
+Terror Geoffroy hid in the neighbourhood of Paris, only returning in
+1799. An attempt to revive the _Année littéraire_ failed, and Geoffroy
+undertook the dramatic feuilleton of the _Journal des débats_. His
+scathing criticisms had a success of notoriety, but their popularity was
+ephemeral, and the publication of them (5 vols., 1819-1820) as _Cours de
+littérature dramatique_ proved a failure. He was also the author of a
+perfunctory _Commentaire_ on the works of Racine prefixed to Lenormant's
+edition (1808). He died in Paris on the 27th of February 1814.
+
+
+
+
+GEOFFROY SAINT-HILAIRE, ÉTIENNE (1772-1844), French naturalist, was the
+son of Jean Gčrard Geoffroy, procurator and magistrate of Étampes,
+Seine-et-Oise, where he was born on the 15th of April 1772. Destined for
+the church he entered the college of Navarre, in Paris, where he studied
+natural philosophy under M.J. Brisson; and in 1788 he obtained one of
+the canonicates of the chapter of Sainte Croix at Étampes, and also a
+benefice. Science, however, offered him a more congenial career, and he
+gained from his father permission to remain in Paris, and to attend the
+lectures at the Collčge de France and the Jardin des Plantes, on the
+condition that he should also read law. He accordingly took up his
+residence at Cardinal Lemoine's college, and there became the pupil and
+soon the esteemed associate of Brisson's friend, the abbé Haüy, the
+mineralogist. Having, before the close of the year 1790, taken the
+degree of bachelor in law, he became a student of medicine, and attended
+the lectures of A.F. de Fourcroy at the Jardin des Plantes, and of
+L.J.M. Daubenton at the Collčge de France. His studies at Paris were at
+length suddenly interrupted, for, in August 1792, Haüy and the other
+professors of Lemoine's college, as also those of the college of
+Navarre, were arrested by the revolutionists as priests, and confined in
+the prison of St Firmin. Through the influence of Daubenton and others
+Geoffroy on the 14th of August obtained an order for the release of Haüy
+in the name of the Academy; still the other professors of the two
+colleges, save C.F. Lhomond, who had been rescued by his pupil J.L.
+Tallien, remained in confinement. Geoffroy, foreseeing their certain
+destruction if they remained in the hands of the revolutionists,
+determined if possible to secure their liberty by stratagem. By bribing
+one of the officials at St Firmin, and disguising himself as a
+commissioner of prisons, he gained admission to his friends, and
+entreated them to effect their escape by following him. All, however,
+dreading lest their deliverance should render the doom of their
+fellow-captives the more certain, refused the offer, and one priest
+only, who was unknown to Geoffroy, left the prison. Already on the night
+of the 2nd of September the massacre of the proscribed had begun, when
+Geoffroy, yet intent on saving the life of his friends and teachers,
+repaired to St Firmin. At 4 o'clock on the morning of the 3rd of
+September, after eight hours' waiting, he by means of a ladder assisted
+the escape of twelve ecclesiastics, not of the number of his
+acquaintance, and then the approach of dawn and the discharge of a gun
+directed at him warned him, his chief purpose unaccomplished, to return
+to his lodgings. Leaving Paris he retired to Étampes, where, in
+consequence of the anxieties of which he had lately been the prey, and
+the horrors which he had witnessed, he was for some time seriously ill.
+At the beginning of the winter of 1792 he returned to his studies in
+Paris, and in March of the following year Daubenton, through the
+interest of Bernardin de Saint Pierre, procured him the office of
+sub-keeper and assistant demonstrator of the cabinet of natural history,
+vacant by the resignation of B.G.E. Lacépčde. By a law passed in June
+1793, Geoffroy was appointed one of the twelve professors of the newly
+constituted museum of natural history, being assigned the chair of
+zoology. In the same year he busied himself with the formation of a
+menagerie at that institution.
+
+In 1794 through the introduction of A.H. Tessier he entered into
+correspondence with Georges Cuvier, to whom, after the perusal of some
+of his manuscripts, he wrote: "Venez jouer parmi nous le rōle de Linné,
+d'un autre législateur de l'histoire naturelle." Shortly after the
+appointment of Cuvier as assistant at the Muséum d'Histoire Naturelle,
+Geoffroy received him into his house. The two friends wrote together
+five memoirs on natural history, one of which, on the classification of
+mammals, puts forward the idea of the subordination of characters upon
+which Cuvier based his zoological system. It was in a paper entitled
+"Histoire des Makis, ou singes de Madagascar," written in 1795, that
+Geoffroy first gave expression to his views on "the unity of organic
+composition," the influence of which is perceptible in all his
+subsequent writings; nature, he observes, presents us with only one plan
+of construction, the same in principle, but varied in its accessory
+parts.
+
+In 1798 Geoffroy was chosen a member of the great scientific expedition
+to Egypt, and on the capitulation of Alexandria in August 1801, he took
+part in resisting the claim made by the British general to the
+collections of the expedition, declaring that, were that demand
+persisted in, history would have to record that he also had burnt a
+library in Alexandria. Early in January 1802 Geoffroy returned to his
+accustomed labours in Paris. He was elected a member of the academy of
+sciences of that city in September 1807. In March of the following year
+the emperor, who had already recognized his national services by the
+award of the cross of the legion of honour, selected him to visit the
+museums of Portugal, for the purpose of procuring collections from them,
+and in the face of considerable opposition from the British he
+eventually was successful in retaining them as a permanent possession
+for his country. In 1809, the year after his return to France, he was
+made professor of zoology at the faculty of sciences at Paris, and from
+that period he devoted himself more exclusively than before to
+anatomical study. In 1818 he gave to the world the first part of his
+celebrated _Philosophie anatomique_, the second volume of which,
+published in 1822, and subsequent memoirs account for the formation of
+monstrosities on the principle of arrest of development, and of the
+attraction of similar parts. When, in 1830, Geoffroy proceeded to apply
+to the invertebrata his views as to the unity of animal composition, he
+found a vigorous opponent in Georges Cuvier, and the discussion between
+them, continued up to the time of the death of the latter, soon
+attracted the attention of the scientific throughout Europe. Geoffroy, a
+synthesist, contended, in accordance with his theory of unity of plan in
+organic composition, that all animals are formed of the same elements,
+in the same number, and with the same connexions: homologous parts,
+however they differ in form and size, must remain associated in the same
+invariable order. With Goethe he held that there is in nature a law of
+compensation or balancing of growth, so that if one organ take on an
+excess of development, it is at the expense of some other part; and he
+maintained that, since nature takes no sudden leaps, even organs which
+are superfluous in any given species, if they have played an important
+part in other species of the same family, are retained as rudiments,
+which testify to the permanence of the general plan of creation. It was
+his conviction that, owing to the conditions of life, the same forms had
+not been perpetuated since the origin of all things, although it was not
+his belief that existing species are becoming modified. Cuvier, who was
+an analytical observer of facts, admitted only the prevalence of "laws
+of co-existence" or "harmony" in animal organs, and maintained the
+absolute invariability of species, which he declared had been created
+with a regard to the circumstances in which they were placed, each organ
+contrived with a view to the function it had to fulfil, thus putting, in
+Geoffroy's considerations, the effect for the cause.
+
+In July 1840 Geoffroy became blind, and some months later he had a
+paralytic attack. From that time his strength gradually failed him. He
+resigned his chair at the museum in 1841, and died at Paris on the 19th
+of June 1844.
+
+  Geoffroy wrote: _Catalogue des mammifčres du Muséum National
+  d'Histoire Naturelle_ (1813), not quite completed; _Philosophie
+  anatomique_--t. i., _Des organes respiratoires_ (1818), and t. ii.,
+  _Des monstruosités humaines_ (1822); _Systčme dentaire des mammifčres
+  et des oiseaux_ (1st pt., 1824); _Sur le principe de l'unité de
+  composition organique_ (1828); _Cours de l'histoire naturelle des
+  mammifčres_ (1829); _Principes de philosophie zoologique_ (1830);
+  _Études progressives d'un naturaliste_ (1835); _Fragments
+  biographiques_ (1832); _Notions synthétiques, historiques et
+  physiologiques de philosophie naturelle_ (1838), and other works; also
+  part of the _Description de l'Égypte par la commission des sciences_
+  (1821-1830); and, with Frédéric Cuvier (1773-1838), a younger brother
+  of G. Cuvier, _Histoire naturelle des mammifčres_ (4 vols.,
+  1820-1842); besides numerous papers on such subjects as the anatomy of
+  marsupials, ruminants and electrical fishes, the vertebrate theory of
+  the skull, the opercula of fishes, teratology, palaeontology and the
+  influence of surrounding conditions in modifying animal forms.
+
+  See _Vie, travaux, et doctrine scientifique d'Étienne Geoffroy
+  Saint-Hilaire, par son fils M. Isidore Geoffroy Saint-Hilaire_ (Paris
+  and Strasburg, 1847), to which is appended a list of Geoffroy's works;
+  and Joly, in _Biog. universelle_, t. xvi. (1856).
+
+
+
+
+
+GEOFFROY SAINT-HILAIRE, ISIDORE (1805-1861), French zoologist, son of
+the preceding, was born at Paris on the 16th of December 1805. In his
+earlier years he showed an aptitude for mathematics, but eventually he
+devoted himself to the study of natural history and of medicine, and in
+1824 he was appointed assistant naturalist to his father. On the
+occasion of his taking the degree of doctor of medicine in September
+1829, he read a thesis entitled _Propositions sur la monstruosité,
+considérée chez l'homme et les animaux_; and in 1832-1837 was published
+his great teratological work, _Histoire générale et particuličre des
+anomalies de l'organisation chez l'homme et les animaux_, 3 vols. 8vo.
+with 20 plates. In 1829 he delivered for his father the second part of a
+course of lectures on ornithology, and during the three following years
+he taught zoology at the Athénée, and teratology at the École pratique.
+He was elected a member of the academy of sciences at Paris in 1833, was
+in 1837 appointed to act as deputy for his father at the faculty of
+sciences in Paris, and in the following year was sent to Bordeaux to
+organize a similar faculty there. He became successively inspector of
+the academy of Paris (1840), professor of the museum on the retirement
+of his father (1841), inspector-general of the university (1844), a
+member of the royal council for public instruction (1845), and on the
+death of H.M.D. de Blainville, professor of zoology at the faculty of
+sciences (1850). In 1854 he founded the Acclimatization Society of
+Paris, of which he was president. He died at Paris on the 10th of
+November 1861.
+
+  Besides the above-mentioned works, he wrote: _Essais de zoologie
+  générale_ (1841); _Vie ... d'Étienne Geoffroy Saint-Hilaire_ (1847);
+  _Acclimatation et domestication des animaux utiles_ (1849; 4th ed.,
+  1861); _Lettres sur les substances alimentaires et particuličrement
+  sur la viande de cheval_ (1856); and _Histoire naturelle générale des
+  rčgnes organiques_ (3 vols., 1854-1862), which was not quite
+  completed. He was the author also of various papers on zoology,
+  comparative anatomy and palaeontology.
+
+
+
+
+GEOGRAPHY (Gr. [Greek: gź], earth, and [Greek: graphein], to write), the
+exact and organized knowledge of the distribution of phenomena on the
+surface of the earth. The fundamental basis of geography is the vertical
+relief of the earth's crust, which controls all mobile distributions.
+The grander features of the relief of the lithosphere or stony crust of
+the earth control the distribution of the hydrosphere or collected
+waters which gather into the hollows, filling them up to a height
+corresponding to the volume, and thus producing the important practical
+division of the surface into land and water. The distribution of the
+mass of the atmosphere over the surface of the earth is also controlled
+by the relief of the crust, its greater or lesser density at the surface
+corresponding to the lesser or greater elevation of the surface. The
+simplicity of the zonal distribution of solar energy on the earth's
+surface, which would characterize a uniform globe, is entirely destroyed
+by the dissimilar action of land and water with regard to radiant heat,
+and by the influence of crust-forms on the direction of the resulting
+circulation. The influence of physical environment becomes clearer and
+stronger when the distribution of plant and animal life is considered,
+and if it is less distinct in the case of man, the reason is found in
+the modifications of environment consciously produced by human effort.
+Geography is a synthetic science, dependent for the data with which it
+deals on the results of specialized sciences such as astronomy, geology,
+oceanography, meteorology, biology and anthropology, as well as on
+topographical description. The physical and natural sciences are
+concerned in geography only so far as they deal with the forms of the
+earth's surface, or as regards the distribution of phenomena. The
+distinctive task of geography as a science is to investigate the control
+exercised by the crust-forms directly or indirectly upon the various
+mobile distributions. This gives to it unity and definiteness, and
+renders superfluous the attempts that have been made from time to time
+to define the limits which divide geography from geology on the one hand
+and from history on the other. It is essential to classify the
+subject-matter of geography in such a manner as to give prominence not
+only to facts, but to their mutual relations and their natural and
+inevitable order.
+
+The fundamental conception of geography is form, including the figure of
+the earth and the varieties of crustal relief. Hence mathematical
+geography (see MAP), including cartography as a practical application,
+comes first. It merges into physical geography, which takes account of the
+forms of the lithosphere (geomorphology), and also of the distribution of
+the hydrosphere and the rearrangements resulting from the workings of
+solar energy throughout the hydrosphere and atmosphere (oceanography and
+climatology). Next follows the distribution of plants and animals
+(biogeography), and finally the distribution of mankind and the various
+artificial boundaries and redistributions (anthropogeography). The
+applications of anthropogeography to human uses give rise to political and
+commercial geography, in the elucidation of which all the earlier
+departments or stages have to be considered, together with historical and
+other purely human conditions. The evolutionary idea has revolutionized
+and unified geography as it did biology, breaking down the old
+hard-and-fast partitions between the various departments, and substituting
+the study of the nature and influence of actual terrestrial environments
+for the earlier motive, the discovery and exploration of new lands.
+
+
+  HISTORY OF GEOGRAPHICAL THEORY
+
+  The earliest conceptions of the earth, like those held by the
+  primitive peoples of the present day, are difficult to discover and
+  almost impossible fully to grasp. Early generalizations, as far as
+  they were made from known facts, were usually expressed in symbolic
+  language, and for our present purpose it is not profitable to
+  speculate on the underlying truths which may sometimes be suspected in
+  the old mythological cosmogonies.
+
+
+    Early Greek ideas.
+
+    Flat earth of Homer.
+
+    Hecataeus.
+
+    Herodotus.
+
+    The idea of symmetry.
+
+  The first definite geographical theories to affect the western world
+  were those evolved, or at least first expressed, by the Greeks.[1] The
+  earliest theoretical problem of geography was the form of the earth.
+  The natural supposition that the earth is a flat disk, circular or
+  elliptical in outline, had in the time of Homer acquired a special
+  definiteness by the introduction of the idea of the ocean river
+  bounding the whole, an application of imperfectly understood
+  observations. Thales of Miletus is claimed as the first exponent of
+  the idea of a spherical earth; but, although this does not appear to
+  be warranted, his disciple Anaximander (c. 580 B.C.) put forward the
+  theory that the earth had the figure of a solid body hanging freely in
+  the centre of the hollow sphere of the starry heavens. The Pythagorean
+  school of philosophers adopted the theory of a spherical earth, but
+  from metaphysical rather than scientific reasons; their convincing
+  argument was that a sphere being the most perfect solid figure was the
+  only one worthy to circumscribe the dwelling-place of man. The
+  division of the sphere into parallel zones and some of the
+  consequences of this generalization seem to have presented themselves
+  to Parmenides (c. 450 B.C.); but these ideas did not influence the
+  Ionian school of philosophers, who in their treatment of geography
+  preferred to deal with facts demonstrable by travel rather than with
+  speculations. Thus Hecataeus, claimed by H.F. Tozer[2] as the father
+  of geography on account of his _Periodos_, or general treatise on the
+  earth, did not advance beyond the primitive conception of a circular
+  disk. He systematized the form of the land within the ring of
+  ocean--the [Greek: oikoumenź], or habitable world--by recognizing two
+  continents: Europe to the north, and Asia to the south of the midland
+  sea. Herodotus, equally oblivious of the sphere, criticized and
+  ridiculed the circular outline of the _oekumene_, which he knew to be
+  longer from east to west than it was broad from north to south. He
+  also pointed out reasons for accepting a division of the land into
+  three continents--Europe, Asia and Africa. Beyond the limits of his
+  personal travels Herodotus applied the characteristically Greek theory
+  of symmetry to complete, in the unknown, outlines of lands and rivers
+  analogous to those which had been explored. Symmetry was in fact the
+  first geographical theory, and the effect of Herodotus's hypothesis
+  that the Nile must flow from west to east before turning north in
+  order to balance the Danube running from west to east before turning
+  south lingered in the maps of Africa down to the time of Mungo
+  Park.[3]
+
+
+    Aristotle and the sphere.
+
+  To Aristotle (384-322 B.C.) must be given the distinction of founding
+  scientific geography. He demonstrated the sphericity of the earth by
+  three arguments, two of which could be tested by observation. These
+  were: (1) that the earth must be spherical, because of the tendency of
+  matter to fall together towards a common centre; (2) that only a
+  sphere could always throw a circular shadow on the moon during an
+  eclipse; and (3) that the shifting of the horizon and the appearance
+  of new constellations, or the disappearance of familiar stars, as one
+  travelled from north to south, could only be explained on the
+  hypothesis that the earth was a sphere. Aristotle, too, gave greater
+  definiteness to the idea of zones conceived by Parmenides, who had
+  pictured a torrid zone uninhabitable by reason of heat, two frigid
+  zones uninhabitable by reason of cold, and two intermediate temperate
+  zones fit for human occupation. Aristotle defined the temperate zone
+  as extending from the tropic to the arctic circle, but there is some
+  uncertainty as to the precise meaning he gave to the term "arctic
+  circle." Soon after his time, however, this conception was clearly
+  established, and with so large a generalization the mental horizon was
+  widened to conceive of a geography which was a science. Aristotle had
+  himself shown that in the southern temperate zone winds similar to
+  those of the northern temperate zone should blow, but from the
+  opposite direction.
+
+
+    Fitting the oekumene to the sphere.
+
+  While the theory of the sphere was being elaborated the efforts of
+  practical geographers were steadily directed towards ascertaining the
+  outline and configuration of the _oekumene_, or habitable world, the
+  only portion of the terrestrial surface known to the ancients and to
+  the medieval peoples, and still retaining a shadow of its old monopoly
+  of geographical attention in its modern name of the "Old World." The
+  fitting of the _oekumene_ to the sphere was the second theoretical
+  problem. The circular outline had given way in geographical opinion to
+  the elliptical with the long axis lying east and west, and Aristotle
+  was inclined to view it as a very long and relatively narrow band
+  almost encircling the globe in the temperate zone. His argument as to
+  the narrowness of the sea between West Africa and East Asia, from the
+  occurrence of elephants at both extremities, is difficult to
+  understand, although it shows that he looked on the distribution of
+  animals as a problem of geography.
+
+
+    Problem of the Antipodes.
+
+  Pythagoras had speculated as to the existence of antipodes, but it was
+  not until the first approximately accurate measurements of the globe
+  and estimates of the length and breadth of the _oekumene_ were made by
+  Eratosthenes (c. 250 B.C.) that the fact that, as then known, it
+  occupied less than a quarter of the surface of the sphere was clearly
+  recognized. It was natural, if not strictly logical, that the ocean
+  river should be extended from a narrow stream to a world-embracing
+  sea, and here again Greek theory, or rather fancy, gave its modern
+  name to the greatest feature of the globe. The old instinctive idea of
+  symmetry must often have suggested other _oekumene_ balancing the
+  known world in the other quarters of the globe. The Stoic
+  philosophers, especially Crates of Mallus, arguing from the love of
+  nature for life, placed an _oekumene_ in each quarter of the sphere,
+  the three unknown world-islands being those of the Antoeci, Perioeci
+  and Antipodes. This was a theory not only attractive to the
+  philosophical mind, but eminently adapted to promote exploration. It
+  had its opponents, however, for Herodotus showed that sea-basins
+  existed cut off from the ocean, and it is still a matter of
+  controversy how far the pre-Ptolemaic geographers believed in a
+  water-connexion between the Atlantic and Indian oceans. It is quite
+  clear that Pomponius Mela (c. A.D. 40), following Strabo, held that
+  the southern temperate zone contained a habitable land, which he
+  designated by the name _Antichthones_.
+
+
+    Aristotle's geographical views.
+
+  Aristotle left no work on geography, so that it is impossible to know
+  what facts he associated with the science of the earth's surface. The
+  word geography did not appear before Aristotle, the first use of it
+  being in the [Greek: Peri kosmōn], which is one of the writings
+  doubtfully ascribed to him, and H. Berger considers that the
+  expression was introduced by Eratosthenes.[4] Aristotle was certainly
+  conversant with many facts, such as the formation of deltas,
+  coast-erosion, and to a certain extent the dependence of plants and
+  animals on their physical surroundings. He formed a comprehensive
+  theory of the variations of climate with latitude and season, and was
+  convinced of the necessity of a circulation of water between the sea
+  and rivers, though, like Plato, he held that this took place by water
+  rising from the sea through crevices in the rocks, losing its
+  dissolved salts in the process. He speculated on the differences in
+  the character of races of mankind living in different climates, and
+  correlated the political forms of communities with their situation on
+  a seashore, or in the neighbourhood of natural strongholds.
+
+
+    Strabo.
+
+  Strabo (c. 50 B.C.-A.D. 24) followed Eratosthenes rather than
+  Aristotle, but with sympathies which went out more to the human
+  interests than the mathematical basis of geography. He compiled a very
+  remarkable work dealing, in large measure from personal travel, with
+  the countries surrounding the Mediterranean. He may be said to have
+  set the pattern which was followed in succeeding ages by the compilers
+  of "political geographies" dealing less with theories than with
+  facts, and illustrating rather than formulating the principles of the
+  science.
+
+
+    Ptolemy.
+
+  Claudius Ptolemaeus (c. A.D. 150) concentrated in his writings the
+  final outcome of all Greek geographical learning, and passed it across
+  the gulf of the middle ages by the hands of the Arabs, to form the
+  starting-point of the science in modern times. His geography was based
+  more immediately on the work of his predecessor, Marinus of Tyre, and
+  on that of Hipparchus, the follower and critic of Eratosthenes. It was
+  the ambition of Ptolemy to describe and represent accurately the
+  surface of the _oekumene_, for which purpose he took immense trouble
+  to collect all existing determinations of the latitude of places, all
+  estimates of longitude, and to make every possible rectification in
+  the estimates of distances by land or sea. His work was mainly
+  cartographical in its aim, and theory was as far as possible excluded.
+  The symmetrically placed hypothetical islands in the great continuous
+  ocean disappeared, and the _oekumene_ acquired a new form by the
+  representation of the Indian Ocean as a larger Mediterranean
+  completely cut off by land from the Atlantic. The _terra incognita_
+  uniting Africa and Farther Asia was an unfortunate hypothesis which
+  helped to retard exploration. Ptolemy used the word _geography_ to
+  signify the description of the whole _oekumene_ on mathematical
+  principles, while _chorography_ signified the fuller description of a
+  particular region, and _topography_ the very detailed description of a
+  smaller locality. He introduced the simile that geography represented
+  an artist's sketch of a whole portrait, while chorography corresponded
+  to the careful and detailed drawing of an eye or an ear.[5]
+
+  The Caliph al-Mam[ = u]n (c. A.D. 815), the son and successor of H[ =
+  a]r[ = u]n al-Rash[ = i]d, caused an Arabic version of Ptolemy's great
+  astronomical work ([Greek: Suntaxis megistź]) to be made, which is
+  known as the _Almagest_, the word being nothing more than the Gr.
+  [Greek: megistź] with the Arabic article _al_ prefixed. The geography
+  of Ptolemy was also known and is constantly referred to by Arab
+  writers. The Arab astronomers measured a degree on the plains of
+  Mesopotamia, thereby deducing a fair approximation to the size of the
+  earth. The caliph's librarian, Abu Jafar Muhammad Ben Musa, wrote a
+  geographical work, now unfortunately lost, entitled _Rasm el Arsi_ ("A
+  Description of the World"), which is often referred to by subsequent
+  writers as having been composed on the model of that of Ptolemy.
+
+
+    Geography in the middle ages.
+
+  The middle ages saw geographical knowledge die out in Christendom,
+  although it retained, through the Arabic translations of Ptolemy, a
+  certain vitality in Islam. The verbal interpretation of Scripture led
+  Lactantius (c. A.D. 320) and other ecclesiastics to denounce the
+  spherical theory of the earth as heretical. The wretched subterfuge of
+  Cosmas (c. A.D. 550) to explain the phenomena of the apparent
+  movements of the sun by means of an earth modelled on the plan of the
+  Jewish Tabernacle gave place ultimately to the wheel-maps--the T in an
+  O--which reverted to the primitive ignorance of the times of Homer and
+  Hecataeus.[6]
+
+  The journey of Marco Polo, the increasing trade to the East and the
+  voyages of the Arabs in the Indian Ocean prepared the way for the
+  reacceptance of Ptolemy's ideas when the sealed books of the Greek
+  original were translated into Latin by Angelus in 1410.
+
+
+    Revival of geography.
+
+  The old arguments of Aristotle and the old measurements of Ptolemy
+  were used by Toscanelli and Columbus in urging a westward voyage to
+  India; and mainly on this account did the crossing of the Atlantic
+  rank higher in the history of scientific geography than the laborious
+  feeling out of the coast-line of Africa. But not until the voyage of
+  Magellan shook the scales from the eyes of Europe did modern geography
+  begin to advance. Discovery had outrun theory; the rush of new facts
+  made Ptolemy practically obsolete in a generation, after having been
+  the fount and origin of all geography for a millennium.
+
+
+    Apianus.
+
+  The earliest evidence of the reincarnation of a sound theoretical
+  geography is to be found in the text-books by Peter Apian and
+  Sebastian Münster. Apian in his _Cosmographicus liber_, published in
+  1524, and subsequently edited and added to by Gemma Frisius under the
+  title of _Cosmographia_, based the whole science on mathematics and
+  measurement. He followed Ptolemy closely, enlarging on his distinction
+  between geography and chorography, and expressing the artistic analogy
+  in a rough diagram. This slender distinction was made much of by most
+  subsequent writers until Nathanael Carpenter in 1625 pointed out that
+  the difference between geography and chorography was simply one of
+  degree, not of kind.
+
+
+    Münster.
+
+  Sebastian Münster, on the other hand, in his _Cosmographia
+  universalis_ of 1544, paid no regard to the mathematical basis of
+  geography, but, following the model of Strabo, described the world
+  according to its different political divisions, and entered with great
+  zest into the question of the productions of countries, and into the
+  manners and costumes of the various peoples. Thus early commenced the
+  separation between what were long called mathematical and political
+  geography, the one subject appealing mainly to mathematicians, the
+  other to historians.
+
+  Throughout the 16th and 17th centuries the rapidly accumulating store
+  of facts as to the extent, outline and mountain and river systems of
+  the lands of the earth were put in order by the generation of
+  cartographers of which Mercator was the chief; but the writings of
+  Apian and Münster held the field for a hundred years without a serious
+  rival, unless the many annotated editions of Ptolemy might be so
+  considered. Meanwhile the new facts were the subject of original study
+  by philosophers and by practical men without reference to classical
+  traditions. Bacon argued keenly on geographical matters and was a
+  lover of maps, in which he observed and reasoned upon such
+  resemblances as that between the outlines of South America and Africa.
+
+
+    Cluverius.
+
+  Philip Cluver's _Introductio in geographiam universam tam veterem quam
+  novam_ was published in 1624. Geography he defined as "the description
+  of the whole earth, so far as it is known to us." It is distinguished
+  from cosmography by dealing with the earth alone, not with the
+  universe, and from chorography and topography by dealing with the
+  whole earth, not with a country or a place. The first book, of
+  fourteen short chapters, is concerned with the general properties of
+  the globe; the remaining six books treat in considerable detail of the
+  countries of Europe and of the other continents. Each country is
+  described with particular regard to its people as well as to its
+  surface, and the prominence given to the human element is of special
+  interest.
+
+
+    Carpenter.
+
+  A little-known book which appears to have escaped the attention of
+  most writers on the history of modern geography was published at
+  Oxford in 1625 by Nathanael Carpenter, fellow of Exeter College, with
+  the title _Geographie delineated forth in Two Bookes, containing the
+  Sphericall and Topicall parts thereof_. It is discursive in its style
+  and verbose; but, considering the period at which it appeared, it is
+  remarkable for the strong common sense displayed by the author, his
+  comparative freedom from prejudice, and his firm application of the
+  methods of scientific reasoning to the interpretation of phenomena.
+  Basing his work on the principles of Ptolemy, he brings together
+  illustrations from the most recent travellers, and does not hesitate
+  to take as illustrative examples the familiar city of Oxford and his
+  native county of Devon. He divides geography into _The Spherical
+  Part_, or that for the study of which mathematics alone is required,
+  and _The Topical Part_, or the description of the physical relations
+  of parts of the earth's surface, preferring this division to that
+  favoured by the ancient geographers--into general and special. It is
+  distinguished from other English geographical books of the period by
+  confining attention to the principles of geography, and not describing
+  the countries of the world.
+
+
+    Varenius.
+
+  A much more important work in the history of geographical method is
+  the _Geographia generalis_ of Bernhard Varenius, a German medical
+  doctor of Leiden, who died at the age of twenty-eight in 1650, the
+  year of the publication of his book. Although for a time it was lost
+  sight of on the continent, Sir Isaac Newton thought so highly of this
+  book that he prepared an annotated edition which was published in
+  Cambridge in 1672, with the addition of the plates which had been
+  planned by Varenius, but not produced by the original publishers. "The
+  reason why this great man took so much care in correcting and
+  publishing our author was, because he thought him necessary to be read
+  by his audience, the young gentlemen of Cambridge, while he was
+  delivering lectures on the same subject from the Lucasian Chair."[7]
+  The treatise of Varenius is a model of logical arrangement and terse
+  expression; it is a work of science and of genius; one of the few of
+  that age which can still be studied with profit. The English
+  translation renders the definition thus: "Geography is that part of
+  _mixed mathematics_ which explains the state of the earth and of its
+  parts, depending on quantity, viz. its figure, place, magnitude and
+  motion, with the celestial appearances, &c. By some it is taken in too
+  limited a sense, for a bare description of the several countries; and
+  by others too extensively, who along with such a description would
+  have their political constitution."
+
+  Varenius was reluctant to include the human side of geography in his
+  system, and only allowed it as a concession to custom, and in order to
+  attract readers by imparting interest to the sterner details of the
+  science. His division of geography was into two parts--(i.) General or
+  universal, dealing with the earth in general, and explaining its
+  properties without regard to particular countries; and (ii.) Special
+  or particular, dealing with each country in turn from the
+  chorographical or topographical point of view. General geography was
+  divided into--(1) the _Absolute_ part, dealing with the form,
+  dimensions, position and substance of the earth, the distribution of
+  land and water, mountains, woods and deserts, hydrography (including
+  all the waters of the earth) and the atmosphere; (2) the _Relative_
+  part, including the celestial properties, i.e. latitude, climate
+  zones, longitude, &c.; and (3) the _Comparative_ part, which
+  "considers the particulars arising from comparing one part with
+  another"; but under this head the questions discussed were longitude,
+  the situation and distances of places, and navigation. Varenius does
+  not treat of special geography, but gives a scheme for it under three
+  heads--(1) _Terrestrial_, including position, outline, boundaries,
+  mountains, mines, woods and deserts, waters, fertility and fruits, and
+  living creatures; (2) _Celestial_, including appearance of the heavens
+  and the climate; (3) _Human_, but this was added out of deference to
+  popular usage.
+
+  This system of geography founded a new epoch, and the book--translated
+  into English, Dutch and French--was the unchallenged standard for more
+  than a century. The framework was capable of accommodating itself to
+  new facts, and was indeed far in advance of the knowledge of the
+  period. The method included a recognition of the causes and effects of
+  phenomena as well as the mere fact of their occurrence, and for the
+  first time the importance of the vertical relief of the land was
+  fairly recognized.
+
+  The physical side of geography continued to be elaborated after
+  Varenius's methods, while the historical side was developed
+  separately. Both branches, although enriched by new facts, remained
+  stationary so far as method is concerned until nearly the end of the
+  18th century. The compilation of "geography books" by uninstructed
+  writers led to the pernicious habit, which is not yet wholly overcome,
+  of reducing the general or "physical" part to a few pages of
+  concentrated information, and expanding the particular or "political"
+  part by including unrevised travellers' stories and uncritical
+  descriptions of the various countries of the world. Such books were in
+  fact not geography, but merely compressed travel.
+
+
+    Bergman.
+
+  The next marked advance in the theory of geography may be taken as the
+  nearly simultaneous studies of the physical earth carried out by the
+  Swedish chemist, Torbern Bergman, acting under the impulse of
+  Linnaeus, and by the German philosopher, Immanuel Kant. Bergman's
+  _Physical Description of the Earth_ was published in Swedish in 1766,
+  and translated into English in 1772 and into German in 1774. It is a
+  plain, straightforward description of the globe, and of the various
+  phenomena of the surface, dealing only with definitely ascertained
+  facts in the natural order of their relationships, but avoiding any
+  systematic classification or even definitions of terms.
+
+
+    Kant.
+
+  The problems of geography had been lightened by the destructive
+  criticism of the French cartographer D'Anville (who had purged the map
+  of the world of the last remnants of traditional fact unverified by
+  modern observations) and rendered richer by the dawn of the new era of
+  scientific travel, when Kant brought his logical powers to bear upon
+  them. Kant's lectures on physical geography were delivered in the
+  university of Königsberg from 1765 onwards.[8] Geography appealed to
+  him as a valuable educational discipline, the joint foundation with
+  anthropology of that "knowledge of the world" which was the result of
+  reason and experience. In this connexion he divided the communication
+  of experience from one person to another into two categories--the
+  narrative or historical and the descriptive or geographical; both
+  history and geography being viewed as descriptions, the former a
+  description in order of time, the latter a description in order of
+  space.
+
+  Physical geography he viewed as a summary of nature, the basis not
+  only of history but also of "all the other possible geographies," of
+  which he enumerates five, viz. (1) _Mathematical geography_, which
+  deals with the form, size and movements of the earth and its place in
+  the solar system; (2) _Moral geography_, or an account of the
+  different customs and characters of mankind according to the region
+  they inhabit; (3) _Political geography_, the divisions according to
+  their organized governments; (4) _Mercantile geography_, dealing with
+  the trade in the surplus products of countries; (5) _Theological
+  geography_, or the distribution of religions. Here there is a clear
+  and formal statement of the interaction and causal relation of all the
+  phenomena of distribution on the earth's surface, including the
+  influence of physical geography upon the various activities of mankind
+  from the lowest to the highest. Notwithstanding the form of this
+  classification, Kant himself treats mathematical geography as
+  preliminary to, and therefore not dependent on, physical geography.
+  Physical geography itself is divided into two parts: a general, which
+  has to do with the earth and all that belongs to it--water, air and
+  land; and a particular, which deals with special products of the
+  earth--mankind, animals, plants and minerals. Particular importance is
+  given to the vertical relief of the land, on which the various
+  branches of human geography are shown to depend.
+
+
+    Humboldt.
+
+  Alexander von Humboldt (1769-1859) was the first modern geographer to
+  become a great traveller, and thus to acquire an extensive stock of
+  first-hand information on which an improved system of geography might
+  be founded. The impulse given to the study of natural history by the
+  example of Linnaeus; the results brought back by Sir Joseph Banks, Dr
+  Solander and the two Forsters, who accompanied Cook in his voyages of
+  discovery; the studies of De Saussure in the Alps, and the lists of
+  desiderata in physical geography drawn up by that investigator,
+  combined to prepare the way for Humboldt. The theory of geography was
+  advanced by Humboldt mainly by his insistence on the great principle
+  of the unity of nature. He brought all the "observable things," which
+  the eager collectors of the previous century had been heaping together
+  regardless of order or system, into relation with the vertical relief
+  and the horizontal forms of the earth's surface. Thus he demonstrated
+  that the forms of the land exercise a directive and determining
+  influence on climate, plant life, animal life and on man himself. This
+  was no new idea; it had been familiar for centuries in a less definite
+  form, deduced from a priori considerations, and so far as regards the
+  influence of surrounding circumstances upon man, Kant had already
+  given it full expression. Humboldt's concrete illustrations and the
+  remarkable power of his personality enabled him to enforce these
+  principles in a way that produced an immediate and lasting effect. The
+  treatises on physical geography by Mrs Mary Somerville and Sir John
+  Herschel (the latter written for the eighth edition of the
+  _Encyclopaedia Britannica_) showed the effect produced in Great
+  Britain by the stimulus of Humboldt's work.
+
+
+    Ritter.
+
+  Humboldt's contemporary, Carl Ritter (1779-1859), extended and
+  disseminated the same views, and in his interpretation of "Comparative
+  Geography" he laid stress on the importance of forming conclusions,
+  not from the study of one region by itself, but from the comparison of
+  the phenomena of many places. Impressed by the influence of
+  terrestrial relief and climate on human movements, Ritter was led
+  deeper and deeper into the study of history and archaeology. His
+  monumental _Vergleichende Geographie_, which was to have made the
+  whole world its theme, died out in a wilderness of detail in
+  twenty-one volumes before it had covered more of the earth's surface
+  than Asia and a portion of Africa. Some of his followers showed a
+  tendency to look on geography rather as an auxiliary to history than
+  as a study of intrinsic worth.
+
+
+    Geography as a natural science.
+
+  During the rapid development of physical geography many branches of
+  the study of nature, which had been included in the cosmography of the
+  early writers, the physiography of Linnaeus and even the _Erdkunde_ of
+  Ritter, had been so much advanced by the labours of specialists that
+  their connexion was apt to be forgotten. Thus geology, meteorology,
+  oceanography and anthropology developed into distinct sciences. The
+  absurd attempt was, and sometimes is still, made by geographers to
+  include all natural science in geography; but it is more common for
+  specialists in the various detailed sciences to think, and sometimes
+  to assert, that the ground of physical geography is now fully occupied
+  by these sciences. Political geography has been too often looked on
+  from both sides as a mere summary of guide-book knowledge, useful in
+  the schoolroom, a poor relation of physical geography that it was
+  rarely necessary to recognize.
+
+  The science of geography, passed on from antiquity by Ptolemy,
+  re-established by Varenius and Newton, and systematized by Kant,
+  included within itself definite aspects of all those terrestrial
+  phenomena which are now treated exhaustively under the heads of
+  geology, meteorology, oceanography and anthropology; and the inclusion
+  of the requisite portions of the perfected results of these sciences
+  in geography is simply the gathering in of fruit matured from the seed
+  scattered by geography itself.
+
+  The study of geography was advanced by improvements in cartography
+  (see MAP), not only in the methods of survey and projection, but in
+  the representation of the third dimension by means of contour lines
+  introduced by Philippe Buache in 1737, and the more remarkable because
+  less obvious invention of isotherms introduced by Humboldt in 1817.
+
+
+    The teleological argument in geography.
+
+  The "argument from design" had been a favourite form of reasoning
+  amongst Christian theologians, and, as worked out by Paley in his
+  _Natural Theology_, it served the useful purpose of emphasizing the
+  fitness which exists between all the inhabitants of the earth and
+  their physical environment. It was held that the earth had been
+  created so as to fit the wants of man in every particular. This
+  argument was tacitly accepted or explicitly avowed by almost every
+  writer on the theory of geography, and Carl Ritter distinctly
+  recognized and adopted it as the unifying principle of his system. As
+  a student of nature, however, he did not fail to see, and as professor
+  of geography he always taught, that man was in very large measure
+  conditioned by his physical environment. The apparent opposition of
+  the observed fact to the assigned theory he overcame by looking upon
+  the forms of the land and the arrangement of land and sea as
+  instruments of Divine Providence for guiding the destiny as well as
+  for supplying the requirements of man. This was the central theme of
+  Ritter's philosophy; his religion and his geography were one, and the
+  consequent fervour with which he pursued his mission goes far to
+  account for the immense influence he acquired in Germany.
+
+
+    The theory of evolution in geography.
+
+  The evolutionary theory, more than hinted at in Kant's "Physical
+  Geography," has, since the writings of Charles Darwin, become the
+  unifying principle in geography. The conception of the development of
+  the plan of the earth from the first cooling of the surface of the
+  planet throughout the long geological periods, the guiding power of
+  environment on the circulation of water and of air, on the
+  distribution of plants and animals, and finally on the movements of
+  man, give to geography a philosophical dignity and a scientific
+  completeness which it never previously possessed. The influence of
+  environment on the organism may not be quite so potent as it was once
+  believed to be, in the writings of Buckle, for instance,[9] and
+  certainly man, the ultimate term in the series, reacts upon and
+  greatly modifies his environment; yet the fact that environment does
+  influence all distributions is established beyond the possibility of
+  doubt. In this way also the position of geography, at the point where
+  physical science meets and mingles with mental science, is explained
+  and justified. The change which took place during the 19th century in
+  the substance and style of geography may be well seen by comparing the
+  eight volumes of Malte-Brun's _Géographie universelle_ (Paris,
+  1812-1829) with the twenty-one volumes of Reclus's _Géographie
+  universelle_ (Paris, 1876-1895).
+
+  In estimating the influence of recent writers on geography it is usual
+  to assign to Oscar Peschel (1826-1875) the credit of having corrected
+  the preponderance which Ritter gave to the historical element, and of
+  restoring physical geography to its old pre-eminence.[10] As a matter
+  of fact, each of the leading modern exponents of theoretical
+  geography--such as Ferdinand von Richthofen, Hermann Wagner, Friedrich
+  Ratzel, William M. Davis, A. Penck, A. de Lapparent and Elisée
+  Reclus--has his individual point of view, one devoting more attention
+  to the results of geological processes, another to anthropological
+  conditions, and the rest viewing the subject in various blendings of
+  the extreme lights.
+
+  The two conceptions which may now be said to animate the theory of
+  geography are the genetic, which depends upon processes of origin, and
+  the morphological, which depends on facts of form and distribution.
+
+
+  PROGRESS OF GEOGRAPHICAL DISCOVERY
+
+  Exploration and geographical discovery must have started from more
+  than one centre, and to deal justly with the matter one ought to treat
+  of these separately in the early ages before the whole civilized world
+  was bound together by the bonds of modern intercommunication. At the
+  least there should be some consideration of four separate systems of
+  discovery--the Eastern, in which Chinese and Japanese explorers
+  acquired knowledge of the geography of Asia, and felt their way
+  towards Europe and America; the Western, in which the dominant races
+  of the Mexican and South American plateaus extended their knowledge of
+  the American continent before Columbus; the Polynesian, in which the
+  conquering races of the Pacific Islands found their way from group to
+  group; and the Mediterranean. For some of these we have no certain
+  information, and regarding others the tales narrated in the early
+  records are so hard to reconcile with present knowledge that they are
+  better fitted to be the battle-ground of scholars championing rival
+  theories than the basis of definite history. So it has come about that
+  the only practicable history of geographical exploration starts from
+  the Mediterranean centre, the first home of that civilization which
+  has come to be known as European, though its field of activity has
+  long since overspread the habitable land of both temperate zones,
+  eastern Asia alone in part excepted.
+
+  From all centres the leading motives of exploration were probably the
+  same--commercial intercourse, warlike operations, whether resulting in
+  conquest or in flight, religious zeal expressed in pilgrimages or
+  missionary journeys, or, from the other side, the avoidance of
+  persecution, and, more particularly in later years, the advancement of
+  knowledge for its own sake. At different times one or the other motive
+  predominated.
+
+  Before the 14th century B.C. the warrior kings of Egypt had carried
+  the power of their arms southward from the delta of the Nile well-nigh
+  to its source, and eastward to the confines of Assyria. The
+  hieroglyphic inscriptions of Egypt and the cuneiform inscriptions of
+  Assyria are rich in records of the movements and achievements of
+  armies, the conquest of towns and the subjugation of peoples; but
+  though many of the recorded sites have been identified, their
+  discovery by wandering armies was isolated from their subsequent
+  history and need not concern us here.
+
+
+    The Phoenicians.
+
+  The Phoenicians are the earliest Mediterranean people in the
+  consecutive chain of geographical discovery which joins pre-historic
+  time with the present. From Sidon, and later from its more famous
+  rival Tyre, the merchant adventurers of Phoenicia explored and
+  colonized the coasts of the Mediterranean and fared forth into the
+  ocean beyond. They traded also on the Red sea, and opened up regular
+  traffic with India as well as with the ports of the south and west, so
+  that it was natural for Solomon to employ the merchant navies of Tyre
+  in his oversea trade. The western emporium known in the scriptures as
+  Tarshish was probably situated in the south of Spain, possibly at
+  Cadiz, although some writers contend that it was Carthage in North
+  Africa. Still more diversity of opinion prevails as to the southern
+  gold-exporting port of Ophir, which some scholars place in Arabia,
+  others at one or another point on the east coast of Africa. Whether
+  associated with the exploitation of Ophir (q.v.) or not the first
+  great voyage of African discovery appears to have been accomplished by
+  the Phoenicians sailing the Red Sea. Herodotus (himself a notable
+  traveller in the 5th century B.C.) relates that the Egyptian king
+  Necho of the XXVIth Dynasty (c. 600 B.C.) built a fleet on the Red
+  Sea, and confided it to Phoenician sailors with the orders to sail
+  southward and return to Egypt by the Pillars of Hercules and the
+  Mediterranean sea. According to the tradition, which Herodotus quotes
+  sceptically, this was accomplished; but the story is too vague to be
+  accepted as more than a possibility.
+
+  The great Phoenician colony of Carthage, founded before 800 B.C.,
+  perpetuated the commercial enterprise of the parent state, and
+  extended the sphere of practical trade to the ocean shores of Africa
+  and Europe. The most celebrated voyage of antiquity undertaken for the
+  express purpose of discovery was that fitted out by the senate of
+  Carthage under the command of Hanno, with the intention of founding
+  new colonies along the west coast of Africa. According to Pliny, the
+  only authority on this point, the period of the voyage was that of the
+  greatest prosperity of Carthage, which may be taken as somewhere
+  between 570 and 480 B.C. The extent of this voyage is doubtful, but it
+  seems probable that the farthest point reached was on the east-running
+  coast which bounds the Gulf of Guinea on the north. Himilco, a
+  contemporary of Hanno, was charged with an expedition along the west
+  coast of Iberia northward, and as far as the uncertain references to
+  this voyage can be understood, he seems to have passed the Bay of
+  Biscay and possibly sighted the coast of England.
+
+
+    The Greeks.
+
+  The sea power of the Greek communities on the coast of Asia Minor and
+  in the Archipelago began to be a formidable rival to the Phoenician
+  soon after the time of Hanno and Himilco, and peculiar interest
+  attaches to the first recorded Greek voyage beyond the Pillars of
+  Hercules. Pytheas, a navigator of the Phocean colony of Massilia
+  (Marseilles), determined the latitude of that port with considerable
+  precision by the somewhat clumsy method of ascertaining the length of
+  the longest day, and when, about 330 B.C., he set out on exploration
+  to the northward in search of the lands whence came gold, tin and
+  amber, he followed this system of ascertaining his position from time
+  to time. If on each occasion he himself made the observations his
+  voyage must have extended over six years; but it is not impossible
+  that he ascertained the approximate length of the longest day in some
+  cases by questioning the natives. Pytheas, whose own narrative is not
+  preserved, coasted the Bay of Biscay, sailed up the English Channel
+  and followed the coast of Britain to its most northerly point. Beyond
+  this he spoke of a land called _Thule_, which, if his estimate of the
+  length of the longest day is correct, may have been Shetland, but was
+  possibly Iceland; and from some confused statements as to a sea which
+  could not be sailed through, it has been assumed that Pytheas was the
+  first of the Greeks to obtain direct knowledge of the Arctic regions.
+  During this or a second voyage Pytheas entered the Baltic, discovered
+  the coasts where amber is obtained and returned to the Mediterranean.
+  It does not seem that any maritime trade followed these discoveries,
+  and indeed it is doubtful whether his contemporaries accepted the
+  truth of Pytheas's narrative; Strabo four hundred years later
+  certainly did not, but the critical studies of modern scholars have
+  rehabilitated the Massilian explorer.
+
+
+    Alexander the Great.
+
+  The Greco-Persian wars had made the remoter parts of Asia Minor more
+  than a name to the Greek geographers before the time of Alexander the
+  Great, but the campaigns of that conqueror from 329 to 325 B.C. opened
+  up the greater Asia to the knowledge of Europe. His armies crossed the
+  plains beyond the Caspian, penetrated the wild mountain passes
+  north-west of India, and did not turn back until they had entered on
+  the Indo-Gangetic plain. This was one of the few great epochs of
+  geographical discovery.
+
+  The world was henceforth viewed as a very large place stretching far
+  on every side beyond the Midland or Mediterranean Sea, and the land
+  journey of Alexander resulted in a voyage of discovery in the outer
+  ocean from the mouth of the Indus to that of the Tigris, thus opening
+  direct intercourse between Grecian and Hindu civilization. The Greeks
+  who accompanied Alexander described with care the towns and villages,
+  the products and the aspect of the country. The conqueror also
+  intended to open up trade by sea between Europe and India, and the
+  narrative of his general Nearchus records this famous voyage of
+  discovery, the detailed accounts of the chief pilot Onesicritus being
+  lost. At the beginning of October 326 B.C. Nearchus left the Indus
+  with his fleet, and the anchorages sought for each night are carefully
+  recorded. He entered the Persian Gulf, and rejoined Alexander at Susa,
+  when he was ordered to prepare another expedition for the
+  circumnavigation of Arabia. Alexander died at Babylon in 323 B.C., and
+  the fleet was dispersed without making the voyage.
+
+  The dynasties founded by Alexander's generals, Seleucus, Antiochus and
+  Ptolemy, encouraged the same spirit of enterprise which their master
+  had fostered, and extended geographical knowledge in several
+  directions. Seleucus Nicator established the Greco-Bactrian empire and
+  continued the intercourse with India. Authentic information respecting
+  the great valley of the Ganges was supplied by Megasthenes, an
+  ambassador sent by Seleucus, who reached the remote city of
+  Patali-putra, the modern Patna.
+
+
+    The Ptolemies.
+
+  The Ptolemies in Egypt showed equal anxiety to extend the bounds of
+  geographical knowledge. Ptolemy Euergetes (247-222 B.C.) rendered the
+  greatest service to geography by the protection and encouragement of
+  Eratosthenes, whose labours gave the first approximate knowledge of
+  the true size of the spherical earth. The second Euergetes and his
+  successor Ptolemy Lathyrus (118-115 B.C.) furnished Eudoxus with a
+  fleet to explore the Arabian sea. After two successful voyages,
+  Eudoxus, impressed with the idea that Africa was surrounded by ocean
+  on the south, left the Egyptian service, and proceeded to Cadiz and
+  other Mediterranean centres of trade seeking a patron who would
+  finance an expedition for the purpose of African discovery; and we
+  learn from Strabo that the veteran explorer made at least two voyages
+  southward along the coast of Africa. The Ptolemies continued to send
+  fleets annually from their Red Sea ports of Berenice and Myos Hormus
+  to Arabia, as well as to ports on the coasts of Africa and India.
+
+
+    The Romans.
+
+  The Romans did not encourage navigation and commerce with the same
+  ardour as their predecessors; still the luxury of Rome, which gave
+  rise to demands for the varied products of all the countries of the
+  known world, led to an active trade both by ships and caravans. But it
+  was the military genius of Rome, and the ambition for universal
+  empire, which led, not only to the discovery, but also to the survey
+  of nearly all Europe, and of large tracts in Asia and Africa. Every
+  new war produced a new survey and itinerary of the countries which
+  were conquered, and added one more to the imperishable roads that led
+  from every quarter of the known world to Rome. In the height of their
+  power the Romans had surveyed and explored all the coasts of the
+  Mediterranean, Italy, Greece, the Balkan Peninsula, Spain, Gaul,
+  western Germany and southern Britain. In Africa their empire included
+  Egypt, Carthage, Numidia and Mauritania. In Asia they held Asia Minor
+  and Syria, had sent expeditions into Arabia, and were acquainted with
+  the more distant countries formerly invaded by Alexander, including
+  Persia, Scythia, Bactria and India. Roman intercourse with India
+  especially led to the extension of geographical knowledge.
+
+  Before the Roman legions were sent into a new region to extend the
+  limits of the empire, it was usual to send out exploring expeditions
+  to report as to the nature of the country. It is narrated by Pliny and
+  Seneca that the emperor Nero sent out two centurions on such a mission
+  towards the source of the Nile (probably about A.D. 60), and that the
+  travellers pushed southwards until they reached vast marshes through
+  which they could not make their way either on foot or in boats. This
+  seems to indicate that they had penetrated to about 9° N. Shortly
+  before A.D. 79 Hippalus took advantage of the regular alternation of
+  the monsoons to make the voyage from the Red Sea to India across the
+  open ocean out of sight of land. Even though this sea-route was known,
+  the author of the _Periplus of the Erythraean Sea_, published after
+  the time of Pliny, recites the old itinerary around the coast of the
+  Arabian Gulf. It was, however, in the reigns of Severus and his
+  immediate successors that Roman intercourse with India was at its
+  height, and from the writings of Pausanias (c. 174) it appears that
+  direct communication between Rome and China had already taken place.
+
+  After the division of the Roman empire, Constantinople became the last
+  refuge of learning, arts and taste; while Alexandria continued to be
+  the emporium whence were imported the commodities of the East. The
+  emperor Justinian (483-565), in whose reign the greatness of the
+  Eastern empire culminated, sent two Nestorian monks to China, who
+  returned with eggs of the silkworm concealed in a hollow cane, and
+  thus silk manufactures were established in the Peloponnesus and the
+  Greek islands. It was also in the reign of Justinian that Cosmas
+  Indicopleustes, an Egyptian merchant, made several voyages, and
+  afterwards composed his [Greek: Christianikź topographia] (Christian
+  Topography), containing, in addition to his absurd cosmogony, a
+  tolerable description of India.
+
+
+    The Arabs.
+
+  The great outburst of Mahommedan conquest in the 7th century was
+  followed by the Arab civilization, having its centres at Bagdad and
+  Cordova, in connexion with which geography again received a share of
+  attention. The works of the ancient Greek geographers were translated
+  into Arabic, and starting with a sound basis of theoretical knowledge,
+  exploration once more made progress. From the 9th to the 13th century
+  intelligent Arab travellers wrote accounts of what they had seen and
+  heard in distant lands. The earliest Arabian traveller whose
+  observations have come down to us is the merchant Sulaiman, who
+  embarked in the Persian Gulf and made several voyages to India and
+  China, in the middle of the 9th century. Abu Zaid also wrote on India,
+  and his work is the most important that we possess before the
+  epoch-making discoveries of Marco Polo. Masudi, a great traveller who
+  knew from personal experience all the countries between Spain and
+  China, described the plains, mountains and seas, the dynasties and
+  peoples, in his _Meadows of Gold_, an abstract made by himself of his
+  larger work _News of the Time_. He died in 956, and was known, from
+  the comprehensiveness of his survey, as the Pliny of the East. Amongst
+  his contemporaries were Istakhri, who travelled through all the
+  Mahommedan countries and wrote his _Book of Climates_ in 950, and Ibn
+  Haukal, whose _Book of Roads and Kingdoms_, based on the work of
+  Istakhri, was written in 976. Idrisi, the best known of the Arabian
+  geographical authors, after travelling far and wide in the first half
+  of the 12th century, settled in Sicily, where he wrote a treatise
+  descriptive of an armillary sphere which he had constructed for Roger
+  II., the Norman king, and in this work he incorporated all accessible
+  results of contemporary travel.
+
+
+    The Northmen.
+
+  The Northmen of Denmark and Norway, whose piratical adventures were
+  the terror of all the coasts of Europe, and who established themselves
+  in Great Britain and Ireland, in France and Sicily, were also
+  geographical explorers in their rough but practical way during the
+  darkest period of the middle ages. All Northmen were not bent on
+  rapine and plunder; many were peaceful merchants. Alfred the Great,
+  king of the Saxons in England, not only educated his people in the
+  learning of the past ages; he inserted in the geographical works he
+  translated many narratives of the travel of his own time. Thus he
+  placed on record the voyages of the merchant Ulfsten in the Baltic,
+  including particulars of the geography of Germany. And in particular
+  he told of the remarkable voyage of Other, a Norwegian of Helgeland,
+  who was the first authentic Arctic explorer, the first to tell of the
+  rounding of the North Cape and the sight of the midnight sun. This
+  voyage of the middle of the 9th century deserves to be held in happy
+  memory, for it unites the first Norwegian polar explorer with the
+  first English collector of travels. Scandinavian merchants brought the
+  products of India to England and Ireland. From the 8th to the 11th
+  century a commercial route from India passed through Novgorod to the
+  Baltic, and Arabian coins found in Sweden, and particularly in the
+  island of Gotland, prove how closely the enterprise of the Northmen
+  and of the Arabs intertwined. Five-sixths of these coins preserved at
+  Stockholm were from the mints of the Samanian dynasty, which reigned
+  in Khorasan and Transoxiana from about A.D. 900 to 1000. It was the
+  trade with the East that originally gave importance to the city of
+  Visby in Gotland.
+
+  In the end of the 9th century Iceland was colonized from Norway; and
+  about 985 the intrepid viking, Eric the Red, discovered Greenland, and
+  induced some of his Icelandic countrymen to settle on its inhospitable
+  shores. His son, Leif Ericsson, and others of his followers were
+  concerned in the discovery of the North American coast (see VINLAND),
+  which, but for the isolation of Iceland from the centres of European
+  awakening, would have had momentous consequences. As things were, the
+  importance of this discovery passed unrecognized. The story of two
+  Venetians, Nicolo and Antonio Zeno, who gave a vague account of
+  voyages in the northern seas in the end of the 13th century, is no
+  longer to be accepted as history.
+
+
+    Close of the dark ages.
+
+  At length the long period of barbarism which accompanied and followed
+  the fall of the Roman empire drew to a close in Europe. The Crusades
+  had a favourable influence on the intellectual state of the Western
+  nations. Interesting regions, known only by the scant reports of
+  pilgrims, were made the objects of attention and study; while
+  religious zeal, and the hope of gain, combined with motives of mere
+  curiosity, induced several persons to travel by land into remote
+  regions of the East, far beyond the countries to which the operations
+  of the crusaders extended. Among these was Benjamin of Tudela, who set
+  out from Spain in 1160, travelled by land to Constantinople, and
+  having visited India and some of the eastern islands, returned to
+  Europe by way of Egypt after an absence of thirteen years.
+
+
+    Asiatic journeys.
+
+  Joannes de Plano Carpini, a Franciscan monk, was the head of one of
+  the missions despatched by Pope Innocent to call the chief and people
+  of the Tatars to a better mind. He reached the headquarters of Batu,
+  on the Volga, in February 1246; and, after some stay, went on to the
+  camp of the great khan near Karakorum in central Asia, and returned
+  safely in the autumn of 1247. A few years afterwards, a Fleming named
+  Rubruquis was sent on a similar mission, and had the merit of being
+  the first traveller of this era who gave a correct account of the
+  Caspian Sea. He ascertained that it had no outlet. At nearly the same
+  time Hayton, king of Armenia, made a journey to Karakorum in 1254, by
+  a route far to the north of that followed by Carpini and Rubruquis. He
+  was treated with honour and hospitality, and returned by way of
+  Samarkand and Tabriz, to his own territory. The curious narrative of
+  King Hayton was translated by Klaproth.
+
+  While the republics of Italy, and above all the state of Venice, were
+  engaged in distributing the rich products of India and the Far East
+  over the Western world, it was impossible that motives of curiosity,
+  as well as a desire of commercial advantage, should not be awakened to
+  such a degree as to impel some of the merchants to visit those remote
+  lands. Among these were the brothers Polo, who traded with the East
+  and themselves visited Tatary. The recital of their travels fired the
+  youthful imagination of young Marco Polo, son of Nicolo, and he set
+  out for the court of Kublai Khan, with his father and uncle, in 1265.
+  Marco remained for seventeen years in the service of the Great Khan,
+  and was employed on many important missions. Besides what he learnt
+  from his own observation, he collected much information from others
+  concerning countries which he did not visit. He returned to Europe
+  possessed of a vast store of knowledge respecting the eastern parts of
+  the world, and, being afterwards made a prisoner by the Genoese, he
+  dictated the narrative of his travels during his captivity. The work
+  of Marco Polo is the most valuable narrative of travels that appeared
+  during the middle ages, and despite a cold reception and many denials
+  of the accuracy of the record, its substantial truthfulness has been
+  abundantly proved.
+
+  Missionaries continued to do useful geographical work. Among them were
+  John of Monte Corvino, a Franciscan monk, Andrew of Perugia, John
+  Marignioli and Friar Jordanus, who visited the west coast of India,
+  and above all Friar Odoric of Pordenone. Odoric set out on his travels
+  about 1318, and his journeys embraced parts of India, the Malay
+  Archipelago, China and even Tibet, where he was the first European to
+  enter Lhasa, not yet a forbidden city.
+
+  Ibn Batuta, the great Arab traveller, is separated by a wide space of
+  time from his countrymen already mentioned, and he finds his proper
+  place in a chronological notice after the days of Marco Polo, for he
+  did not begin his wanderings until 1325, his career thus coinciding in
+  time with the fabled journeyings of Sir John Mandeville. While Arab
+  learning flourished during the darkest ages of European ignorance, the
+  last of the Arab geographers lived to see the dawn of the great period
+  of the European awakening. Ibn Batuta went by land from Tangier to
+  Cairo, then visited Syria, and performed the pilgrimages to Medina and
+  Mecca. After exploring Persia, and again residing for some time at
+  Mecca, he made a voyage down the Red sea to Yemen, and travelled
+  through that country to Aden. Thence he visited the African coast,
+  touching at Mombasa and Quiloa, and then sailed across to Ormuz and
+  the Persian Gulf. He crossed Arabia from Bahrein to Jidda, traversed
+  the Red sea and the desert to Syene, and descended the Nile to Cairo.
+  After this he revisited Syria and Asia Minor, and crossed the Black
+  sea, the desert from Astrakhan to Bokhara, and the Hindu Kush. He was
+  in the service of Muhammad Tughluk, ruler of Delhi, about eight years,
+  and was sent on an embassy to China, in the course of which the
+  ambassadors sailed down the west coast of India to Calicut, and then
+  visited the Maldive Islands and Ceylon. Ibn Batuta made the voyage
+  through the Malay Archipelago to China, and on his return he proceeded
+  from Malabar to Bagdad and Damascus, ultimately reaching Fez, the
+  capital of his native country, in November 1349. After a journey into
+  Spain he set out once more for Central Africa in 1352, and reached
+  Timbuktu and the Niger, returning to Fez in 1353. His narrative was
+  committed to writing from his dictation.
+
+
+    Spanish exploration.
+
+  The European country which had come the most completely under the
+  influence of Arab culture now began to send forth explorers to distant
+  lands, though the impulse came not from the Moors but from Italian
+  merchant navigators in Spanish service. The peaceful reign of Henry
+  III. of Castile is famous for the attempts of that prince to extend
+  the diplomatic relations of Spain to the remotest parts of the earth.
+  He sent embassies to all the princes of Christendom and to the Moors.
+  In 1403 the Spanish king sent a knight of Madrid, Ruy Gonzalez de
+  Clavijo, to the distant court of Timur, at Samarkand. He returned in
+  1406, and wrote a valuable narrative of his travels.
+
+  Italians continued to make important journeys in the East during the
+  15th century. Among them was Nicolo Conti, who passed through Persia,
+  sailed along the coast of Malabar, visited Sumatra, Java and the south
+  of China, returned by the Red sea, and got home to Venice in 1444
+  after an absence of twenty-five years. He related his adventures to
+  Poggio Bracciolini, secretary to Pope Eugenius IV.; and the narrative
+  contains much interesting information. One of the most remarkable of
+  the Italian travellers was Ludovico di Varthema, who left his native
+  land in 1502. He went to Egypt and Syria, and for the sake of visiting
+  the holy cities became a Mahommedan. He was the first European who
+  gave an account of the interior of Yemen. He afterwards visited and
+  described many places in Persia, India and the Malay Archipelago,
+  returning to Europe in a Portuguese ship after an absence of five
+  years.
+
+
+    Portuguese exploration--Prince Henry the Navigator.
+
+  In the 15th century the time was approaching when the discovery of the
+  Cape of Good Hope was to widen the scope of geographical enterprise.
+  This great event was preceded by the general utilization in Europe of
+  the polarity of the magnetic needle in the construction of the
+  mariner's compass. Portugal took the lead along this new path, and
+  foremost among her pioneers stands Prince Henry the Navigator
+  (1394-1460), who was a patron both of exploration and of the study of
+  geographical theory. The great westward projection of the coast of
+  Africa, and the islands to the north-west of that continent, were the
+  principal scene of the work of the mariners sent out at his expense;
+  but his object was to push onward and reach India from the Atlantic.
+  The progress of discovery received a check on his death, but only for
+  a time. In 1462 Pedro de Cintra extended Portuguese exploration along
+  the African coast and discovered Sierra Leone. Fernan Gomez followed
+  in 1469, and opened trade with the Gold Coast; and in 1484 Diogo Cćo
+  discovered the mouth of the Congo. The king of Portugal next
+  despatched Bartolomeu Diaz in 1486 to continue discoveries southwards;
+  while, in the following year, he sent Pedro de Covilhćo and Affonso de
+  Payva to discover the country of Prester John. Diaz succeeded in
+  rounding the southern point of Africa, which he named Cabo
+  Tormentoso--the Cape of Storms--but King Joćo II., foreseeing the
+  realization of the long-sought passage to India, gave it the
+  stimulating and enduring name of the Cape of Good Hope. Payva died at
+  Cairo; but Covilhćo, having heard that a Christian ruler reigned in
+  the mountains of Ethiopia, penetrated into Abyssinia in 1490. He
+  delivered the letter which Joćo II. had addressed to Prester John to
+  the Negus Alexander of Abyssinia, but he was detained by that prince
+  and never allowed to leave the country.
+
+
+    Columbus.
+
+  The Portuguese, following the lead of Prince Henry, continued to look
+  for the road to India by the Cape of Good Hope. The same end was
+  sought by Christopher Columbus, following the suggestion of
+  Toscanelli, and under-estimating the diameter of the globe, by sailing
+  due west. The voyages of Columbus (1492-1498) resulted in the
+  discovery of the West Indies and North America which barred the way to
+  the Far East. In 1493 the pope, Alexander VI., issued a bull
+  instituting the famous "line of demarcation" running from N. to S. 100
+  leagues W. of the Azores, to the west of which the Spaniards were
+  authorized to explore and to the east of which the Portuguese received
+  the monopoly of discovery. The direct line of Portuguese exploration
+  resulted in the discovery of the Cape route to India by Vasco da Gama
+  (1498), and in 1500 to the independent discovery of South America by
+  Pedro Alvarez Cabral. The voyages of Columbus and of Vasco da Gama
+  were so important that it is unnecessary to detail their results in
+  this place. See COLUMBUS, CHRISTOPHER; GAMA, VASCO DA.
+
+
+    Vasco da Gama.
+
+  The three voyages of Vasco da Gama (who died on the scene of his
+  labours, at Cochin, in 1524) revolutionized the commerce of the East.
+  Until then the Venetians held the carrying trade of India, which was
+  brought by the Persian Gulf and Red sea into Syria and Egypt, the
+  Venetians receiving the products of the East at Alexandria and Beirut
+  and distributing them over Europe. This commerce was a great source of
+  wealth to Venice; but after the discovery of the new passage round the
+  Cape, and the conquests of the Portuguese, the trade of the East
+  passed into other hands.
+
+
+    Spaniards in America.
+
+  The discoveries of Columbus awakened a spirit of enterprise in Spain
+  which continued in full force for a century; adventurers flocked
+  eagerly across the Atlantic, and discovery followed discovery in rapid
+  succession. Many of the companions of Columbus continued his work.
+  Vicente Yańez Pinzon in 1500 reached the mouth of the Amazon. In the
+  same year Alonso de Ojeda, accompanied by Juan de la Cosa, from whose
+  maps we learn much of the discoveries of the 16th century navigators,
+  and by a Florentine named Amerigo Vespucci, touched the coast of South
+  America somewhere near Surinam, following the shore as far as the Gulf
+  of Maracaibo. Vespucci afterwards made three voyages to the Brazilian
+  coast; and in 1504 he wrote an account of his four voyages, which was
+  widely circulated, and became the means of procuring for its author at
+  the hands of the cartographer Waldseemüller in 1507 the
+  disproportionate distinction of giving his name to the whole
+  continent. In 1508 Alonso de Ojeda obtained the government of the
+  coast of South America from Cabo de la Vela to the Gulf of Darien;
+  Ojeda landed at Cartagena in 1510, and sustained a defeat from the
+  natives, in which his lieutenant, Juan de la Cosa, was killed. After
+  another reverse on the east side of the Gulf of Darien Ojeda returned
+  to Hispaniola and died there. The Spaniards in the Gulf of Darien were
+  left by Ojeda under the command of Francisco Pizarro, the future
+  conqueror of Peru. After suffering much from famine and disease,
+  Pizarro resolved to leave, and embarked the survivors in small
+  vessels, but outside the harbour they met a ship which proved to be
+  that of Martin Fernandez Enciso, Ojeda's partner, coming with
+  provisions and reinforcements. One of the crew of Enciso's ship, Vasco
+  Nuńez de Balboa, the future discoverer of the Pacific Ocean, induced
+  his commander to form a settlement on the other side of the Gulf of
+  Darien. The soldiers became discontented and deposed Enciso, who was a
+  man of learning and an accomplished cosmographer. His work _Suma de
+  Geografia_, which was printed in 1519, is the first Spanish book which
+  gives an account of America. Vasco Nuńez, the new commander, entered
+  upon a career of conquest in the neighbourhood of Darien, which ended
+  in the discovery of the Pacific Ocean on the 25th of September 1513.
+  Vasco Nuńez was beheaded in 1517 by Pedrarias de Avila, who was sent
+  out to supersede him. This was one of the greatest calamities that
+  could have happened to South America; for the discoverer of the South
+  sea was on the point of sailing with a little fleet into his unknown
+  ocean, and a humane and judicious man would probably have been the
+  conqueror of Peru, instead of the cruel and ignorant Pizarro. In the
+  year 1519 Panama was founded by Pedrarias; and the conquest of Peru by
+  Pizarro followed a few years afterwards. Hernan Cortes overran and
+  conquered Mexico from 1518 to 1521, and the discovery and conquest of
+  Guatemala by Alvarado, the invasion of Florida by De Soto, and of
+  Nueva Granada by Quesada, followed in rapid succession. The first
+  detailed account of the west coast of South America was written by a
+  keenly observant old soldier, Pedro de Cieza de Leon, who was
+  travelling in South America from 1533 to 1550, and published his story
+  at Seville in 1553.
+
+
+    Pacific Ocean.
+
+  The great desire of the Spanish government at that time was to find a
+  westward route to the Moluccas. For this purpose Juan Diaz de Solis
+  was despatched in October 1515, and in January 1516 he discovered the
+  mouth of the Rio de la Plata. He was, however, killed by the natives,
+  and his ships returned. In the following year the Portuguese
+  Ferdinando Magalhćes, familiarly known as Magellan, laid before
+  Charles V., at Valladolid, a scheme for reaching the Spice Islands by
+  sailing westward. He started on the 21st of September 1519, entered
+  the strait which now bears his name in October 1520, worked his way
+  through between Patagonia and Tierra del Fuego, and entered on the
+  vast Pacific which he crossed without sighting any of its innumerable
+  island groups. This was unquestionably the greatest of the voyages
+  which followed from the impulse of Prince Henry, and it was rendered
+  possible only by the magnificent courage of the commander in spite of
+  rebellion, mutiny and starvation. It was the 6th of March 1521 when he
+  reached the Ladrone Islands. Thence Magellan proceeded to the
+  Philippines, and there his career ended in an unimportant encounter
+  with hostile natives. Eventually a Biscayan named Sebastian del Cano,
+  sailing home by way of the Cape of Good Hope, reached San Lucar in
+  command of the "Victoria" on the 6th of September 1522, with eighteen
+  survivors; this one ship of the squadron which sailed on the quest
+  succeeded in accomplishing the first circumnavigation of the globe.
+  Del Cano was received with great distinction by the emperor, who
+  granted him a globe for his crest, and the motto _Primus circumdedisti
+  me_.
+
+
+    Portuguese in Africa and the East.
+
+  While the Spaniards were circumnavigating the world and completing
+  their knowledge of the coasts of Central and South America, the
+  Portuguese were actively engaged on similar work as regards Africa and
+  the East Indies.
+
+  With Abyssinia the mission of Covilhćo led to further intercourse. In
+  April 1520 Vasco da Gama, as viceroy of the Indies, took a fleet into
+  the Red sea, and landed an embassy consisting of Dom Rodriguez de Lima
+  and Father Francisco Alvarez, a priest whose detailed narrative is the
+  earliest and not the least interesting account we possess of
+  Abyssinia. It was not until 1526 that the embassy was dismissed; and
+  not many years afterwards the negus entreated the help of the
+  Portuguese against Mahommedan invaders, and the viceroy sent an
+  expeditionary force, commanded by his brother Cristoforo da Gama, with
+  450 musketeers. Da Gama was taken prisoner and killed, but his
+  followers enabled the Christians of Abyssinia to regain their power,
+  and a Jesuit mission remained in the country. The Portuguese also
+  established a close connexion with the kingdom of Congo on the west
+  side of Africa, and obtained much information respecting the interior
+  of the continent. Duarte Lopez, a Portuguese settled in the country,
+  was sent on a mission to Rome by the king of Congo, and Pope Sixtus V.
+  caused him to recount to his chamberlain, Felipe Pigafetta, all he had
+  learned during the nine years he had been in Africa, from 1578 to
+  1587. This narrative, under the title of _Description of the Kingdom
+  of Congo_, was published at Rome by Pigafetta in 1591. A map was
+  attached on which several great equatorial lakes are shown, and the
+  empire of Monomwezi or Unyamwezi is laid down. The most valuable work
+  on Africa about this time is, however, that written by the Moor Leo
+  Africanus in the early part of the 16th century. Leo travelled
+  extensively in the north and west of Africa, and was eventually taken
+  by pirates and sold to a master who presented him to Pope Leo X. At
+  the pope's desire he translated his work on Africa into Italian.
+
+  In Further India and the Malay Archipelago the Portuguese acquired
+  predominating influence at sea, establishing factories on the Malabar
+  coast, in the Persian Gulf, at Malacca, and in the Spice Islands, and
+  extending their commercial enterprises from the Red sea to China.
+  Their missionaries were received at the court of Akbar, and Benedict
+  Goes, a native of the Azores, was despatched on a journey overland
+  from Agra to China. He started in 1603, and, after traversing the
+  least-known parts of Central Asia, he reached the confines of China.
+  He appears to have ascended from Kabul to the plateau of the Pamir,
+  and thence onwards by Yarkand, Khotan and Aksu. He died on the journey
+  in March 1607; and thus, as one of the brethren pronounced his
+  epitaph, "seeking Cathay he found heaven."
+
+
+    English, Dutch and French.
+
+  The activity and love of adventure, which became a passion for two or
+  three generations in Spain and Portugal, spread to other countries. It
+  was the spirit of the age; and England, Holland and France were fired
+  by it. English enterprise was first aroused by John and Sebastian
+  Cabot, father and son, who came from Venice and settled at Bristol in
+  the time of Henry VII. The Cabots received a patent in 1496,
+  empowering them to seek unknown lands; and John Cabot discovered
+  Newfoundland and part of the coast of America. Sebastian afterwards
+  made a voyage to Rio de la Plata in the service of Spain, but he
+  returned to England in 1548 and received a pension from Edward VI. At
+  his suggestion a voyage was undertaken for the discovery of a
+  north-east passage to Cathay, with Sir Hugh Willoughby as
+  captain-general of the fleet and Richard Chancellor as pilot-major.
+  They sailed in May 1553, but Willoughby and all his crew perished on
+  the Lapland coast. Chancellor, however, was more fortunate. He reached
+  the White Sea, performed the journey overland to Moscow, where he was
+  well received, and may be said to have been the founder of the trade
+  between Russia and England. He returned to Archangel and brought his
+  ship back in safety to England. On a second voyage, in 1556,
+  Chancellor was drowned; and three subsequent voyages, led by Stephen
+  Burrough, Arthur Pet and Charles Jackman, in small craft of 50 tons
+  and under, carried on an examination of the straits which lead into
+  the Kara sea.
+
+  The French followed closely on the track of John Cabot, and Norman and
+  Breton fishermen frequented the banks of Newfoundland at the beginning
+  of the 16th century. In 1524 Francis I. sent Giovanni da Verazzano of
+  Florence on an expedition of discovery to the coast of North America;
+  and the details of his voyage were embodied in a letter addressed by
+  him to the king of France from Dieppe, in July 1524. In 1534 Jacques
+  Cartier set out to continue the discoveries of Verazzano, and visited
+  Newfoundland and the Gulf of St Lawrence. In the following year he
+  made another voyage, discovered the island of Anticosti, and ascended
+  the St Lawrence to Hochelaga, now Montreal. He returned, after passing
+  two winters in Canada; and on another occasion he also failed to
+  establish a colony. Admiral de Coligny made several unsuccessful
+  endeavours to form a colony in Florida under Jean Ribault of Dieppe,
+  René de Laudonničre and others, but the settlers were furiously
+  assailed by the Spaniards and the attempt was abandoned.
+
+
+    The Elizabethan era.
+
+  The reign of Elizabeth is famous for the gallant enterprises that were
+  undertaken by sea and land to discover and bring to light the unknown
+  parts of the earth. The great promoter of geographical discovery in
+  the Elizabethan period was Richard Hakluyt (1553-1616), who was active
+  in the formation of the two companies for colonizing Virginia in 1606;
+  and devoted his life to encouraging and recording similar
+  undertakings. He published much, and left many valuable papers at his
+  death, most of which, together with many other narratives, were
+  published in 1622 in the great work of the Rev. Samuel Purchas,
+  entitled _Hakluytus Posthumus, or Purchas his Pilgrimes_.
+
+  It is from these works that our knowledge of the gallant deeds of the
+  English and other explorers of the Elizabethan age is mainly derived.
+  The great and splendidly illustrated collections of voyages and
+  travels of Theodorus de Bry and Hulsius served a similar useful
+  purpose on the continent of Europe. One important object of English
+  maritime adventurers of those days was to discover a route to Cathay
+  by the north-west, a second was to settle Virginia, and a third was to
+  raid the Spanish settlements in the West Indies. Nor was the trade to
+  Muscovy and Turkey neglected; while latterly a resolute and successful
+  attempt was made to establish direct commercial relations with India.
+
+  The conception of the north-western route to Cathay now leads the
+  story of exploration, for the first time as far as important and
+  sustained efforts are concerned, towards the Arctic seas. This part of
+  the story is fully told under the heading of POLAR REGIONS, and only
+  the names of Martin Frobisher (1576), John Davis (1585), Henry Hudson
+  (1607) and William Baffin (1616) need be mentioned here in order to
+  preserve the complete conspectus of the history of discovery. The
+  Dutch emulated the British in the Arctic seas during this period,
+  directing their efforts mainly towards the discovery of a north-east
+  passage round the northern end of Novaya Zemlya; and William Barents
+  or Barendsz (1594-1597) is the most famous name in this connexion, his
+  boat voyage along the coast of Novaya Zemlya after losing his ship and
+  wintering in a high latitude, being one of the most remarkable
+  achievements in polar annals.
+
+  Many English voyages were also made to Guinea and the West Indies, and
+  twice English vessels followed in the track of Magellan, and
+  circumnavigated the globe. In 1577 Francis Drake, who had previously
+  served with Hawkins in the West Indies, undertook his celebrated
+  voyage round the world. Reaching the Pacific through the Strait of
+  Magellan, Drake proceeded northward along the west coast of America,
+  resolved to attempt the discovery of a northern passage from the
+  Pacific to the Atlantic. The coast from the southern extremity of the
+  Californian peninsula to Cape Mendocino had been discovered by Juan
+  Rodriguez Cabrillo and Francisco de Ulloa in 1539. Drake's discoveries
+  extended from Cape Mendocino to 48° N., in which latitude he gave up
+  his quest, sailed across the Pacific and reached the Philippine
+  Islands, returning home round the Cape of Good Hope in 1580.
+
+  Thomas Cavendish, emulous of Drake's example, fitted out three vessels
+  for an expedition to the South sea in 1586. He took the same route as
+  Drake along the west coast of America. From Cape San Lucas Cavendish
+  steered across the Pacific, seeing no land until he reached the
+  Ladrone Islands. He returned to England in 1588. The third English
+  voyage into the Pacific was not so fortunate. Sir Richard Hawkins
+  (1593) on reaching the bay of Atacames, in 1°N. in 1594, was attacked
+  by a Spanish fleet, and, after a desperate naval engagement, was
+  forced to surrender. Hawkins declared his object to be discovery and
+  the survey of unknown lands, and his voyage, though terminating in
+  disaster, bore good fruit. _The Observations of Sir Richard Hawkins in
+  his Voyage into the South Sea_, published in 1622, are very valuable.
+  It was long before another British ship entered the Pacific Ocean. Sir
+  John Narborough took two ships through the Strait of Magellan in 1670
+  and touched on the coast of Chile, but it was not until 1685 that
+  Dampier sailed over the part of the Pacific where Hawkins met his
+  defeat.
+
+  The exploring enterprise of the Spanish nation did not wane after the
+  conquest of Peru and Mexico, and the acquisition of the vast empire of
+  the Indies. It was spurred into renewed activity by the audacity of
+  Sir John Hawkins in the West Indies, and by the appearance of Drake,
+  Cavendish and Richard Hawkins in the Pacific.
+
+  In the interior of South America the Spanish conquerors had explored
+  the region of the Andes from the isthmus of Panama to Chile. Pedro de
+  Valdivia in 1540 made an expedition into the country of the Araucanian
+  Indians of Chile, and was the first to explore the eastern base of
+  the Andes in what is now Argentine Patagonia. In 1541 Francisco de
+  Orellana discovered the whole course of the Amazon from its source in
+  the Andes to the Atlantic. A second voyage on the Amazon was made in
+  1561 by the mad pirate Lope de Aguirre; but it was not until 1639 that
+  a full account was written of the great river by Father Cristoval de
+  Acuńa, who ascended it from its mouth and reached the city of Quito.
+
+
+    Spaniards in the Pacific.
+
+  The voyage of Drake across the Pacific was preceded by that of Alvaro
+  de Mendańa, who was despatched from Peru in 1567 to discover the great
+  Antarctic continent which was believed to extend far northward into
+  the South sea, the search for which now became one of the leading
+  motives of exploration. After a voyage of eighty days across the
+  Pacific, Mendańa discovered the Solomon Islands; and the expedition
+  returned in safety to Callao. The appearance of Drake on the Peruvian
+  coast led to an expedition being fitted out at Callao, to go in chase
+  of him, under the command of Pedro Sarmiento. He sailed from Callao in
+  October 1579, and made a careful survey of the Strait of Magellan,
+  with the object of fortifying that entrance to the South sea. The
+  colony which he afterwards took out from Spain was a complete failure,
+  and is only remembered now from the name of "Port Famine," which
+  Cavendish gave to the site at which he found the starving remnant of
+  Sarmiento's settlers. In June 1595 Mendańa sailed from the coast of
+  Peru in command of a second expedition to colonize the Solomon
+  Islands. After discovering the Marquesas, he reached the island of
+  Santa Cruz of evil memory, where he and many of the settlers died. His
+  young widow took command of the survivors and brought them safely to
+  Manila. The viceroys of Peru still persevered in their attempts to
+  plant a colony in the hypothetical southern continent. Pedro Fernandez
+  de Quiros, who was pilot under Mendańa and Luis Vaez de Torres, were
+  sent in command of two ships to continue the work of exploration. They
+  sailed from Callao in December 1605, and discovered several islands of
+  the New Hebrides group. They anchored in a bay of a large island which
+  Quiros named "Australia del Espiritu Santo." From this place Quiros
+  returned to America, but Torres continued the voyage, passed through
+  the strait between Australia and New Guinea which bears his name, and
+  explored and mapped the southern and eastern coasts of New Guinea.
+
+  The Portuguese, in the early part of the 17th century (1578-1640),
+  were under the dominion of Spain, and their enterprise was to some
+  extent damped; but their missionaries extended geographical knowledge
+  in Africa. Father Francisco Paez acquired great influence in
+  Abyssinia, and explored its highlands from 1600 to 1622. Fathers
+  Mendez and Lobo traversed the deserts between the coast of the Red sea
+  and the mountains, became acquainted with Lake Tsana, and discovered
+  the sources of the Blue Nile in 1624-1633.
+
+
+    Rivalry in the East.
+
+  But the attention of the Portuguese was mainly devoted to vain
+  attempts to maintain their monopoly of the trade of India against the
+  powerful rivalry of the English and Dutch. The English enterprises
+  were persevering, continuous and successful. James Lancaster made a
+  voyage to the Indian Ocean from 1591 to 1594; and in 1599 the
+  merchants and adventurers of London resolved to form a company, with
+  the object of establishing a trade with the East Indies. On the 31st
+  of December 1599 Queen Elizabeth granted the charter of incorporation
+  to the East India Company, and Sir James Lancaster, one of the
+  directors, was appointed general of their first fleet. He was
+  accompanied by John Davis, the great Arctic navigator, as pilot-major.
+  This voyage was eminently successful. The ships touched at Achin in
+  Sumatra and at Java, returning with full ladings of pepper in 1603.
+  The second voyage was commanded by Sir Henry Middleton; but it was in
+  the third voyage, under Keelinge and Hawkins, that the mainland of
+  India was first reached in 1607. Captain Hawkins landed at Surat and
+  travelled overland to Agra, passing some time at the court of the
+  Great Mogul. In the voyage of Sir Edward Michelborne in 1605, John
+  Davis lost his life in a fight with a Japanese junk. The eighth
+  voyage, led by Captain Saris, extended the operations of the company
+  to Japan; and in 1613 the Japanese government granted privileges to
+  the company; but the British retired in 1623, giving up their factory.
+  The chief result of this early intercourse between Great Britain and
+  Japan was the interesting series of letters written by William Adams
+  from 1611 to 1617. From the tenth voyage of the East India Company,
+  commanded by Captain Best, who left England in 1612, dates the
+  establishment of permanent British factories on the coast of India. It
+  was Captain Best who secured a regular _firman_ for trade from the
+  Great Mogul. From that time a fleet was despatched every year, and the
+  company's operations greatly increased geographical knowledge of India
+  and the Eastern Archipelago. British visits to Eastern countries, at
+  this time, were not confined to the voyages of the company. Journeys
+  were also made by land, and, among others, the entertaining author of
+  the _Crudities_, Thomas Coryate, of Odcombe in Somersetshire, wandered
+  on foot from France to India, and died (1617) in the company's factory
+  at Surat. In 1561 Anthony Jenkinson arrived in Persia with a letter
+  from Queen Elizabeth to the shah. He travelled through Russia to
+  Bokhara, and returned by the Caspian and Volga. In 1579 Christopher
+  Burroughs built a ship at Nizhniy Novgorod and traded across the
+  Caspian to Baku; and in 1598 Sir Anthony and Robert Shirley arrived in
+  Persia, and Robert was afterwards sent by the shah to Europe as his
+  ambassador. He was followed by a Spanish mission under Garcia de
+  Silva, who wrote an interesting account of his travels; and to Sir
+  Dormer Cotton's mission, in 1628, we are indebted for Sir Thomas
+  Herbert's charming narrative. In like manner Sir Thomas Roe's mission
+  to India resulted not only in a large collection of valuable reports
+  and letters of his own, but also in the detailed account of his
+  chaplain Terry. But the most learned and intelligent traveller in the
+  East, during the 17th century, was the German, Engelbrecht Kaempfer,
+  who accompanied an embassy to Persia, in 1684, and was afterwards a
+  surgeon in the service of the Dutch East India Company. He was in the
+  Persian Gulf, India and Java, and resided for more than two years in
+  Japan, of which he wrote a history.
+
+
+    Dutch exploration, 16th-17th centuries.
+
+  The Dutch nation, as soon as it was emancipated from Spanish tyranny,
+  displayed an amount of enterprise, which, for a long time, was fully
+  equal to that of the British. The Arctic voyages of Barents were
+  quickly followed by the establishment of a Dutch East India Company;
+  and the Dutch, ousting the Portuguese, not only established factories
+  on the mainland of India and in Japan, but acquired a preponderating
+  influence throughout the Malay Archipelago. In 1583 Jan Hugen van
+  Linschoten made a voyage to India with a Portuguese fleet, and his
+  full and graphic descriptions of India, Africa, China and the Malay
+  Archipelago must have been of no small use to his countrymen in their
+  distant voyages. The first of the Dutch Indian voyages was performed
+  by ships which sailed in April 1595, and rounded the Cape of Good
+  Hope. A second large Dutch fleet sailed in 1598; and, so eager was the
+  republic to extend her commerce over the world that another fleet,
+  consisting of five ships of Rotterdam, was sent in the same year by
+  way of Magellan's Strait, under Jacob Mahu as admiral, with William
+  Adams as pilot. Mahu died on the passage out, and was succeeded by
+  Simon de Cordes, who was killed on the coast of Chile. In September
+  1599 the fleet had entered the Pacific. The ships were then steered
+  direct for Japan, and anchored off Bungo in April 1600. In the same
+  year, 1598, a third expedition was despatched under Oliver van Noort,
+  a native of Utrecht, but the voyage contributed nothing to geography.
+  The Dutch Company in 1614 again resolved to send a fleet to the
+  Moluccas by the westward route, and Joris Spilbergen was appointed to
+  the command as admiral, with a commission from the States-General. He
+  was furnished with four ships of Amsterdam, two of Rotterdam and one
+  from Zeeland. On the 6th of May 1615 Spilbergen entered the Pacific
+  Ocean, and touched at several places on the coast of Chile and Peru,
+  defeating the Spanish fleet in a naval engagement off Chilca. After
+  plundering Payta and making requisitions at Acapulco, the Dutch fleet
+  crossed the Pacific and reached the Moluccas in March 1616.
+
+  The Dutch now resolved to discover a passage into the Pacific to the
+  south of Tierra del Fuego, the insular nature of which had been
+  ascertained by Sir Francis Drake. The vessels fitted out for this
+  purpose were the "Eendracht," of 360 tons, commanded by Jacob Lemaire,
+  and the "Hoorn," of 110 tons, under Willem Schouten. They sailed from
+  the Texel on the 14th of June 1615, and by the 20th of January 1616
+  they were south of the entrance of Magellan's Strait. Passing through
+  the strait of Lemaire they came to the southern extremity of Tierra
+  del Fuego, which was named Cape Horn, in honour of the town of Hoorn
+  in West Friesland, of which Schouten was a native. They passed the
+  cape on the 31st of January, encountering the usual westerly winds.
+  The great merit of this discovery of a second passage into the South
+  sea lies in the fact that it was not accidental or unforeseen, but was
+  due to the sagacity of those who designed the voyage. On the 1st of
+  March the Dutch fleet sighted the island of Juan Fernandez; and,
+  having crossed the Pacific, the explorers sailed along the north coast
+  of New Guinea and arrived at the Moluccas on the 17th of September
+  1616.
+
+  There were several early indications of the existence of the great
+  Australian continent, and the Dutch endeavoured to obtain further
+  knowledge concerning the country and its extent; but only its northern
+  and western coasts had been visited before the time of Governor van
+  Diemen. Dirk Hartog had been on the west coast in latitude 26° 30' S.
+  in 1616. Pelsert struck on a reef called "Houtman's Abrolhos" on the
+  4th of June 1629. In 1697 the Dutch captain Vlamingh landed on the
+  west coast of Australia, then called New Holland, in 31° 43' S., and
+  named the Swan river from the black swans he discovered there. In 1642
+  the governor and council of Batavia fitted out two ships to prosecute
+  the discovery of the south land, then believed to be part of a vast
+  Antarctic continent, and entrusted the command to Captain Abel Jansen
+  Tasman. This voyage proved to be the most important to geography that
+  had been undertaken since the first circumnavigation of the globe.
+  Tasman sailed from Batavia in 1642, and on the 24th of November
+  sighted high land in 42° 30' S., which was named van Diemen's Land,
+  and after landing there proceeded to the discovery of the western
+  coast of New Zealand; at first called Staten Land, and supposed to be
+  connected with the Antarctic continent from which this voyage proved
+  New Holland to be separated. He then reached Tongatabu, one of the
+  Friendly Islands of Cook; and returned by the north coast of New
+  Guinea to Batavia. In 1644 Tasman made a second voyage to effect a
+  fuller discovery of New Guinea.
+
+
+    French in North America.
+
+  The French directed their enterprise more in the direction of North
+  America than of the Indies. One of their most distinguished explorers
+  was Samuel Champlain, a captain in the navy, who, after a remarkable
+  journey through Mexico and the West Indies from 1599 to 1602,
+  established his historic connexion with Canada, to the geographical
+  knowledge of which he made a very large addition.
+
+
+    Missionaries in the East.
+
+  The principles and methods of surveying and position finding had by
+  this time become well advanced, and the most remarkable example of the
+  early application of these improvements is to be found in the survey
+  of China by Jesuit missionaries. They first prepared a map of the
+  country round Peking, which was submitted to the emperor Kang-hi, and,
+  being satisfied with the accuracy of the European method of surveying,
+  he resolved to have a survey made of the whole empire on the same
+  principles. This great work was begun in July 1708, and the completed
+  maps were presented to the emperor in 1718. The records preserved in
+  each city were examined, topographical information was diligently
+  collected, and the Jesuit fathers checked their triangulation by
+  meridian altitudes of the sun and pole star and by a system of
+  remeasurements. The result was a more accurate map of China than
+  existed, at that time, of any country in Europe. Kang-hi next ordered
+  a similar map to be made of Tibet, the survey being executed by two
+  lamas who were carefully trained as surveyors by the Jesuits at
+  Peking. From these surveys were constructed the well-known maps which
+  were forwarded to Duhalde, and which D'Anville utilized for his atlas.
+
+
+    The 18th century.
+
+    Asia.
+
+  Several European missionaries had previously found their way from
+  India to Tibet. Antonio Andrada, in 1624, was the first European to
+  enter Tibet since the visit of Friar Odoric in 1325. The next journey
+  was that of Fathers Grueber and Dorville about 1660, who succeeded in
+  passing from China, through Tibet, into India. In 1715 Fathers
+  Desideri and Freyre made their way from Agra, across the Himalayas, to
+  Lhasa, and the Capuchin Friar Orazio della Penna resided in that city
+  from 1735 until 1747. But the most remarkable journey in this
+  direction was performed by a Dutch traveller named Samuel van de
+  Putte. He left Holland in 1718, went by land through Persia to India,
+  and eventually made his way to Lhasa, where he resided for a long
+  time. He went thence to China, returned to Lhasa, and was in India in
+  time to be an eye-witness of the sack of Delhi by Nadir Shah in 1737.
+  In 1743 he left India and died at Batavia on the 27th of September
+  1745. The premature death of this illustrious traveller is the more to
+  be lamented because his vast knowledge died with him. Two English
+  missions sent by Warren Hastings to Tibet, one led by George Bogle in
+  1774, and the other by Captain Turner in 1783, complete Tibetan
+  exploration in the 18th century.
+
+  From Persia much new information was supplied by Jean Chardin, Jean
+  Tavernier, Charles Hamilton, Jean de Thévenot and Father Jude
+  Krusinski, and by English traders on the Caspian. In 1738 John Elton
+  traded between Astrakhan and the Persian port of Enzelī on the
+  Caspian, and undertook to build a fleet for Nadir Shah. Another
+  English merchant, named Jonas Hanway, arrived at Astrabad from Russia,
+  and travelled to the camp of Nadir at Kazvin. One lasting and valuable
+  result of Hanway's wanderings was a charming book of travels. In 1700
+  Guillaume Delisle published his map of the continents of the Old
+  World; and his successor D'Anville produced his map of India in 1752.
+  D'Anville's map contained all that was then known, but ten years
+  afterwards Major Rennell began his surveying labours, which extended
+  over the period from 1763 to 1782. His survey covered an area 900 m.
+  long by 300 wide, from the eastern confines of Bengal to Agra, and
+  from the Himalayas to Calpi. Rennell was indefatigable in collecting
+  geographical information; his Bengal atlas appeared in 1781, his
+  famous map of India in 1788 and the memoir in 1792. Surveys were also
+  made along the Indian coasts.
+
+  Arabia received very careful attention, in the 18th century, from the
+  Danish scientific mission, which included Carsten Niebuhr among its
+  members. Niebuhr landed at Loheia, on the coast of Yemen, in December
+  1762, and went by land to Sana. All the other members of the mission
+  died, but he proceeded from Mokha to Bombay. He then made a journey
+  through Persia and Syria to Constantinople, returning to Copenhagen in
+  1767. His valuable work, the _Description of Arabia_, was published in
+  1772, and was followed in 1774-1778 by two volumes of travels in Asia.
+  The great traveller survived until 1815, when he died at the age of
+  eighty-two.
+
+
+    Africa.
+
+  James Bruce of Kinnaird, the contemporary of Niebuhr, was equally
+  devoted to Eastern travel; and his principal geographical work was the
+  tracing of the Blue Nile from its source to its junction with the
+  White Nile. Before the death of Bruce an African Association was
+  formed, in 1788, for collecting information respecting the interior of
+  that continent, with Major Rennell and Sir Joseph Banks as leading
+  members. The association first employed John Ledyard (who had
+  previously made an extraordinary journey into Siberia) to cross Africa
+  from east to west on the parallel of the Niger, and William Lucas to
+  cross the Sahara to Fezzan. Lucas went from Tripoli to Mesurata,
+  obtained some information respecting Fezzan and returned in 1789. One
+  of the chief problems the association wished to solve was that of the
+  existence and course of the river Niger, which was believed by some
+  authorities to be identical with the Congo. Mungo Park, then an
+  assistant surgeon of an Indiaman, volunteered his services, which were
+  accepted by the association, and in 1795 he succeeded in reaching the
+  town of Segu on the Niger, but was prevented from continuing his
+  journey to Timbuktu. Five years later he accepted an offer from the
+  government to command an expedition into the interior of Africa, the
+  plan being to cross from the Gambia to the Niger and descend the
+  latter river to the sea. After losing most of his companions he
+  himself and the rest perished in a rapid on the Niger at Busa, having
+  been attacked from the shore by order of a chief who thought he had
+  not received suitable presents. His work, however, had established the
+  fact that the Niger was not identical with the Congo.
+
+  While the British were at work in the direction of the Niger, the
+  Portuguese were not unmindful of their old exploring fame. In 1798 Dr
+  F.J.M. de Lacerda, an accomplished astronomer, was appointed to
+  command a scientific expedition of discovery to the north of the
+  Zambesi. He started in July, crossed the Muchenja Mountains, and
+  reached the capital of the Cazembe, where he died of fever. Lacerda
+  left a valuable record of his adventurous journey; but with Mungo Park
+  and Lacerda the history of African exploration in the 18th century
+  closes.
+
+
+    South America.
+
+  In South America scientific exploration was active during this period.
+  The great geographical event of the century, as regards that
+  continent, was the measurement of an arc of the meridian. The
+  undertaking was proposed by the French Academy as part of an
+  investigation with the object of ascertaining the length of the degree
+  near the equator and near the pole respectively so as to determine the
+  figure of the earth. A commission left Paris in 1735, consisting of
+  Charles Marie de la Condamine, Pierre Bouguer, Louis Godin and Joseph
+  de Jussieu the naturalist. Spain appointed two accomplished naval
+  officers, the brothers Ulloa, as coadjutors. The operations were
+  carried on during eight years on a plain to the south of Quito; and,
+  in addition to his memoir on this memorable measurement, La Condamine
+  collected much valuable geographical information during a voyage down
+  the Amazon. The arc measured was 3° 7' 3" in length; and the work
+  consisted of two measured bases connected by a series of triangles,
+  one north and the other south of the equator, on the meridian of
+  Quito. Contemporaneously, in 1738, Pierre Louis Moreau de Maupertuis,
+  Alexis Claude Clairaut, Charles Etienne Louis Camus, Pierre Charles
+  Lemonnier and the Swedish physicist Celsius measured an arc of the
+  meridian in Lapland.
+
+
+    The Pacific Ocean.
+
+  The British and French governments despatched several expeditions of
+  discovery into the Pacific and round the world during the 18th
+  century. They were preceded by the wonderful and romantic voyages of
+  the buccaneers. The narratives of such men as Woodes Rogers, Edward
+  Davis, George Shelvocke, Clipperton and William Dampier, can never
+  fail to interest, while they are not without geographical value. The
+  works of Dampier are especially valuable, and the narratives of
+  William Funnell and Lionel Wafer furnished the best accounts then
+  extant of the Isthmus of Darien. Dampier's literary ability eventually
+  secured for him a commission in the king's service; and he was sent on
+  a voyage of discovery, during which he explored part of the coasts of
+  Australia and New Guinea, and discovered the strait which bears his
+  name between New Guinea and New Britain, returning in 1701. In 1721
+  Jacob Roggewein was despatched on a voyage of some importance across
+  the Pacific by the Dutch West India Company, during which he
+  discovered Easter Island on the 6th of April 1722.
+
+  The voyage of Lord Anson to the Pacific in 1740-1744 was of a
+  predatory character, and he lost more than half his men from scurvy;
+  while it is not pleasant to reflect that at the very time when the
+  French and Spaniards were measuring an arc of the meridian at Quito,
+  the British under Anson were pillaging along the coast of the Pacific
+  and burning the town of Payta. But a romantic interest attaches to the
+  wreck of the "Wager," one of Anson's fleet, on a desert island near
+  Chiloe, for it bore fruit in the charming narrative of Captain John
+  Byron, which will endure for all time. In 1764 Byron himself was sent
+  on a voyage of discovery round the world, which led immediately after
+  his return to the despatch of another to complete his work, under the
+  command of Captain Samuel Wallis.
+
+  The expedition, consisting of the "Dolphin" commanded by Wallis, and
+  the "Swallow" under Captain Philip Carteret, sailed in September 1766,
+  but the ships were separated on entering the Pacific from the Strait
+  of Magellan. Wallis discovered Tahiti on the 19th of June 1767, and he
+  gave a detailed account of that island. He returned to England in May
+  1768. Carteret discovered the Charlotte and Gloucester Islands, and
+  Pitcairn Island on the 2nd of July 1767; revisited the Santa Cruz
+  group, which was discovered by Mendańa and Quiros; and discovered the
+  strait separating New Britain from New Ireland. He reached Spithead
+  again in February 1769. Wallis and Carteret were followed very closely
+  by the French expedition of Bougainville, which sailed from Nantes in
+  November 1766. Bougainville had first to perform the unpleasant task
+  of delivering up the Falkland Islands, where he had encouraged the
+  formation of a French settlement, to the Spaniards. He then entered
+  the Pacific, and reached Tahiti in April 1768. Passing through the New
+  Hebrides group he touched at Batavia, and arrived at St Malo after an
+  absence of two years and four months.
+
+
+    Captain Cook.
+
+  The three voyages of Captain James Cook form an era in the history of
+  geographical discovery. In 1767 he sailed for Tahiti, with the object
+  of observing the transit of Venus, accompanied by two naturalists, Sir
+  Joseph Banks and Dr Solander, a pupil of Linnaeus, as well as by two
+  astronomers. The transit was observed on the 3rd of June 1769. After
+  exploring Tahiti and the Society group, Cook spent six months
+  surveying New Zealand, which he discovered to be an island, and the
+  coast of New South Wales from latitude 38° S. to the northern
+  extremity. The belief in a vast Antarctic continent stretching far
+  into the temperate zone had never been abandoned, and was vehemently
+  asserted by Charles Dalrymple, a disappointed candidate nominated by
+  the Royal Society for the command of the Transit expedition of 1769.
+  In 1772 the French explorer Yves Kerguelen de Tremarec had discovered
+  the land that bears his name in the South Indian Ocean without
+  recognizing it to be an island, and naturally believed it to be part
+  of the southern continent.
+
+  Cook's second voyage was mainly intended to settle the question of the
+  existence of such a continent once for all, and to define the limits
+  of any land that might exist in navigable seas towards the Antarctic
+  circle. James Cook at his first attempt reached a south latitude of
+  57° 15'. On a second cruise from the Society Islands, in 1773, he,
+  first of all men, crossed the Antarctic circle, and was stopped by ice
+  in 71° 10' S. During the second voyage Cook visited Easter Island,
+  discovered several islands of the New Hebrides and New Caledonia; and
+  on his way home by Cape Horn, in March 1774, he discovered the
+  Sandwich Island group and described South Georgia. He proved
+  conclusively that any southern continent that might exist lay under
+  the polar ice. The third voyage was intended to attempt the passage
+  from the Pacific to the Atlantic by the north-east. The "Resolution"
+  and "Discovery" sailed in 1776, and Cook again took the route by the
+  Cape of Good Hope. On reaching the North American coast, he proceeded
+  northward, fixed the position of the western extremity of America and
+  surveyed Bering Strait. He was stopped by the ice in 70° 41' N., and
+  named the farthest visible point on the American shore Icy Cape. He
+  then visited the Asiatic shore and discovered Cape North. Returning to
+  Hawaii, Cook was murdered by the natives. On the 14th of February
+  1779, his second, Captain Edward Clerke, took command, and proceeding
+  to Petropavlovsk in the following summer, he again examined the edge
+  of the ice, but only got as far as 70° 33' N. The ships returned to
+  England in October 1780.
+
+  In 1785 the French government carefully fitted out an expedition of
+  discovery at Brest, which was placed under the command of Franēois La
+  Pérouse, an accomplished and experienced officer. After touching at
+  Concepcion in Chile and at Easter Island, La Pérouse proceeded to
+  Hawaii and thence to the coast of California, of which he has given a
+  very interesting account. He then crossed the Pacific to Macao, and in
+  July 1787 he proceeded to explore the Gulf of Tartary and the shores
+  of Sakhalin, remaining some time at Castries Bay, so named after the
+  French minister of marine. Thence he went to the Kurile Islands and
+  Kamchatka, and sailed from the far north down the meridian to the
+  Navigator and Friendly Islands. He was in Botany Bay in January 1788;
+  and sailing thence, the explorer, his ship and crew were never seen
+  again. Their fate was long uncertain. In September 1791 Captain
+  Antoine d'Entrecasteaux sailed from Brest with two vessels to seek for
+  tidings. He visited the New Hebrides, Santa Cruz, New Caledonia and
+  Solomon Islands, and made careful though rough surveys of the
+  Louisiade Archipelago, islands north of New Britain and part of New
+  Guinea. D'Entrecasteaux died on board his ship on the 20th of July
+  1793, without ascertaining the fate of La Pérouse. Captain Peter
+  Dillon at length ascertained, in 1828, that the ships of La Pérouse
+  had been wrecked on the island of Vanikoro during a hurricane.
+
+  The work of Captain Cook bore fruit in many ways. His master, Captain
+  William Bligh, was sent in the "Bounty" to convey breadfruit plants
+  from Tahiti to the West Indies. He reached Tahiti in October 1788, and
+  in April 1789 a mutiny broke out, and he, with several officers and
+  men, was thrust into an open boat in mid-ocean. During the remarkable
+  voyage he then made to Timor, Bligh passed amongst the northern
+  islands of the New Hebrides, which he named the Banks Group, and made
+  several running surveys. He reached England in March 1790. The
+  "Pandora," under Captain Edwards, was sent out in search of the
+  "Bounty," and discovered the islands of Cherry and Mitre, east of the
+  Santa Cruz group, but she was eventually lost on a reef in Torres
+  Strait. In 1796-1797 Captain Wilson, in the missionary ship "Duff,"
+  discovered the Gambier and other islands, and rediscovered the islands
+  known to and seen by Quiros, but since called the Duff Group. Another
+  result of Captain Cook's work was the colonization of Australia. On
+  the 18th of January 1788 Admiral Phillip and Captain Hunter arrived in
+  Botany Bay in the "Supply" and "Sirius," followed by six transports,
+  and established a colony at Port Jackson. Surveys were then undertaken
+  in several directions. In 1795 and 1796 Matthew Flinders and George
+  Bass were engaged on exploring work in a small boat called the "Tom
+  Thumb." In 1797 Bass, who had been a surgeon, made an expedition
+  southwards, continued the work of Cook from Ram Head, and explored the
+  strait which bears his name, and in 1798 he and Flinders were
+  surveying on the east coast of Van Diemen's land.
+
+  Yet another outcome of Captain Cook's work was the voyage of George
+  Vancouver, who had served as a midshipman in Cook's second and third
+  voyages. The Spaniards under Quadra had begun a survey of
+  north-western America and occupied Nootka Sound, which their
+  government eventually agreed to surrender. Captain Vancouver was sent
+  out to receive the cession, and to survey the coast from Cape
+  Mendocino northwards. He commanded the old "Discovery," and was at
+  work during the seasons of 1792, 1793 and 1794, wintering at Hawaii.
+  Returning home in 1795, he completed his narrative and a valuable
+  series of charts.
+
+
+    Arctic regions.
+
+  The 18th century saw the Arctic coast of North America reached at two
+  points, as well as the first scientific attempt to reach the North
+  Pole. The Hudson Bay Company had been incorporated in 1670, and its
+  servants soon extended their operations over a wide area to the north
+  and west of Canada. In 1741 Captain Christopher Middleton was ordered
+  to solve the question of a passage from Hudson Bay to the westward.
+  Leaving Fort Churchill in July 1742, he discovered the Wager river and
+  Repulse Bay. He was followed by Captain W. Moor in 1746, and Captain
+  Coats in 1751, who examined the Wager Inlet up to the end. In November
+  1769 Samuel Hearne was sent by the Hudson Bay Company to discover the
+  sea on the north side of America, but was obliged to return. In
+  February 1770 he set out again from Fort Prince of Wales; but, after
+  great hardships, he was again forced to return to the fort. He started
+  once more in December 1771, and at length reached the Coppermine
+  river, which he surveyed to its mouth, but his observations are
+  unreliable. With the same object Alexander Mackenzie, with a party of
+  Canadians, set out from Fort Chippewyan on the 3rd of June 1789, and
+  descending the great river which now bears the explorer's name reached
+  the Arctic sea.
+
+  In February 1773 the Royal Society submitted a proposal to the king
+  for an expedition towards the North Pole. The expedition was fitted
+  out under Captains Constantine Phipps and Skeffington Lutwidge, and
+  the highest latitude reached was 80° 48' N., but no opening was
+  discovered in the heavy Polar pack. The most important Arctic work in
+  the 18th century was performed by the Russians, for they succeeded in
+  delineating the whole of the northern coast of Siberia. Some of this
+  work was possibly done at a still earlier date. The Cossack Simon
+  Dezhneff is thought to have made a voyage, in the summer of 1648, from
+  the river Kolyma, through Bering Strait (which was rediscovered by
+  Vitus Bering in 1728) to Anadyr. Between 1738 and 1750 Manin and
+  Sterlegoff made their way in small sloops from the mouth of the
+  Yenesei as far north as 75° 15' N. The land from Taimyr to Cape
+  Chelyuskin, the most northern extremity of Siberia, was mapped in many
+  years of patient exploration by Chelyuskin, who reached the extreme
+  point (77° 34' N.) in May 1742. To the east of Cape Chelyuskin the
+  Russians encountered greater difficulties. They built small vessels at
+  Yakutsk on the Lena, 900 m. from its mouth, whence the first
+  expedition was despatched under Lieut. Prontschichev in 1735. He
+  sailed from the mouth of the Lena to the mouth of the Olonek, where he
+  wintered, and on the 1st of September 1736 he got as far as 77° 29'
+  N., within 5 m. of Cape Chelyuskin. Both he and his young wife died of
+  scurvy, and the vessel returned. A second expedition, under Lieut.
+  Laptyev, started from the Lena in 1739, but encountered masses of
+  drift ice in Chatanga bay, and with this ended the voyages to the
+  westward of the Lena. Several attempts were also made to navigate the
+  sea from the Lena to the Kolyma. In 1736 Lieut. Laptyev sailed, but
+  was stopped by the drift ice in August, and in 1739, during another
+  trial, he reached the mouth of the Indigirka, where he wintered. In
+  the season of 1740 he continued his voyage to beyond the Kolyma,
+  wintering at Nizhni Kolymsk. In September 1740 Vitus Bering sailed
+  from Okhotsk on a second Arctic voyage with George William Steller on
+  board as naturalist. In June 1741 he named the magnificent peak on the
+  coast of North America Mount St Elias and explored the Aleutian
+  Islands. In November the ship was wrecked on Bering Island; and the
+  gallant Dane, worn out with scurvy, died there on the 8th of December
+  1741. In March 1770 a merchant named Liakhov saw a large herd of
+  reindeer coming from the north to the Siberian coast, which induced
+  him to start in a sledge in the direction whence they came. Thus he
+  reached the New Siberian or Liakhov Islands, and for years afterwards
+  the seekers for fossil ivory resorted to them. The Russian Captain
+  Vassili Chitschakov in 1765 and 1766 made two persevering attempts to
+  penetrate the ice north of Spitsbergen, and reached 80° 30' N., while
+  Russian parties twice wintered at Bell Sound.
+
+
+    Geographical societies.
+
+  In reviewing the progress of geographical discovery thus far, it has
+  been possible to keep fairly closely to a chronological order. But in
+  the 19th century and after exploring work was so generally and
+  steadily maintained in all directions, and was in so many cases
+  narrowed down from long journeys to detailed surveys within relatively
+  small areas, that it becomes desirable to cover the whole period at
+  one view for certain great divisions of the world. (See AFRICA; ASIA;
+  AUSTRALIA; POLAR REGIONS; &c.) Here, however, may be noticed the
+  development of geographical societies devoted to the encouragement of
+  exploration and research. The first of the existing geographical
+  societies was that of Paris, founded in 1825 under the title of La
+  Société de Géographie. The Berlin Geographical Society (Gesellschaft
+  für Erdkunde) is second in order of seniority, having been founded in
+  1827. The Royal Geographical Society, which was founded in London in
+  1830, comes third on the list; but it may be viewed as a direct result
+  of the earlier African Association founded in 1788. Sir John Barrow,
+  Sir John Cam Hobhouse (Lord Broughton), Sir Roderick Murchison, Mr
+  Robert Brown and Mr Bartle Frere formed the foundation committee of
+  the Royal Geographical Society, and the first president was Lord
+  Goderich. The action of the society in supplying practical instruction
+  to intending travellers, in astronomy, surveying and the various
+  branches of science useful to collectors, has had much to do with
+  advancement of discovery. Since the war of 1870 many geographical
+  societies have been established on the continent of Europe. At the
+  close of the 19th century there were upwards of 100 such societies in
+  the world, with more than 50,000 members, and over 150 journals were
+  devoted entirely to geographical subjects.[11] The great development
+  of photography has been a notable aid to explorers, not only by
+  placing at their disposal a faithful and ready means of recording the
+  features of a country and the types of inhabitants, but by supplying a
+  method of quick and accurate topographical surveying.
+
+
+  THE PRINCIPLES OF GEOGRAPHY
+
+  As regards the scope of geography, the order of the various
+  departments and their inter-relation, there is little difference of
+  opinion, and the principles of geography[12] are now generally
+  accepted by modern geographers. The order in which the various
+  subjects are treated in the following sketch is the natural succession
+  from fundamental to dependent facts, which corresponds also to the
+  evolution of the diversities of the earth's crust and of its
+  inhabitants.
+
+
+    Mathematical geography.
+
+  The fundamental geographical conceptions are mathematical, the
+  relations of space and form. The figure and dimensions of the earth
+  are the first of these. They are ascertained by a combination of
+  actual measurement of the highest precision on the surface and angular
+  observations of the positions of the heavenly bodies. The science of
+  geodesy is part of mathematical geography, of which the arts of
+  surveying and cartography are applications. The motions of the earth
+  as a planet must be taken into account, as they render possible the
+  determination of position and direction by observations of the
+  heavenly bodies. The diurnal rotation of the earth furnishes two fixed
+  points or poles, the axis joining which is fixed or nearly so in its
+  direction in space. The rotation of the earth thus fixes the
+  directions of north and south and defines those of east and west. The
+  angle which the earth's axis makes with the plane in which the planet
+  revolves round the sun determines the varying seasonal distribution of
+  solar radiation over the surface and the mathematical zones of
+  climate. Another important consequence of rotation is the deviation
+  produced in moving bodies relatively to the surface. In the form known
+  as Ferrell's Law this runs: "If a body moves in any direction on the
+  earth's surface, there is a deflecting force which arises from the
+  earth's rotation which tends to deflect it to the right in the
+  northern hemisphere but to the left in the southern hemisphere." The
+  deviation is of importance in the movement of air, of ocean currents,
+  and to some extent of rivers.[13]
+
+
+    Physical geography.
+
+  In popular usage the words "physical geography" have come to mean
+  geography viewed from a particular standpoint rather than any special
+  department of the subject. The popular meaning is better conveyed by
+  the word physiography, a term which appears to have been introduced by
+  Linnaeus, and was reinvented as a substitute for the cosmography of
+  the middle ages by Professor Huxley. Although the term has since been
+  limited by some writers to one particular part of the subject, it
+  seems best to maintain the original and literal meaning. In the
+  stricter sense, physical geography is that part of geography which
+  involves the processes of contemporary change in the crust and the
+  circulation of the fluid envelopes. It thus draws upon physics for the
+  explanation of the phenomena with the space-relations of which it is
+  specially concerned. Physical geography naturally falls into three
+  divisions, dealing respectively with the surface of the
+  lithosphere--geomorphology; the hydrosphere--oceanography; and the
+  atmosphere--climatology. All these rest upon the facts of mathematical
+  geography, and the three are so closely inter-related that they cannot
+  be rigidly separated in any discussion.
+
+
+    Geomorphology.
+
+  Geomorphology is the part of geography which deals with terrestrial
+  relief, including the submarine as well as the subaėrial portions of
+  the crust. The history of the origin of the various forms belongs to
+  geology, and can be completely studied only by geological methods. But
+  the relief of the crust is not a finished piece of sculpture; the
+  forms are for the most part transitional, owing their characteristic
+  outlines to the process by which they are produced; therefore the
+  geographer must, for strictly geographical purposes, take some account
+  of the processes which are now in action modifying the forms of the
+  crust. Opinion still differs as to the extent to which the
+  geographer's work should overlap that of the geologist.
+
+  The primary distinction of the forms of the crust is that between
+  elevations and depressions. Granting that the geoid or mean surface of
+  the ocean is a uniform spheroid, the distribution of land and water
+  approximately indicates a division of the surface of the globe into
+  two areas, one of elevation and one of depression. The increasing
+  number of measurements of the height of land in all continents and
+  islands, and the very detailed levellings in those countries which
+  have been thoroughly surveyed, enable the average elevation of the
+  land above sea-level to be fairly estimated, although many vast gaps
+  in accurate knowledge remain, and the estimate is not an exact one.
+  The only part of the sea-bed the configuration of which is at all well
+  known is the zone bordering the coasts where the depth is less than
+  about 100 fathoms or 200 metres, i.e. those parts which sailors speak
+  of as "in soundings." Actual or projected routes for telegraph cables
+  across the deep sea have also been sounded with extreme accuracy in
+  many cases; but beyond these lines of sounding the vast spaces of the
+  ocean remain unplumbed save for the rare researches of scientific
+  expeditions, such as those of the "Challenger," the "Valdivia," the
+  "Albatross" and the "Scotia." Thus the best approximation to the
+  average depth of the ocean is little more than an expert guess; yet a
+  fair approximation is probable for the features of sub-oceanic relief
+  are so much more uniform than those of the land that a smaller number
+  of fixed points is required to determine them.
+
+
+    Crustal relief.
+
+  The chief element of uncertainty as to the largest features of the
+  relief of the earth's crust is due to the unexplored area in the
+  Arctic region and the larger regions of the Antarctic, of which we
+  know nothing. We know that the earth's surface if unveiled of water
+  would exhibit a great region of elevation arranged with a certain
+  rough radiate symmetry round the north pole, and extending southwards
+  in three unequal arms which taper to points in the south. A depression
+  surrounds the little-known south polar region in a continuous ring and
+  extends northwards in three vast hollows lying between the arms of the
+  elevated area. So far only is it possible to speak with certainty, but
+  it is permissible to take a few steps into the twilight of dawning
+  knowledge and indicate the chief subdivisions which are likely to be
+  established in the great crust-hollow and the great crust-heap. The
+  boundary between these should obviously be the mean surface of the
+  sphere.
+
+  Sir John Murray deduced the mean height of the land of the globe as
+  about 2250 ft. above sea-level, and the mean depth of the oceans as
+  2080 fathoms or 12,480 ft. below sea-level.[14] Calculating the area
+  of the land at 55,000,000 sq. m. (or 28.6% of the surface), and that
+  of the oceans as 137,200,000 sq. m. (or 71.4% of the surface), he
+  found that the volume of the land above sea-level was 23,450,000 cub.
+  m., the volume of water below sea-level 323,800,000, and the total
+  volume of the water equal to about 1/666th of the volume of the whole
+  globe. From these data, as revised by A. Supan,[15] H.R. Mill
+  calculated the position of mean sphere-level at about 10,000 ft. or
+  1700 fathoms below sea-level. He showed that an imaginary spheroidal
+  shell, concentric with the earth and cutting the slope between the
+  elevated and depressed areas at the contour-line of 1700 fathoms,
+  would not only leave above it a volume of the crust equal to the
+  volume of the hollow left below it, but would also divide the surface
+  of the earth so that the area of the elevated region was equal to that
+  of the depressed region.[16]
+
+
+    Areas of the crust according to Murray.
+
+  A similar observation was made almost simultaneously by Romieux,[17]
+  who further speculated on the equilibrium between the weight of the
+  elevated land mass and that of the total waters of the ocean, and
+  deduced some interesting relations between them. Murray, as the result
+  of his study, divided the earth's surface into three zones--the
+  _continental area_ containing all dry land, the _transitional area_
+  including the submarine slopes down to 1000 fathoms, and the _abysmal
+  area_ consisting of the floor of the ocean beyond that depth; and Mill
+  proposed to take the line of mean-sphere level, instead of the
+  empirical depth of 1000 fathoms, as the boundary between the
+  transitional and abysmal areas.
+
+  An elaborate criticism of all the existing data regarding the volume
+  relations of the vertical relief of the globe was made in 1894 by
+  Professor Hermann Wagner, whose recalculations of volumes and mean
+  heights--the best results which have yet been obtained--led to the
+  following conclusions.[18]
+
+
+    Areas of the crust according to Wagner.
+
+  The area of the dry land was taken as 28.3% of the surface of the
+  globe, and that of the oceans as 71.7%. The mean height deduced for
+  the land was 2300 ft. above sea-level, the mean depth of the sea
+  11,500 ft. below, while the position of mean-sphere level comes out as
+  7500 ft. (1250 fathoms) below sea-level. From this it would appear
+  that 43% of the earth's surface was above and 57% below the mean
+  level. It must be noted, however, that since 1895 the soundings of
+  Nansen in the north polar area, of the "Valdivia," "Belgica," "Gauss"
+  and "Scotia" in the Southern Ocean, and of various surveying ships in
+  the North and South Pacific, have proved that the mean depth of the
+  ocean is considerably greater than had been supposed, and mean-sphere
+  level must therefore lie deeper than the calculations of 1895 show;
+  possibly not far from the position deduced from the freer estimate of
+  1888. The whole of the available data were utilized by the prince of
+  Monaco in 1905 in the preparation of a complete bathymetrical map of
+  the oceans on a uniform scale, which must long remain the standard
+  work for reference on ocean depths.
+
+  By the device of a hypsographic curve co-ordinating the vertical
+  relief and the areas of the earth's surface occupied by each zone of
+  elevation, according to the system introduced by Supan,[19] Wagner
+  showed his results graphically.
+
+  This curve with the values reduced from metres to feet is reproduced
+  below.
+
+  Wagner subdivides the earth's surface, according to elevation, into
+  the following five regions:
+
+
+    _Wagner's Divisions of the Earth's Crust:_
+
+    +---------------------+-----------+-------------+-------------+
+    |        Name.        |Per cent of|    From     |     To      |
+    |                     |  Surface. |             |             |
+    +---------------------+-----------+-------------+-------------+
+    | Depressed area      |     3     |   Deepest.  |-16,400 feet.|
+    | Oceanic plateau     |    54     |-16,400 feet.|- 7,400   "  |
+    | Continental slope   |     9     |- 7,400   "  |-   660   "  |
+    | Continental plateau |    28     |-   660   "  |+ 3,000   "  |
+    | Culminating area    |     6     |+ 3,300   "  |   Highest.  |
+    +---------------------+-----------+-------------+-------------+
+
+  [Illustration]
+
+  The continental plateau might for purposes of detailed study be
+  divided into the _continental shelf_ from -660 ft. to sea-level, and
+  _lowlands_ from sea-level to +660 ft. (corresponding to the mean level
+  of the whole globe).[20] _Uplands_ reaching from 660 ft. to 2300 (the
+  approximate mean level of the land), and _highlands_, from 2300
+  upwards, might also be distinguished.
+
+
+    Arrangement of world-ridges and hollows.
+
+  A striking fact in the configuration of the crust is that each
+  continent, or elevated mass of the crust, is diametrically opposite to
+  an ocean basin or great depression; the only partial exception being
+  in the case of southern South America, which is antipodal to eastern
+  Asia. Professor C. Lapworth has generalized the grand features of
+  crustal relief in a scheme of attractive simplicity. He sees
+  throughout all the chaos of irregular crust-forms the recurrence of a
+  certain harmony, a succession of folds or waves which build up all the
+  minor features.[21] One great series of crust waves from east to west
+  is crossed by a second great series of crust waves from north to
+  south, giving rise by their interference to six great elevated masses
+  (the continents), arranged in three groups, each consisting of a
+  northern and a southern member separated by a minor depression. These
+  elevated masses are divided from one another by similar great
+  depressions.
+
+
+    Lapworth's fold-theory.
+
+  He says: "The surface of each of our great continental masses of land
+  resembles that of a long and broad arch-like form, of which we see the
+  simplest type in the New World. The surface of the North American arch
+  is sagged downwards in the middle into a central depression which lies
+  between two long marginal plateaus, and these plateaus are finally
+  crowned by the wrinkled crests which form its two modern mountain
+  systems. The surface of each of our ocean floors exactly resembles
+  that of a continent turned upside down. Taking the Atlantic as our
+  simplest type, we may say that the surface of an ocean basin resembles
+  that of a mighty trough or syncline, buckled up more or less centrally
+  in a medial ridge, which is bounded by two long and deep marginal
+  hollows, in the cores of which still deeper grooves sink to the
+  profoundest depths. This complementary relationship descends even to
+  the minor features of the two. Where the great continental sag sinks
+  below the ocean level, we have our gulfs and our Mediterraneans, seen
+  in our type continent, as the Mexican Gulf and Hudson Bay. Where the
+  central oceanic buckle attains the water-line we have our oceanic
+  islands, seen in our type ocean, as St Helena and the Azores. Although
+  the apparent crust-waves are neither equal in size nor symmetrical in
+  form, this complementary relationship between them is always
+  discernible. The broad Pacific depression seems to answer to the broad
+  elevation of the Old World--the narrow trough of the Atlantic to the
+  narrow continent of America."
+
+
+    Suess's theory.
+
+  The most thorough discussion of the great features of terrestrial
+  relief in the light of their origin is that by Professor E. Suess,[22]
+  who points out that the plan of the earth is the result of two
+  movements of the crust--one, subsidence over wide areas, giving rise
+  to oceanic depressions and leaving the continents protuberant; the
+  other, folding along comparatively narrow belts, giving rise to
+  mountain ranges. This theory of crust blocks dropped by subsidence is
+  opposed to Lapworth's theory of vast crust-folds, but geology is the
+  science which has to decide between them.
+
+  Geomorphology is concerned, however, in the suggestions which have
+  been made as to the cause of the distribution of heap and hollow in
+  the larger features of the crust. Élie de Beaumont, in his
+  speculations on the relation between the direction of mountain ranges
+  and their geological age and character, was feeling towards a
+  comprehensive theory of the forms of crustal relief; but his ideas
+  were too geometrical, and his theory that the earth is a spheroid
+  built up on a rhombic dodecahedron, the pentagonal faces of which
+  determined the direction of mountain ranges, could not be proved.[23]
+  The "tetrahedral theory" brought forward by Lowthian Green,[24] that
+  the form of the earth is a spheroid based on a regular tetrahedron, is
+  more serviceable, because it accounts for three very interesting facts
+  of the terrestrial plan--(1) the antipodal position of continents and
+  ocean basins; (2) the triangular outline of the continents; and (3)
+  the excess of sea in the southern hemisphere. Recent investigations
+  have recalled attention to the work of Lowthian Green, but the
+  question is still in the controversial stage.[25] The study of tidal
+  strain in the earth's crust by Sir George Darwin has led that
+  physicist to indicate the possibility of the triangular form and
+  southerly direction of the continents being a result of the
+  differential or tidal attraction of the sun and moon. More recently
+  Professor A.E.H. Love has shown that the great features of the relief
+  of the lithosphere may be expressed by spherical harmonics of the
+  first, second and third degrees, and their formation related to
+  gravitational action in a sphere of unequal density.[26]
+
+  In any case it is fully recognized that the plan of the earth is so
+  clear as to leave no doubt as to its being due to some general cause
+  which should be capable of detection.
+
+
+    The continents.
+
+  If the level of the sea were to become coincident with the mean level
+  of the lithosphere, there would result one tri-radiate land-mass of
+  nearly uniform outline and one continuous sheet of water broken by
+  few islands. The actual position of sea-level lies so near the summit
+  of the crust-heap that the varied relief of the upper portion leads to
+  the formation of a complicated coast-line and a great number of
+  detached portions of land. The hydrosphere is, in fact, continuous,
+  and the land is all in insular masses: the largest is the Old World of
+  Europe, Asia and Africa; the next in size, America; the third,
+  possibly, Antarctica; the fourth, Australia; the fifth, Greenland.
+  After this there is a considerable gap before New Guinea, Borneo,
+  Madagascar, Sumatra and the vast multitude of smaller islands
+  descending in size by regular gradations to mere rocks. The contrast
+  between island and mainland was natural enough in the days before the
+  discovery of Australia, and the mainland of the Old World was
+  traditionally divided into three continents. These "continents,"
+  "parts of the earth," or "quarters of the globe," proved to be
+  convenient divisions; America was added as a fourth, and subsequently
+  divided into two, while Australia on its discovery was classed
+  sometimes as a new continent, sometimes merely as an island, sometimes
+  compromisingly as an island-continent, according to individual
+  opinion. The discovery of the insularity of Greenland might again give
+  rise to the argument as to the distinction between island and
+  continent. Although the name of continent was not applied to large
+  portions of land for any physical reasons, it so happens that there is
+  a certain physical similarity or homology between them which is not
+  shared by the smaller islands or peninsulas.
+
+
+    Homology of continents.
+
+  The typical continental form is triangular as regards its sea-level
+  outline. The relief of the surface typically includes a central plain,
+  sometimes dipping below sea-level, bounded by lateral highlands or
+  mountain ranges, loftier on one side than on the other, the higher
+  enclosing a plateau shut in by mountains. South America and North
+  America follow this type most closely; Eurasia (the land mass of
+  Europe and Asia) comes next, while Africa and Australia are farther
+  removed from the type, and the structure of Antarctica and Greenland
+  is unknown.
+
+  If the continuous, unbroken, horizontal extent of land in a continent
+  is termed its _trunk_,[27] and the portions cut up by inlets or
+  channels of the sea into islands and peninsulas the _limbs_, it is
+  possible to compare the continents in an instructive manner.
+
+  The following table is from the statistics of Professor H. Wagner,[28]
+  his metric measurements being transposed into British units:
+
+
+    _Comparison of the Continents._
+
+    +---------------+-------+-------+-------+------+--------+------+------+
+    |               |       |       |       | Area |        |      |      |
+    |               | Area  | Mean  |  Area |penin-|  Area  | Area | Area |
+    |               | total |height,| trunk,|sulas,|islands,|limbs,|limbs,|
+    |               |  mil. | feet. |  mil. | mil. |  mil.  | mil. | per  |
+    |               | sq. m.|       | sq. m.|sq. m.| sq. m. |sq. m.| cent.|
+    +---------------+-------+-------+-------+------+--------+------+------+
+    | Old World     | 35.8  | 2360  |       |      |        |      |      |
+    | New World     | 16.2  | 2230  |       |      |        |      |      |
+    | Eurasia       | 20.85 | 2620  | 15.42 | 4.09 |  1.34  | 5.43 | 26   |
+    | Africa        | 11.46 | 2130  | 11.22 |  ..  |  0.24  | 0.24 |  2.1 |
+    | North America |  9.26 | 2300  |  6.92 | 0.78 |  1.56  | 2.34 | 25   |
+    | South America |  6.84 | 1970  |  6.76 | 0.02 |  0.06  | 0.08 |  1.1 |
+    | Australia     |  3.43 | 1310  |  2.77 | 0.16 |  0.50  | 0.66 | 19   |
+    | Asia          | 17.02 | 3120  | 12.93 | 3.05 |  1.04  | 4.09 | 24   |
+    | Europe        |  3.83 |  980  |  2.49 | 1.04 |  0.30  | 1.34 | 35   |
+    +---------------+-------+-------+-------+------+--------+------+------+
+
+
+    Islands.
+
+  The usual classification of islands is into continental and oceanic.
+  The former class includes all those which rise from the continental
+  shelf, or show evidence in the character of their rocks of having at
+  one time been continuous with a neighbouring continent. The latter
+  rise abruptly from the oceanic abysses. Oceanic islands are divided
+  according to their geological character into volcanic islands and
+  those of organic origin, including coral islands. More elaborate
+  subdivisions according to structure, origin and position have been
+  proposed.[29] In some cases a piece of land is only an island at high
+  water, and by imperceptible gradation the form passes into a
+  peninsula. The typical peninsula is connected with the mainland by a
+  relatively narrow isthmus; the name is, however, extended to any limb
+  projecting from the trunk of the mainland, even when, as in the Indian
+  peninsula, it is connected by its widest part.
+
+
+    Coasts.
+
+  Small peninsulas are known as promontories or headlands, and the
+  extremity as a cape. The opposite form, an inlet of the sea, is known
+  when wide as a gulf, bay or bight, according to size and degree of
+  inflection, or as a fjord or ria when long and narrow. It is
+  convenient to employ a specific name for a projection of a coast-line
+  less pronounced than a peninsula, and for an inlet less pronounced
+  than a bay or bight; outcurve and incurve may serve the turn. The
+  varieties of coast-lines were reduced to an exact classification by
+  Richthofen, who grouped them according to the height and slope of the
+  land into cliff-coasts (_Steilküsten_)--narrow beach coasts with
+  cliffs, wide beach coasts with cliffs, and low coasts, subdividing
+  each group according as the coast-line runs parallel to or crosses the
+  line of strike of the mountains, or is not related to mountain
+  structure. A further subdivision depends on the character of the
+  inter-relation of land and sea along the shore producing such types as
+  a fjord-coast, ria-coast or lagoon-coast. This extremely elaborate
+  subdivision may be reduced, as Wagner points out, to three types--the
+  continental coast where the sea comes up to the solid rock-material of
+  the land; the marine coast, which is formed entirely of soft material
+  sorted out by the sea; and the composite coast, in which both forms
+  are combined.
+
+
+    Coast-lines.
+
+  On large-scale maps it is necessary to show two coast-lines, one for
+  the highest, the other for the lowest tide; but in small-scale maps a
+  single line is usually wider than is required to represent the whole
+  breadth of the inter-tidal zone. The measurement of a coast-line is
+  difficult, because the length will necessarily be greater when
+  measured on a large-scale map where minute irregularities can be taken
+  into account. It is usual to distinguish between the general
+  coast-line measured from point to point of the headlands disregarding
+  the smaller bays, and the detailed coast-line which takes account of
+  every inflection shown by the map employed, and follows up river
+  entrances to the point where tidal action ceases. The ratio between
+  these two coast-lines represents the "coastal development" of any
+  region.
+
+
+    Submarine forms.
+
+  While the forms of the sea-bed are not yet sufficiently well known to
+  admit of exact classification, they are recognized to be as a rule
+  distinct from the forms of the land, and the importance of using a
+  distinctive terminology is felt. Efforts have been made to arrive at a
+  definite international agreement on this subject, and certain terms
+  suggested by a committee were adopted by the Eighth International
+  Geographical Congress at New York in 1904.[30] The forms of the ocean
+  floor include the "shelf," or shallow sea margin, the "depression," a
+  general term applied to all submarine hollows, and the "elevation." A
+  depression when of great extent is termed a "basin," when it is of a
+  more or less round form with approximately equal diameters, a "trough"
+  when it is wide and elongated with gently sloping borders, and a
+  "trench" when narrow and elongated with steeply sloping borders, one
+  of which rises higher than the other. The extension of a trough or
+  basin penetrating the land or an elevation is termed an "embayment"
+  when wide, and a "gully" when long and narrow; and the deepest part of
+  a depression is termed a "deep." A depression of small extent when
+  steep-sided is termed a "caldron," and a long narrow depression
+  crossing a part of the continental border is termed a "furrow." An
+  elevation of great extent which rises at a very gentle angle from a
+  surrounding depression is termed a "rise," one which is relatively
+  narrow and steep-sided a "ridge," and one which is approximately equal
+  in length and breadth but steep-sided a "plateau," whether it springs
+  direct from a depression or from a rise. An elevation of small extent
+  is distinguished as a "dome" when it is more than 100 fathoms from the
+  surface, a "bank" when it is nearer the surface than 100 fathoms but
+  deeper than 6 fathoms, and a "shoal" when it comes within 6 fathoms of
+  the surface and so becomes a serious danger to shipping. The highest
+  point of an elevation is termed a "height," if it does not form an
+  island or one of the minor forms.
+
+
+    Land forms.
+
+  The forms of the dry land are of infinite variety, and have been
+  studied in great detail.[31] From the descriptive or topographical
+  point of view, geometrical form alone should be considered; but the
+  origin and geological structure of land forms must in many cases be
+  taken into account when dealing with the function they exercise in the
+  control of mobile distributions. The geographers who have hitherto
+  given most attention to the forms of the land have been trained as
+  geologists, and consequently there is a general tendency to make
+  origin or structure the basis of classification rather than form
+  alone.
+
+
+    The six elementary land forms.
+
+  The fundamental form-elements may be reduced to the six proposed by
+  Professor Penck as the basis of his double system of classification by
+  form and origin.[32] These may be looked upon as being all derived by
+  various modifications or arrangements of the single form-unit, the
+  _slope_ or inclined plane surface. No one form occurs alone, but
+  always grouped together with others in various ways to make up
+  districts, regions and lands of distinctive characters. The
+  form-elements are:
+
+  1. The _plain_ or gently inclined uniform surface.
+
+  2. The _scarp_ or steeply inclined slope; this is necessarily of small
+  extent except in the direction of its length.
+
+  3. The _valley_, composed of two lateral parallel slopes inclined
+  towards a narrow strip of plain at a lower level which itself slopes
+  downwards in the direction of its length. Many varieties of this
+  fundamental form may be distinguished.
+
+  4. The _mount_, composed of a surface falling away on every side from
+  a particular place. This place may either be a point, as in a volcanic
+  cone, or a line, as in a mountain range or ridge of hills.
+
+  5. The _hollow_ or form produced by a land surface sloping inwards
+  from all sides to a particular lowest place, the converse of a mount.
+
+  6. The _cavern_ or space entirely surrounded by a land surface.
+
+
+    Geology and land forms.
+
+  These forms never occur scattered haphazard over a region, but always
+  in an orderly subordination depending on their mode of origin. The
+  dominant forms result from crustal movements, the subsidiary from
+  secondary reactions during the action of the primitive forms on mobile
+  distributions. The geological structure and the mineral composition of
+  the rocks are often the chief causes determining the character of the
+  land forms of a region. Thus the scenery of a limestone country
+  depends on the solubility and permeability of the rocks, leading to
+  the typical Karst-formations of caverns, swallow-holes and underground
+  stream courses, with the contingent phenomena of dry valleys and
+  natural bridges. A sandy beach or desert owes its character to the
+  mobility of its constituent sand-grains, which are readily drifted and
+  piled up in the form of dunes. A region where volcanic activity has
+  led to the embedding of dykes or bosses of hard rock amongst softer
+  strata produces a plain broken by abrupt and isolated eminences.[33]
+
+
+    Classification of mountains.
+
+  It would be impracticable to go fully into the varieties of each
+  specific form; but, partly as an example of modern geographical
+  classification, partly because of the exceptional importance of
+  mountains amongst the features of the land, one exception may be made.
+  The classification of mountains into types has usually had regard
+  rather to geological structure than to external form, so that some
+  geologists would even apply the name of a mountain range to a region
+  not distinguished by relief from the rest of the country if it bear
+  geological evidence of having once been a true range. A mountain may
+  be described (it cannot be defined) as an elevated region of irregular
+  surface rising comparatively abruptly from lower ground. The actual
+  elevation of a summit above sea-level does not necessarily affect its
+  mountainous character; a gentle eminence, for instance, rising a few
+  hundred feet above a tableland, even if at an elevation of say 15,000
+  ft., could only be called a hill.[34] But it may be said that any
+  abrupt slope of 2000 ft. or more in vertical height may justly be
+  called a mountain, while abrupt slopes of lesser height may be called
+  hills. Existing classifications, however, do not take account of any
+  difference in kind between mountain and hills, although it is common
+  in the German language to speak of _Hügelland_, _Mittelgebirge_ and
+  _Hochgebirge_ with a definite significance.
+
+  The simple classification employed by Professor James Geikie[35] into
+  mountains of accumulation, mountains of elevation and mountains of
+  circumdenudation, is not considered sufficiently thorough by German
+  geographers, who, following Richthofen, generally adopt a
+  classification dependent on six primary divisions, each of which is
+  subdivided. The terms employed, especially for the subdivisions,
+  cannot be easily translated into other languages, and the English
+  equivalents in the following table are only put forward tentatively:--
+
+    RICHTHOFEN'S CLASSIFICATION OF MOUNTAINS[36]
+
+      I. _Tektonische Gebirge_--Tectonic mountains.
+        (a) _Bruchgebirge oder Schollengebirge_--Block mountains.
+            1. _Einseitige Schollengebirge oder Schollenrandgebirge_--
+                  Scarp or tilted block mountains.
+                    (i.) _Tafelscholle_--Table blocks.
+                   (ii.) _Abrasionsscholle_--Abraded blocks.
+                  (iii.) _Transgressionsscholle_--Blocks of unconformable
+                              strata.
+            2. _Flexurgebirge_--Flexure mountains.
+            3. _Horstgebirge_--Symmetrical block mountains.
+        (b) _Faltungsgebirge_--Fold mountains.
+            1. _Homöomorphe Faltungsgebirge_--Homomorphic fold mountains.
+            2. _Heteromorphe Faltungsgebirge_--Heteromorphic fold
+                    mountains.
+
+     II. _Rumpfgebirge oder Abrasionsgebirge_--Trunk or abraded mountains.
+
+    III. _Ausbruchsgebirge_--Eruptive mountains.
+
+     IV. _Aufschüttungsgebirge_--Mountains of accumulation.
+
+      V. _Flachböden_--Plateaux.
+        (a) _Abrasionsplatten_--Abraded plateaux.
+        (b) _Marines Flachland_--Plain of marine erosion.
+        (c) _Schichtungstafelland_--Horizontally stratified tableland.
+        (d) _Übergusstafelland_--Lava plain.
+        (e) _Stromflachland_--River plain.
+        (f) _Flachböden der atmosphärischen Aufschüttung_--Plains of
+                   aeolian formation.
+
+     VI. _Erosionsgebirge_--Mountains of erosion.
+
+
+    Mountain forms.
+
+  From the morphological point of view it is more important to
+  distinguish the associations of forms, such as the _mountain mass_ or
+  group of mountains radiating from a centre, with the valleys furrowing
+  their flanks spreading towards every direction; the _mountain chain_
+  or line of heights, forming a long narrow ridge or series of ridges
+  separated by parallel valleys; the _dissected plateau_ or highland,
+  divided into mountains of circumdenudation by a system of deeply-cut
+  valleys; and the _isolated peak_, usually a volcanic cone or a hard
+  rock mass left projecting after the softer strata which embedded it
+  have been worn away (Monadnock of Professor Davis).
+
+
+    Distribution of mountains.
+
+  The geographical distribution of mountains is intimately associated
+  with the great structural lines of the continents of which they form
+  the culminating region. Lofty lines of fold mountains form the
+  "backbones" of North America in the Rocky Mountains and the west coast
+  systems, of South America in the Cordillera of the Andes, of Europe in
+  the Pyrenees, Alps, Carpathians and Caucasus, and of Asia in the
+  mountains of Asia Minor, converging on the Pamirs and diverging thence
+  in the Himalaya and the vast mountain systems of central and eastern
+  Asia. The remarkable line of volcanoes around the whole coast of the
+  Pacific and along the margin of the Caribbean and Mediterranean seas
+  is one of the most conspicuous features of the globe.
+
+
+    Functions of land forms.
+
+    Land waste.
+
+    Glaciers.
+
+  If land forms may be compared to organs, the part they serve in the
+  economy of the earth may, without straining the term, be characterized
+  as functions. The first and simplest function of the land surface is
+  that of guiding loose material to a lower level. The downward pull of
+  gravity suffices to bring about the fall of such material, but the
+  path it will follow and the distance it will travel before coming to
+  rest depend upon the land form. The loose material may, and in an arid
+  region does, consist only of portions of the higher parts of the
+  surface detached by the expansion and contraction produced by heating
+  and cooling due to radiation. Such broken material rolling down a
+  uniform scarp would tend to reduce its steepness by the loss of
+  material in the upper part and by the accumulation of a mound or scree
+  against the lower part of the slope. But where the side is not a
+  uniform scarp, but made up of a series of ridges and valleys, the
+  tendency will be to distribute the detritus in an irregular manner,
+  directing it away from one place and collecting it in great masses in
+  another, so that in time the land form assumes a new appearance. Snow
+  accumulating on the higher portions of the land, when compacted into
+  ice and caused to flow downwards by gravity, gives rise, on account of
+  its more coherent character, to continuous glaciers, which mould
+  themselves to the slopes down which they are guided, different
+  ice-streams converging to send forward a greater volume. Gradually
+  coming to occupy definite beds, which are deepened and polished by the
+  friction, they impress a characteristic appearance on the land, which
+  guides them as they traverse it, and, although the ice melts at lower
+  levels, vast quantities of clay and broken stones are brought down and
+  deposited in terminal moraines where the glacier ends.
+
+
+    Rain.
+
+    River systems.
+
+    Adjustment of rivers to land.
+
+  Rain is by far the most important of the inorganic mobile
+  distributions upon which land forms exercise their function of
+  guidance and control. The precipitation of rain from the aqueous
+  vapour of the atmosphere is caused in part by vertical movements of
+  the atmosphere involving heat changes and apparently independent of
+  the surface upon which precipitation occurs; but in greater part it is
+  dictated by the form and altitude of the land surface and the
+  direction of the prevailing winds, which itself is largely influenced
+  by the land. It is on the windward faces of the highest ground, or
+  just beyond the summit of less dominant heights upon the leeward side,
+  that most rain falls, and all that does not evaporate or percolate
+  into the ground is conducted back to the sea by a route which depends
+  only on the form of the land. More mobile and more searching than ice
+  or rock rubbish, the trickling drops are guided by the deepest lines
+  of the hillside in their incipient flow, and as these lines converge,
+  the stream, gaining strength, proceeds in its torrential course to
+  carve its channel deeper and entrench itself in permanent occupation.
+  Thus the stream-bed, from which at first the water might be blown away
+  into a new channel by a gale of wind, ultimately grows to be the
+  strongest line of the landscape. As the main valley deepens, the
+  tributary stream-beds are deepened also, and gradually cut their way
+  headwards, enlarging the area whence they draw their supplies. Thus
+  new land forms are created--valleys of curious complexity, for
+  example--by the "capture" and diversion of the water of one river by
+  another, leading to a change of watershed.[37] The minor tributaries
+  become more numerous and more constant, until the system of torrents
+  has impressed its own individuality on the mountain side. As the river
+  leaves the mountain, ever growing by the accession of tributaries, it
+  ceases, save in flood time, to be a formidable instrument of
+  destruction; the gentler slope of the land surface gives to it only
+  power sufficient to transport small stones, gravel, sand and
+  ultimately mud. Its valley banks are cut back by the erosion of minor
+  tributaries, or by rain-wash if the climate be moist, or left steep
+  and sharp while the river deepens its bed if the climate be arid. The
+  outline of the curve of a valley's sides ultimately depends on the
+  angle of repose of the detritus which covers them, if there has been
+  no subsequent change, such as the passage of a glacier along the
+  valley, which tends to destroy the regularity of the cross-section.
+  The slope of the river bed diminishes until the plain compels the
+  river to move slowly, swinging in _meanders_ proportioned to its size,
+  and gradually, controlled by the flattening land, ceasing to transport
+  material, but raising its banks and silting up its bed by the dropped
+  sediment, until, split up and shoaled, its distributaries struggle
+  across its delta to the sea. This is the typical river of which there
+  are infinite varieties, yet every variety would, if time were given,
+  and the land remained unchanged in level relatively to the sea,
+  ultimately approach to the type. Movements of the land either of
+  subsidence or elevation, changes in the land by the action of erosion
+  in cutting back an escarpment or cutting through a col, changes in
+  climate by affecting the rainfall and the volume of water, all tend to
+  throw the river valley out of harmony with the actual condition of its
+  stream. There is nothing more striking in geography than the
+  perfection of the adjustment of a great river system to its valleys
+  when the land has remained stable for a very lengthened period. Before
+  full adjustment has been attained the river bed may be broken in
+  places by waterfalls or interrupted by lakes; after adjustment the bed
+  assumes a permanent outline, the slope diminishing more and more
+  gradually, without a break in its symmetrical descent. Excellent
+  examples of the indecisive drainage of a new land surface, on which
+  the river system has not had time to impress itself, are to be seen in
+  northern Canada and in Finland, where rivers are separated by scarcely
+  perceptible divides, and the numerous lakes frequently belong to more
+  than one river system.
+
+
+    The geographical cycle.
+
+  The action of rivers on the land is so important that it has been made
+  the basis of a system of physical geography by Professor W.M. Davis,
+  who classifies land surfaces in terms of the three factors--structure,
+  process and time.[38] Of these time, during which the process is
+  acting on the structure, is the most important. A land may thus be
+  characterized by its position in the "geographical cycle", or cycle of
+  erosion, as young, mature or old, the last term being reached when the
+  base-level of erosion is attained, and the land, however varied its
+  relief may have been in youth or maturity, is reduced to a nearly
+  uniform surface or peneplain. By a re-elevation of a peneplain the
+  rivers of an old land surface may be restored to youthful activity,
+  and resume their shaping action, deepening the old valleys and
+  initiating new ones, starting afresh the whole course of the
+  geographical cycle. It is, however, not the action of the running
+  water on the land, but the function exercised by the land on the
+  running water, that is considered here to be the special province of
+  geography. At every stage of the geographical cycle the land forms, as
+  they exist at that stage, are concerned in guiding the condensation
+  and flow of water in certain definite ways. Thus, for example, in a
+  mountain range at right angles to a prevailing sea-wind, it is the
+  land forms which determine that one side of the range shall be richly
+  watered and deeply dissected by a complete system of valleys, while
+  the other side is dry, indefinite in its valley systems, and sends
+  none of its scanty drainage to the sea. The action of rain, ice and
+  rivers conspires with the movement of land waste to strip the layer of
+  soil from steep slopes as rapidly as it forms, and to cause it to
+  accumulate on the flat valley bottoms, on the graceful flattened cones
+  of alluvial fans at the outlet of the gorges of tributaries, or in the
+  smoothly-spread surface of alluvial plains.
+
+  The whole question of the régime of rivers and lakes is sometimes
+  treated under the name hydrography, a name used by some writers in the
+  sense of marine surveying, and by others as synonymous with
+  oceanography. For the study of rivers alone the name potamology[39]
+  has been suggested by Penck, and the subject being of much practical
+  importance has received a good deal of attention.[40]
+
+
+    Lakes and internal drainage.
+
+  The study of lakes has also been specialized under the name of
+  limnology (see LAKE).[41] The existence of lakes in hollows of the
+  land depends upon the balance between precipitation and evaporation. A
+  stream flowing into a hollow will tend to fill it up, and the water
+  will begin to escape as soon as its level rises high enough to reach
+  the lowest part of the rim. In the case of a large hollow in a very
+  dry climate the rate of evaporation may be sufficient to prevent the
+  water from ever rising to the lip, so that there is no outflow to the
+  sea, and a basin of internal drainage is the result. This is the case,
+  for instance, in the Caspian sea, the Aral and Balkhash lakes, the
+  Tarim basin, the Sahara, inner Australia, the great basin of the
+  United States and the Titicaca basin. These basins of internal
+  drainage are calculated to amount to 22% of the land surface. The
+  percentages of the land surface draining to the different oceans are
+  approximately--Atlantic, 34.3%; Arctic sea, 16.5%; Pacific, 14.4%;
+  Indian Ocean, 12.8%.[42]
+
+
+    Terminology of river systems.
+
+  The parts of a river system have not been so clearly defined as is
+  desirable, hence the exaggerated importance popularly attached to "the
+  source" of a river. A well-developed river system has in fact many
+  equally important and widely-separated sources, the most distant from
+  the mouth, the highest, or even that of largest initial volume not
+  being necessarily of greater geographical interest than the rest. The
+  whole of the land which directs drainage towards one river is known as
+  its basin, catchment area or drainage area--sometimes, by an incorrect
+  expression, as its valley or even its watershed. The boundary line
+  between one drainage area and others is rightly termed the watershed,
+  but on account of the ambiguity which has been tolerated it is better
+  to call it water-parting or, as in America, divide. The only other
+  important term which requires to be noted here is _talweg_, a word
+  introduced from the German into French and English, and meaning the
+  deepest line along the valley, which is necessarily occupied by a
+  stream unless the valley is dry.
+
+  The functions of land forms extend beyond the control of the
+  circulation of the atmosphere, the hydrosphere and the water which is
+  continually being interchanged between them; they are exercised with
+  increased effect in the higher departments of biogeography and
+  anthropogeography.
+
+
+    Biogeography.
+
+  The sum of the organic life on the globe is termed by some geographers
+  the biosphere, and it has been estimated that the whole mass of living
+  substance in existence at one time would cover the surface of the
+  earth to a depth of one-fifth of an inch.[43] The distribution of
+  living organisms is a complex problem, a function of many factors,
+  several of which are yet but little known. They include the biological
+  nature of the organism and its physical environment, the latter
+  involving conditions in which geographical elements, direct or
+  indirect, preponderate. The direct geographical elements are the
+  arrangement of land and sea (continents and islands standing in sharp
+  contrast) and the vertical relief of the globe, which interposes
+  barriers of a less absolute kind between portions of the same land
+  area or oceanic depression. The indirect geographical elements, which,
+  as a rule, act with and intensify the direct, are mainly climatic; the
+  prevailing winds, rainfall, mean and extreme temperatures of every
+  locality depending on the arrangement of land and sea and of land
+  forms. Climate thus guided affects the weathering of rocks, and so
+  determines the kind and arrangement of soil. Different species of
+  organisms come to perfection in different climates; and it may be
+  stated as a general rule that a species, whether of plant or animal,
+  once established at one point, would spread over the whole zone of the
+  climate congenial to it unless some barrier were interposed to its
+  progress. In the case of land and fresh-water organisms the sea is the
+  chief barrier; in the case of marine organisms, the land. Differences
+  in land forms do not exert great influence on the distribution of
+  living creatures directly, but indirectly such land forms as mountain
+  ranges and internal drainage basins are very potent through their
+  action on soil and climate. A snow-capped mountain ridge or an arid
+  desert forms a barrier between different forms of life which is often
+  more effective than an equal breadth of sea. In this way the surface
+  of the land is divided into numerous natural regions, the flora and
+  fauna of each of which include some distinctive species not shared by
+  the others. The distribution of life is discussed in the various
+  articles in this _Encyclopaedia_ dealing with biological, botanical
+  and zoological subjects.[44]
+
+
+    Floral zones.
+
+  The classification of the land surface into areas inhabited by
+  distinctive groups of plants has been attempted by many
+  phyto-geographers, but without resulting in any scheme of general
+  acceptance. The simplest classification is perhaps that of Drude
+  according to climatic zones, subdivided according to continents. This
+  takes account of--(1) the _Arctic-Alpine_ zone, including all the
+  vegetation of the region bordering on perpetual snow; (2) the _Boreal_
+  zone, including the temperate lands of North America, Europe and Asia,
+  all of which are substantially alike in botanical character; (3) the
+  _Tropical_ zone, divided sharply into (a) the tropical zone of the New
+  World, and (b) the tropical zone of the Old World, the forms of which
+  differ in a significant degree; (4) the _Austral_ zone, comprising all
+  continental land south of the equator, and sharply divided into three
+  regions the floras of which are strikingly distinct--(a) South
+  American, (b) South African and (c) Australian; (5) the _Oceanic_,
+  comprising all oceanic islands, the flora of which consists
+  exclusively of forms whose seeds could be drifted undestroyed by ocean
+  currents or carried by birds. To these might be added the antarctic,
+  which is still very imperfectly known. Many subdivisions and
+  transitional zones have been suggested by different authors.
+
+
+    Vegetation areas.
+
+  From the point of view of the economy of the globe this classification
+  by species is perhaps less important than that by mode of life and
+  physiological character in accordance with environment. The following
+  are the chief areas of vegetational activity usually recognized: (1)
+  The ice-deserts of the arctic and antarctic and the highest mountain
+  regions, where there is no vegetation except the lowest forms, like
+  that which causes "red snow." (2) The tundra or region of intensely
+  cold winters, forbidding tree-growth, where mosses and lichens cover
+  most of the ground when unfrozen, and shrubs occur of species which in
+  other conditions are trees, here stunted to the height of a few
+  inches. A similar zone surrounds the permanent snow on lofty mountains
+  in all latitudes. The tundra passes by imperceptible gradations into
+  the moor, bog and heath of warmer climates. (3) The temperate forests
+  of evergreen or deciduous trees, according to circumstances, which
+  occupy those parts of both temperate zones where rainfall and sunlight
+  are both abundant. (4) The grassy steppes or prairies where the
+  rainfall is diminished and temperatures are extreme, and grass is the
+  prevailing form of vegetation. These pass imperceptibly into--(5) the
+  arid desert, where rainfall is at a minimum, and the only plants are
+  those modified to subsist with the smallest supply of water. (6) The
+  tropical forest, which represents the maximum of plant luxuriance,
+  stimulated by the heaviest rainfall, greatest heat and strongest
+  light. These divisions merge one into the other, and admit of almost
+  indefinite subdivision, while they are subject to great modifications
+  by human interference in clearing and cultivating. Plants exhibit the
+  controlling power of environment to a high degree, and thus vegetation
+  is usually in close adjustment to the bolder geographical features of
+  a region.
+
+
+    Faunal realms.
+
+  The divisions of the earth into faunal regions by Dr P.L. Sclater have
+  been found to hold good for a large number of groups of animals as
+  different in their mode of life as birds and mammals, and they may
+  thus be accepted as based on nature. They are six in number: (1)
+  _Palaearctic_, including Europe, Asia north of the Himalaya, and
+  Africa north of the Sahara; (2) _Ethiopian_, consisting of Africa
+  south of the Atlas range, and Madagascar; (3) _Oriental_, including
+  India, Indo-China and the Malay Archipelago north of Wallace's line,
+  which runs between Bali and Lombok; (4) _Australian_, including
+  Australia, New Zealand, New Guinea and Polynesia; (5) _Nearctic_ or
+  North America, north of Mexico; and (6) _Neotropical_ or South
+  America. Each of these divisions is the home of a special fauna, many
+  species of which are confined to it alone; in the Australian region,
+  indeed, practically the whole fauna is peculiar and distinctive,
+  suggesting a prolonged period of complete biological isolation. In
+  some cases, such as the Ethiopian and Neotropical and the Palaearctic
+  and Nearctic regions, the faunas, although distinct, are related,
+  several forms on opposite sides of the Atlantic being analogous, e.g.
+  the lion and puma, ostrich and rhea. Where two of the faunal realms
+  meet there is usually, though not always, a mixing of faunas. These
+  facts have led some naturalists to include the Palaearctic and
+  Nearctic regions in one, termed _Holarctic_, and to suggest
+  transitional regions, such as the _Sonoran_, between North and South
+  America, and the _Mediterranean_, between Europe and Africa, or to
+  create sub-regions, such as Madagascar and New Zealand. Oceanic
+  islands have, as a rule, distinctive faunas and floras which resemble,
+  but are not identical with, those of other islands in similar
+  positions.
+
+
+    Biological distribution as a means of geographical research.
+
+  The study of the evolution of faunas and the comparison of the faunas
+  of distant regions have furnished a trustworthy instrument of
+  pre-historic geographical research, which enables earlier geographical
+  relations of land and sea to be traced out, and the approximate
+  period, or at least the chronological order of the larger changes, to
+  be estimated. In this way, for example, it has been suggested that a
+  land, "Lemuria," once connected Madagascar with the Malay Archipelago,
+  and that a northern extension of the antarctic land once united the
+  three southern continents.
+
+  The distribution of fossils frequently makes it possible to map out
+  approximately the general features of land and sea in long-past
+  geological periods, and so to enable the history of crustal relief to
+  be traced.[45]
+
+
+    Reaction of organisms on environment.
+
+  While the tendency is for the living forms to come into harmony with
+  their environment and to approach the state of equilibrium by
+  successive adjustments if the environment should happen to change, it
+  is to be observed that the action of organisms themselves often tends
+  to change their environment. Corals and other quick-growing calcareous
+  marine organisms are the most powerful in this respect by creating new
+  land in the ocean. Vegetation of all sorts acts in a similar way,
+  either in forming soil and assisting in breaking up rocks, in filling
+  up shallow lakes, and even, like the mangrove, in reclaiming wide
+  stretches of land from the sea. Plant life, utilizing solar light to
+  combine the inorganic elements of water, soil and air into living
+  substance, is the basis of all animal life. This is not by the supply
+  of food alone, but also by the withdrawal of carbonic acid from the
+  atmosphere, by which vegetation maintains the composition of the air
+  in a state fit for the support of animal life. Man in the primitive
+  stages of culture is scarcely to be distinguished from other animals
+  as regards his subjection to environment, but in the higher grades of
+  culture the conditions of control and reaction become much more
+  complicated, and the department of anthropogeography is devoted to
+  their consideration.
+
+
+    Anthropogeography.
+
+  The first requisites of all human beings are food and protection, in
+  their search for which men are brought into intimate relations with
+  the forms and productions of the earth's surface. The degree of
+  dependence of any people upon environment varies inversely as the
+  degree of culture or civilization, which for this purpose may perhaps
+  be defined as the power of an individual to exercise control over the
+  individual and over the environment for the benefit of the community.
+  The development of culture is to a certain extent a question of race,
+  and although forming one species, the varieties of man differ in
+  almost imperceptible gradations with a complexity defying
+  classification (see ANTHROPOLOGY). Professor Keane groups man round
+  four leading types, which may be named the black, yellow, red and
+  white, or the Ethiopic, Mongolic, American and Caucasic. Each may be
+  subdivided, though not with great exactness, into smaller groups,
+  either according to physical characteristics, of which the form of the
+  head is most important, or according to language.
+
+
+    Types of man.
+
+  The black type is found only in tropical or sub-tropical countries,
+  and is usually in a primitive condition of culture, unless educated by
+  contact with people of the white type. They follow the most primitive
+  forms of religion (mainly fetishism), live on products of the woods or
+  of the chase, with the minimum of work, and have only a loose
+  political organization. The red type is peculiar to America,
+  inhabiting every climate from polar to equatorial, and containing
+  representatives of many stages of culture which had apparently
+  developed without the aid or interference of people of any other race
+  until the close of the 15th century. The yellow type is capable of a
+  higher culture, cherishes higher religious beliefs, and inhabits as a
+  rule the temperate zone, although extending to the tropics on one side
+  and to the arctic regions on the other. The white type, originating in
+  the north temperate zone, has spread over the whole world. They have
+  attained the highest culture, profess the purest forms of monotheistic
+  religion, and have brought all the people of the black type and many
+  of those of the yellow under their domination.
+
+  The contrast between the yellow and white types has been softened by
+  the remarkable development of the Japanese following the assimilation
+  of western methods.
+
+  The actual number of human inhabitants in the world has been
+  calculated as follows:
+
+                            By Continents.[46]
+
+    Asia                       875,000,000
+    Europe                     392,000,000
+    Africa                     170,000,000
+    America                    143,000,000
+    Australia and Polynesia      7,000,000
+                             -------------
+      Total                  1,587,000,000
+
+                               By Race.[47]
+
+    White (Caucasic)           770,000,000
+    Yellow (Mong.)             540,000,000
+    Black (Ethiopic)           175,000,000
+    Red (American)              22,000,000
+                             -------------
+      Total                  1,507,000,000
+
+  In round numbers the population of the world is about 1,600,000,000,
+  and, according to an estimate by Ravenstein,[48] the maximum
+  population which it will be possible for the earth to maintain is 6000
+  millions, a number which, if the average rate of increase in 1891
+  continued, would be reached within 200 years.
+
+  While highly civilized communities are able to evade many of the
+  restrictions of environment, to overcome the barriers to
+  intercommunication interposed by land or sea, to counteract the
+  adverse influence of climate, and by the development of trade even to
+  inhabit countries which cannot yield a food-supply, the mass of
+  mankind is still completely under the control of those conditions
+  which in the past determined the distribution and the mode of life of
+  the whole human race.
+
+
+    Influence of environment on man.
+
+  In tropical forests primitive tribes depend on the collection of wild
+  fruits, and in a minor degree on the chase of wild animals, for their
+  food. Clothing is unnecessary; hence there is little occasion for
+  exercising the mental faculties beyond the sense of perception to
+  avoid enemies, or the inventive arts beyond what is required for the
+  simplest weapons and the most primitive fortifications. When the
+  pursuit of game becomes the chief occupation of a people there is of
+  necessity a higher development of courage, skill, powers of
+  observation and invention; and these qualities are still further
+  enhanced in predatory tribes who take by force the food, clothing and
+  other property prepared or collected by a feebler people. The
+  fruit-eating savage cannot stray beyond his woods which bound his life
+  as the water bounds that of a fish; the hunter is free to live on the
+  margin of forests or in open country, while the robber or warrior from
+  some natural stronghold of the mountains sweeps over the adjacent
+  plains and carries his raids into distant lands. Wide grassy steppes
+  lead to the organization of the people as nomads whose wealth consists
+  in flocks and herds, and their dwellings are tents. The nomad not only
+  domesticates and turns to his own use the gentler and more powerful
+  animals, such as sheep, cattle, horses, camels, but even turns some
+  predatory creatures, like the dog, into a means of defending their
+  natural prey. They hunt the beasts of prey destructive to their
+  flocks, and form armed bands for protection against marauders or for
+  purposes of aggression on weaker sedentary neighbours. On the fertile
+  low grounds along the margins of rivers or in clearings of forests,
+  agricultural communities naturally take their rise, dwelling in
+  villages and cultivating the wild grains, which by careful nurture and
+  selection have been turned into rich cereals. The agriculturist as a
+  rule is rooted to the soil. The land he tills he holds, and acquires a
+  closer connexion with a particular patch of ground than either the
+  hunter or the herdsman. In the temperate zone, where the seasons are
+  sharply contrasted, but follow each other with regularity, foresight
+  and self-denial were fostered, because if men did not exercise these
+  qualities seed-time or harvest might pass into lost opportunities and
+  the tribes would suffer. The more extreme climates of arid regions on
+  the margins of the tropics, by the unpredictable succession of
+  droughts and floods, confound the prevision of uninstructed people,
+  and make prudence and industry qualities too uncertain in their
+  results to be worth cultivating. Thus the civilization of agricultural
+  peoples of the temperate zone grew rapidly, yet in each community a
+  special type arose adapted to the soil, the crop and the climate. On
+  the seashore fishing naturally became a means of livelihood, and
+  dwellers by the sea, in virtue of the dangers to which they are
+  exposed from storm and unseaworthy craft, are stimulated to a higher
+  degree of foresight, quicker observation, prompter decision and more
+  energetic action in emergencies than those who live inland. The
+  building and handling of vessels also, and the utilization of such
+  uncontrollable powers of nature as wind and tide, helped forward
+  mechanical invention. To every type of coast there may be related a
+  special type of occupation and even of character; the deep and gloomy
+  fjord, backed by almost impassable mountains, bred bold mariners whose
+  only outlet for enterprise was seawards towards other lands--the
+  _viks_ created the vikings. On the gently sloping margin of the
+  estuary of a great river a view of tranquil inland life was equally
+  presented to the shore-dweller, and the ocean did not present the only
+  prospect of a career. Finally the mountain valley, with its patches of
+  cultivable soil on the alluvial fans of tributary torrents, its narrow
+  pastures on the uplands only left clear of snow in summer, its
+  intensified extremes of climates and its isolation, almost equal to
+  that of an island, has in all countries produced a special type of
+  brave and hardy people, whose utmost effort may bring them comfort,
+  but not wealth, by honest toil, who know little of the outer world,
+  and to whom the natural outlet for ambition is marauding on the
+  fertile plains. The highlander and viking, products of the valleys
+  raised high amid the mountains or half-drowned in the sea, are
+  everywhere of kindred spirit.
+
+  It is in some such manner as these that the natural conditions of
+  regions, which must be conformed to by prudence and utilized by labour
+  to yield shelter and food, have led to the growth of peoples differing
+  in their ways of life, thought and speech. The initial differences so
+  produced are confirmed and perpetuated by the same barriers which
+  divide the faunal or floral regions, the sea, mountains, deserts and
+  the like, and much of the course of past history and present politics
+  becomes clear when the combined results of differing race and
+  differing environment are taken into account.[49]
+
+
+    Density of population.
+
+  The specialization which accompanies the division of labour has
+  important geographical consequences, for it necessitates communication
+  between communities and the interchange of their products. Trade
+  makes it possible to work mineral resources in localities where food
+  can only be grown with great difficulty and expense, or which are even
+  totally barren and waterless, entirely dependent on supplies from
+  distant sources.
+
+  The population which can be permanently supported by a given area of
+  land differs greatly according to the nature of the resources and the
+  requirements of the people. Pastoral communities are always scattered
+  very thinly over large areas; agricultural populations may be almost
+  equally sparse where advanced methods of agriculture and labour-saving
+  machinery are employed; but where a frugal people are situated on a
+  fertile and inexhaustible soil, such as the deltas and river plains of
+  Egypt, India and China, an enormous population may be supported on a
+  small area. In most cases, however, a very dense population can only
+  be maintained in regions where mineral resources have fixed the site
+  of great manufacturing industries. The maximum density of population
+  which a given region can support is very difficult to determine; it
+  depends partly on the race and standard of culture of the people,
+  partly on the nature and origin of the resources on which they depend,
+  partly on the artificial burdens imposed and very largely on the
+  climate. Density of population is measured by the average number of
+  people residing on a unit of area; but in order to compare one part of
+  the world with another the average should, strictly speaking, be taken
+  for regions of equal size or of equal population; and the portions of
+  the country which are permanently uninhabitable ought to be excluded
+  from the calculation.[50] Considering the average density of
+  population within the political limits of countries, the following
+  list is of some value; the figures for a few smaller divisions of
+  large countries are added (in brackets) for comparison:
+
+    _Average Population on 1 sq. m._ (_For 1900 or 1901._)
+
+    +--------------------+---------+-------------------+---------+
+    |      Country.      | Density |      Country.     | Density |
+    |                    | of pop. |                   | of pop. |
+    +--------------------+---------+-------------------+---------+
+    | (Saxony)           |   743*  | Ceylon            |  141**  |
+    | Belgium            |   589*  | Greece            |   97    |
+    | Java               |   568** | European Turkey   |   90    |
+    | (England and Wales)|   558   | Spain             |   97    |
+    | (Bengal)           |   495** | European Russia   |   55**  |
+    | Holland            |   436   | Sweden            |   30    |
+    | United Kingdom     |   344   | United States     |   25    |
+    | Japan              |   317   | Mexico            |   18    |
+    | Italy              |   293   | Norway            |   18    |
+    | China proper       |   270** | Persia            |   15    |
+    | German Empire      |   270   | New Zealand       |    7    |
+    | Austria            |   226   | Argentina         |    5    |
+    | Switzerland        |   207   | Brazil            |    4.5  |
+    | France             |   188   | Eastern States of |         |
+    | Indian Empire      |   167** |   Australia       |    3    |
+    | Denmark            |   160** | Dominion of Canada|    1.5  |
+    | Hungary            |   154** | Siberia           |    1    |
+    | Portugal           |   146   | West Australia    |    0.2  |
+    +--------------------+---------+-------------------+---------+
+      * Almost exclusively industrial.
+     ** Almost exclusively agricultural.
+
+
+    Migration.
+
+  The movement of people from one place to another without the immediate
+  intention of returning is known as migration, and according to its
+  origin it may be classed as centrifugal (directed _from_ a particular
+  area) and centripetal (directed _towards_ a particular area).
+  Centrifugal migration is usually a matter of compulsion; it may be
+  necessitated by natural causes, such as a change of climate leading to
+  the withering of pastures or destruction of agricultural land, to
+  inundation, earthquake, pestilence or to an excess of population over
+  means of support; or to artificial causes, such as the wholesale
+  deportation of a conquered people; or to political or religious
+  persecution. In any case the people are driven out by some adverse
+  change; and when the urgency is great they may require to drive out in
+  turn weaker people who occupy a desirable territory, thus propagating
+  the wave of migration, the direction of which is guided by the forms
+  of the land into inevitable channels. Many of the great historic
+  movements of peoples were doubtless due to the gradual change of
+  geographical or climatic conditions; and the slow desiccation of
+  Central Asia has been plausibly suggested as the real cause of the
+  peopling of modern Europe and of the medieval wars of the Old World,
+  the theatres of which were critical points on the great natural lines
+  of communication between east and west.
+
+  In the case of centripetal migrations people flock to some particular
+  place where exceptionally favourable conditions have been found to
+  exist. The rushes to gold-fields and diamond-fields are typical
+  instances; the growth of towns on coal-fields and near other sources
+  of power, and the rapid settlement of such rich agricultural districts
+  as the wheat-lands of the American prairies and great plains are other
+  examples.
+
+  There is, however, a tendency for people to remain rooted to the land
+  of their birth, when not compelled or induced by powerful external
+  causes to seek a new home.
+
+
+    Political geography.
+
+  Thus arises the spirit of patriotism, a product of purely geographical
+  conditions, thereby differing from the sentiment of loyalty, which is
+  of racial origin. Where race and soil conspire to evoke both loyalty
+  and patriotism in a people, the moral qualities of a great and
+  permanent nation are secured. It is noticeable that the patriotic
+  spirit is strongest in those places where people are brought most
+  intimately into relation with the land; dwellers in the mountain or by
+  the sea, and, above all, the people of rugged coasts and mountainous
+  archipelagoes, have always been renowned for love of country, while
+  the inhabitants of fertile plains and trading communities are
+  frequently less strongly attached to their own land.
+
+  Amongst nomads the tribe is the unit of government, the political bond
+  is personal, and there is no definite territorial association of the
+  people, who may be loyal but cannot be patriotic. The idea of a
+  country arises only when a nation, either homogeneous or composed of
+  several races, establishes itself in a region the boundaries of which
+  may be defined and defended against aggression from without. Political
+  geography takes account of the partition of the earth amongst
+  organized communities, dealing with the relation of races to regions,
+  and of nations to countries, and considering the conditions of
+  territorial equilibrium and instability.
+
+
+    Boundaries.
+
+  The definition of boundaries and their delimitation is one of the most
+  important parts of political geography. Natural boundaries are always
+  the most definite and the strongest, lending themselves most readily
+  to defence against aggression. The sea is the most effective of all,
+  and an island state is recognized as the most stable. Next in
+  importance comes a mountain range, but here there is often difficulty
+  as to the definition of the actual crest-line, and mountain ranges
+  being broad regions, it may happen that a small independent state,
+  like Switzerland or Andorra, occupies the mountain valleys between two
+  or more great countries. Rivers do not form effective international
+  boundaries, although between dependent self-governing communities they
+  are convenient lines of demarcation. A desert, or a belt of country
+  left purposely without inhabitants, like the mark, marches or
+  debatable lands of the middle ages, was once a common means of
+  separating nations which nourished hereditary grievances. The
+  "buffer-state" of modern diplomacy is of the same ineffectual type. A
+  less definite though very practical boundary is that formed by the
+  meeting-line of two languages, or the districts inhabited by two
+  races. The line of fortresses protecting Austria from Italy lies in
+  some places well back from the political boundary, but just inside the
+  linguistic frontier, so as to separate the German and Italian races
+  occupying Austrian territory. Arbitrary lines, either traced from
+  point to point and marked by posts on the ground, or defined as
+  portions of meridians and parallels, are now the most common type of
+  boundaries fixed by treaty. In Europe and Asia frontiers are usually
+  strongly fortified and strictly watched in times of peace as well as
+  during war. In South America strictly defined boundaries are still the
+  exception, and the claims of neighbouring nations have very frequently
+  given rise to war, though now more commonly to arbitration.[51]
+
+
+    Forms of government.
+
+  The modes of government amongst civilized peoples have little
+  influence on political geography; some republics are as arbitrary and
+  exacting in their frontier regulations as some absolute monarchies. It
+  is, however, to be noticed that absolute monarchies are confined to
+  the east of Europe and to Asia, Japan being the only established
+  constitutional monarchy east of the Carpathians. Limited monarchies
+  are (with the exception of Japan) peculiar to Europe, and in these the
+  degree of democratic control may be said to diminish as one passes
+  eastwards from the United Kingdom. Republics, although represented in
+  Europe, are the peculiar form of government of America and are unknown
+  in Asia.
+
+  The forms of government of colonies present a series of transitional
+  types from the autocratic administration of a governor appointed by
+  the home government to complete democratic self-government. The latter
+  occurs only in the temperate possessions of the British empire, in
+  which there is no great preponderance of a coloured native population.
+  New colonial forms have been developed during the partition of Africa
+  amongst European powers, the sphere of influence being especially
+  worthy of notice. This is a vaguer form of control than a
+  protectorate, and frequently amounts merely to an agreement amongst
+  civilized powers to respect the right of one of their number to
+  exercise government within a certain area, if it should decide to do
+  so at any future time.
+
+  The central governments of all civilized countries concerned with
+  external relations are closely similar in their modes of action, but
+  the internal administration may be very varied. In this respect a
+  country is either centralized, like the United Kingdom or France, or
+  federated of distinct self-governing units like Germany (where the
+  units include kingdoms, at least three minor types of monarchies,
+  municipalities and a crown land under a nominated governor), or the
+  United States, where the units are democratic republics. The ultimate
+  cause of the predominant form of federal government may be the
+  geographical diversity of the country, as in the cantons occupying the
+  once isolated mountain valleys of Switzerland, the racial diversity of
+  the people, as in Austria-Hungary, or merely political expediency, as
+  in republics of the American type.
+
+  The minor subdivisions into provinces, counties and parishes, or
+  analogous areas, may also be related in many cases to natural features
+  or racial differences perpetuated by historical causes. The
+  territorial divisions and subdivisions often survive the conditions
+  which led to their origin; hence the study of political geography is
+  allied to history as closely as the study of physical geography is
+  allied to geology, and for the same reason.
+
+
+    Towns.
+
+  The aggregation of population in towns was at one time mainly brought
+  about by the necessity for defence, a fact indicated by the defensive
+  sites of many old towns. In later times, towns have been more often
+  founded in proximity to valuable mineral resources, and at critical
+  points or nodes on lines of communication. These are places where the
+  mode of travelling or of transport is changed, such as seaports, river
+  ports and railway termini, or natural resting-places, such as a ford,
+  the foot of a steep ascent on a road, the entrance of a valley leading
+  up from a plain into the mountains, or a crossing-place of roads or
+  railways.[52] The existence of a good natural harbour is often
+  sufficient to give origin to a town and to fix one end of a line of
+  land communication.
+
+
+    Lines of communication.
+
+  In countries of uniform surface or faint relief, roads and railways
+  may be constructed in any direction without regard to the
+  configuration. In places where the low ground is marshy, roads and
+  railways often follow the ridge-lines of hills, or, as in Finland, the
+  old glacial eskers, which run parallel to the shore. Wherever the
+  relief of the land is pronounced, roads and railways are obliged to
+  occupy the lowest ground winding along the valleys of rivers and
+  through passes in the mountains. In exceptional cases obstructions
+  which it would be impossible or too costly to turn are overcome by a
+  bridge or tunnel, the magnitude of such works increasing with the
+  growth of engineering skill and financial enterprise. Similarly the
+  obstructions offered to water communication by interruption through
+  land or shallows are overcome by cutting canals or dredging out
+  channels. The economy and success of most lines of communication
+  depend on following as far as possible existing natural lines and
+  utilizing existing natural sources of power.[53]
+
+
+    Commercial geography.
+
+  Commercial geography may be defined as the description of the earth's
+  surface with special reference to the discovery, production, transport
+  and exchange of commodities. The transport concerns land routes and
+  sea routes, the latter being the more important. While steam has been
+  said to make a ship independent of wind and tide, it is still true
+  that a long voyage even by steam must be planned so as to encounter
+  the least resistance possible from prevailing winds and permanent
+  currents, and this involves the application of oceanographical and
+  meteorological knowledge. The older navigation by utilizing the power
+  of the wind demands a very intimate knowledge of these conditions, and
+  it is probable that a revival of sailing ships may in the present
+  century vastly increase the importance of the study of maritime
+  meteorology.
+
+  The discovery and production of commodities require a knowledge of the
+  distribution of geological formations for mineral products, of the
+  natural distribution, life-conditions and cultivation or breeding of
+  plants and animals and of the labour market. Attention must also be
+  paid to the artificial restrictions of political geography, to the
+  legislative restrictions bearing on labour and trade as imposed in
+  different countries, and, above all, to the incessant fluctuations of
+  the economic conditions of supply and demand and the combinations of
+  capitalists or workers which affect the market.[54] The term "applied
+  geography" has been employed to designate commercial geography, the
+  fact being that every aspect of scientific geography may be applied to
+  practical purposes, including the purposes of trade. But apart from
+  the applied science, there is an aspect of pure geography which
+  concerns the theory of the relation of economics to the surface of the
+  earth.
+
+
+    Conclusion.
+
+  It will be seen that as each successive aspect of geographical science
+  is considered in its natural sequence the conditions become more
+  numerous, complex, variable and practically important. From the
+  underlying abstract mathematical considerations all through the
+  superimposed physical, biological, anthropological, political and
+  commercial development of the subject runs the determining control
+  exercised by crust-forms acting directly or indirectly on mobile
+  distributions; and this is the essential principle of geography.
+       (H. R. M.)
+
+
+FOOTNOTES:
+
+  [1] A concise sketch of the whole history of geographical method or
+    theory as distinguished from the history of geographical discovery
+    (see later section of this article) is only to be found in the
+    introduction to H. Wagner's _Lehrbuch der Geographie_, vol. i.
+    (Leipzig, 1900), which is in every way the most complete treatise on
+    the principles of geography.
+
+  [2] _History of Ancient Geography_ (Cambridge, 1897), p. 70.
+
+  [3] See J.L. Myres, "An Attempt to reconstruct the Maps used by
+    Herodotus," _Geographical Journal_, viii. (1896), p. 605.
+
+  [4] _Geschichte der wissenschaftlichen Erdkunde der Griechen_
+    (Leipzig, 1891), Abt. 3, p. 60.
+
+  [5] Bunbury's _History of Ancient Geography_ (2 vols., London, 1879),
+    Müller's _Geographi Graeci minores_ (2 vols., Paris, 1855, 1861) and
+    Berger's _Geschichte der wissenschaftlichen Erdkunde der Griechen_ (4
+    vols., Leipzig, 1887-1893) are standard authorities on the Greek
+    geographers.
+
+  [6] The period of the early middle ages is dealt with in Beazley's
+    _Dawn of Modern Geography_ (London; part i., 1897; part ii., 1901;
+    part iii., 1906); see also Winstedt, _Cosmos Indicopleustes_ (1910).
+
+  [7] From translator's preface to the English version by Mr Dugdale
+    (1733), entitled _A Complete System of General Geography_, revised by
+    Dr Peter Shaw (London, 1756).
+
+  [8] Printed in _Schriften zur physischen Geographie_, vol. vi. of
+    Schubert's edition of the collected works of Kant (Leipzig, 1839).
+    First published with notes by Rink in 1802.
+
+  [9] _History of Civilization_, vol. i. (1857).
+
+  [10] See H.J. Mackinder in _British Association Report_ (Ipswich),
+    1895, p. 738, for a summary of German opinion, which has been
+    expressed by many writers in a somewhat voluminous literature.
+
+  [11] H. Wagner's year-book, _Geographische Jahrbuch_, published at
+    Gotha, is the best systematic record of the progress of geography in
+    all departments; and Haack's _Geographen Kalender_, also published
+    annually at Gotha, gives complete lists of the geographical societies
+    and geographers of the world.
+
+  [12] This phrase is old, appearing in one of the earliest English
+    works on geography, William Cuningham's _Cosmographical Glasse
+    conteinyng the pleasant Principles of Cosmographie, Geographie,
+    Hydrographie or Navigation_ (London, 1559).
+
+  [13] See also S. Günther, _Handbuch der mathematischen Geographie_
+    (Stuttgart, 1890).
+
+  [14] "On the Height of the Land and the Depth of the Ocean," _Scot.
+    Geog. Mag._ iv. (1888), p. 1. Estimates had been made previously by
+    Humboldt, De Lapparent, H. Wagner, and subsequently by Penck and
+    Heiderich, and for the oceans by Karstens.
+
+  [15] _Petermanns Mitteilungen_, xxv. (1889), p. 17.
+
+  [16] _Proc. Roy. Soc. Edin._ xvii. (1890) p. 185.
+
+  [17] _Comptes rendus Acad. Sci._ (Paris, 1890), vol. iii. p. 994.
+
+  [18] "Areal und mittlere Erhebung der Landflächen sowie der
+    Erdkruste" in Gerland's _Beiträge zur Geophysik_, ii. (1895) p. 667.
+    See also _Nature_, 54 (1896), p. 112.
+
+  [19] _Petermanns Mitteilungen_, xxxv. (1889) p. 19.
+
+  [20] The areas of the continental shelf and lowlands are
+    approximately equal, and it is an interesting circumstance that,
+    taken as a whole, the actual coast-line comes just midway on the most
+    nearly level belt of the earth's surface, excepting the ocean floor.
+    The configuration of the continental slope has been treated in detail
+    by Nansen in _Scientific Results of Norwegian North Polar
+    Expedition_, vol. iv. (1904), where full references to the literature
+    of the subject will be found.
+
+  [21] _British Association Report_ (Edinburgh, 1892), p. 699.
+
+  [22] _Das Antlitz der Erde_ (4 vols., Leipzig, 1885, 1888, 1901).
+    Translated under the editorship of E. de Margerie, with much
+    additional matter, as _La Face de la terre_, vols. i. and ii. (Paris,
+    1897, 1900), and into English by Dr Hertha Sollas as _The Face of the
+    Earth_, vols. i. and ii. (Oxford, 1904, 1906).
+
+  [23] Élie de Beaumont, _Notice sur les systčmes de montagnes_ (3
+    vols., Paris, 1852).
+
+  [24] _Vestiges of the Molten Globe_ (London, 1875).
+
+  [25] See J.W. Gregory, "The Plan of the Earth and its Causes," _Geog.
+    Journal_, xiii. (1899) p. 225; Lord Avebury, _ibid._ xv. (1900) p.
+    46; Marcel Bertrand, "Déformation tétraédrique de la terre et
+    déplacement du pōle," _Comptes rendus Acad. Sci._ (Paris, 1900), vol.
+    cxxx. p. 449; and A. de Lapparent, _ibid._ p. 614.
+
+  [26] See A.E.H. Love, "Gravitational Stability of the Earth," _Phil.
+    Trans._ ser. A. vol. ccvii. (1907) p. 171.
+
+  [27] _Rumpf_, in German, the language in which this distinction was
+    first made.
+
+  [28] _Lehrbuch der Geographie_ (Hanover and Leipzig, 1900), Bd. i. S.
+    245, 249.
+
+  [29] See, for example, F.G. Hahn's _Insel-Studien_ (Leipzig, 1883).
+
+  [30] See _Geographical Journal_, xxii. (1903) pp. 191-194.
+
+  [31] The most important works on the classification of land forms are
+    F. von Richthofen, _Führer für Forschungsreisende_ (Berlin, 1886); G.
+    de la Noė and E. de Margerie, _Les Formes du terrain_ (Paris, 1888);
+    and above all A. Penck, _Morphologie der Erdoberfläche_ (2 vols.,
+    Stuttgart, 1894). Compare also A. de Lapparent, _Leēons de géographie
+    physique_ (2nd ed., Paris, 1898), and W.M. Davis, _Physical
+    Geography_ (Boston, 1899).
+
+  [32] "Geomorphologie als genetische Wissenschaft," in _Report of
+    Sixth International Geog. Congress_ (London, 1895), p. 735 (English
+    Abstract, p. 748).
+
+  [33] On this subject see J. Geikie, _Earth Sculpture_ (London, 1898);
+    J.E. Marr, _The Scientific Study of Scenery_ (London, 1900); Sir A.
+    Geikie, _The Scenery and Geology of Scotland_ (London, 2nd ed.,
+    1887); Lord Avebury (Sir J. Lubbock), _The Scenery of Switzerland_
+    (London, 1896) and _The Scenery of England_ (London, 1902).
+
+  [34] Some geographers distinguish a mountain from a hill by origin;
+    thus Professor Seeley says "a mountain implies elevation and a hill
+    implies denudation, but the external forms of both are often
+    identical." _Report VI. Int. Geog. Congress_ (London, 1895), p. 751.
+
+  [35] "Mountains," in _Scot. Geog. Mag._ ii. (1896) p. 145.
+
+  [36] _Führer für Forschungsreisende_, pp. 652-685.
+
+  [37] See, for a summary of river-action, A. Phillipson, _Studien über
+    Wasserscheiden_ (Leipzig, 1886); also I.C. Russell, _River
+    Development_, (London, 1898) (published as _The Rivers of North
+    America_, New York, 1898).
+
+  [38] W.M. Davis, "The Geographical Cycle," _Geog. Journ._ xiv. (1899)
+    p. 484.
+
+  [39] A. Penck, "Potamology as a Branch of Physical Geography," _Geog.
+    Journ._ x. (1897) p. 619.
+
+  [40] See, for instance, E. Wisotzki, _Hauptfluss und Nebenfluss_
+    (Stettin, 1889). For practical studies see official reports on the
+    Mississippi, Rhine, Seine, Elbe and other great rivers.
+
+  [41] F.A. Forel, _Handbuch der Seenkunde: allgemeine Limnologie_
+    (Stuttgart, 1901); F.A. Forel, "La Limnologie, branche de la
+    géographie," _Report VI. Int. Geog. Congress_ (London, 1895), p. 593;
+    also _Le Léman_ (2 vols., Lausanne, 1892, 1894); H. Lullies, "Studien
+    über Seen," _Jubiläumsschrift der Albertus-Universität_ (Königsberg,
+    1894); and G.R. Credner, "Die Reliktenseen," _Petermanns
+    Mitteilungen_, Ergänzungshefte 86 and 89 (Gotha., 1887, 1888).
+
+  [42] J. Murray, "Drainage Areas of the Continents," _Scot. Geog.
+    Mag._ ii. (1886) p. 548.
+
+  [43] Wagner, _Lehrbuch der Geographie_ (1900), i. 586.
+
+  [44] For details, see A.R. Wallace, _Geographical Distribution of
+    Animals and Island Life_; A. Heilprin, _Geographical and Geological
+    Distribution of Animals_ (1887); O. Drude, _Handbuch der
+    Pflanzengeographie_; A. Engler, _Entwickelungsgeschichte der
+    Pflanzenwelt_; also Beddard, _Zoogeography_ (Cambridge, 1895); and
+    Sclater, _The Geography of Mammals_ (London, 1899).
+
+  [45] See particularly A. de Lapparent, _Traité de géologie_ (4th ed.,
+    Paris, 1900).
+
+  [46] Estimate for 1900. H. Wagner, _Lehrbuch der Geographie_, i. P.
+    658.
+
+  [47] Estimate for year not stated. A.H. Keane in _International
+    Geography_, p. 108.
+
+  [48] In _Proc. R. G. S._ xiii. (1891) p. 27.
+
+  [49] On the influence of land on people see Shaler, _Nature and Man
+    in America_ (New York and London, 1892); and Ellen C. Semple's
+    _American History and its Geographic Conditions_ (Boston, 1903).
+
+  [50] See maps of density of population in Bartholomew's great
+    large-scale atlases, _Atlas of Scotland_ and _Atlas of England_.
+
+  [51] For the history of territorial changes in Europe, see Freeman,
+    _Historical Geography of Europe_, edited by Bury (Oxford), 1903; and
+    for the official definition of existing boundaries, see Hertslet,
+    _The Map of Europe by Treaty_ (4 vols., London, 1875, 1891); _The Map
+    of Africa by Treaty_ (3 vols., London, 1896). Also Lord Curzon's
+    Oxford address on _Frontiers_ (1907).
+
+  [52] For numerous special instances of the determining causes of town
+    sites, see G.G. Chisholm, "On the Distribution of Towns and Villages
+    in England," _Geographical Journal_ (1897), ix. 76, x. 511.
+
+  [53] The whole subject of anthropogeography is treated in a masterly
+    way by F. Ratzel in his _Anthropogeographie_ (Stuttgart, vol. i. 2nd
+    ed., 1899, vol. ii. 1891), and in his _Politische Geographie_
+    (Leipzig, 1897). The special question of the reaction of man on his
+    environment is handled by G.P. Marsh in _Man and Nature, or Physical
+    Geography as modified by Human Action_ (London, 1864).
+
+  [54] For commercial geography see G.G. Chisholm, _Manual of
+    Commercial Geography_ (1890).
+
+
+
+
+GEOID (from Gr. [Greek: gź], the earth), an imaginary surface employed
+by geodesists which has the property that every element of it is
+perpendicular to the plumb-line where that line cuts it. Compared with
+the "spheroid of reference" the surface of the geoid is in general
+depressed over the oceans and raised over the great land masses. (See
+EARTH, FIGURE OF THE.)
+
+
+
+
+GEOK-TEPE, a former fortress of the Turkomans, in Russian Transcaspia,
+in the oasis of Akhal-tekke, on the Transcaspian railway, 28 m. N.W. of
+Askabad. It consisted of a walled enclosure 1¾ m. in circuit, the wall
+being 18 ft. high and 20 to 30 ft. thick. In December 1880 the place was
+attacked by 6000 Russians under General Skobelev, and after a siege of
+twenty-three days was carried by storm, although the defenders numbered
+25,000. A monument and a small museum commemorate the event.
+
+
+
+
+GEOLOGY (from Gr. [Greek: gź], the earth, and [Greek: logos], science),
+the science which investigates the physical history of the earth. Its
+object is to trace the structural progress of our planet from the
+earliest beginnings of its separate existence, through its various
+stages of growth, down to the present condition of things. It seeks to
+determine the manner in which the evolution of the earth's great surface
+features has been effected. It unravels the complicated processes by
+which each continent has been built up. It follows, even into detail,
+the varied sculpture of mountain and valley, crag and ravine. Nor does
+it confine itself merely to changes in the inorganic world. Geology
+shows that the present races of plants and animals are the descendants
+of other and very different races which once peopled the earth. It
+teaches that there has been a progressive development of the
+inhabitants, as well as one of the globe on which they have dwelt; that
+each successive period in the earth's history, since the introduction of
+living things, has been marked by characteristic types of the animal and
+vegetable kingdoms; and that, however imperfectly the remains of these
+organisms have been preserved or may be deciphered, materials exist for
+a history of life upon the planet. The geographical distribution of
+existing faunas and floras is often made clear and intelligible by
+geological evidence; and in the same way light is thrown upon some of
+the remoter phases in the history of man himself. A subject so
+comprehensive as this must require a wide and varied basis of evidence.
+It is one of the characteristics of geology to gather evidence from
+sources which at first sight seem far removed from its scope, and to
+seek aid from almost every other leading branch of science. Thus, in
+dealing with the earliest conditions of the planet, the geologist must
+fully avail himself of the labours of the astronomer. Whatever is
+ascertainable by telescope, spectroscope or chemical analysis, regarding
+the constitution of other heavenly bodies, has a geological bearing. The
+experiments of the physicist, undertaken to determine conditions of
+matter and of energy, may sometimes be taken as the starting-points of
+geological investigation. The work of the chemical laboratory forms the
+foundation of a vast and increasing mass of geological inquiry. To the
+botanist, the zoologist, even to the unscientific, if observant,
+traveller by land or sea, the geologist turns for information and
+assistance.
+
+But while thus culling freely from the dominions of other sciences,
+geology claims as its peculiar territory the rocky framework of the
+globe. In the materials composing that framework, their composition and
+arrangement, the processes of their formation, the changes which they
+have undergone, and the terrestrial revolutions to which they bear
+witness, lie the main data of geological history. It is the task of the
+geologist to group these elements in such a way that they may be made to
+yield up their evidence as to the march of events in the evolution of
+the planet. He finds that they have in large measure arranged themselves
+in chronological sequence,--the oldest lying at the bottom and the
+newest at the top. Relics of an ancient sea-floor are overlain by traces
+of a vanished land-surface; these are in turn covered by the deposits of
+a former lake, above which once more appear proofs of the return of the
+sea. Among these rocky records lie the lavas and ashes of long-extinct
+volcanoes. The ripple left upon the shore, the cracks formed by the
+sun's heat upon the muddy bottom of a dried-up pool, the very imprint of
+the drops of a passing rainshower, have all been accurately preserved,
+and yield their evidence as to geographical conditions often widely
+different from those which exist where such markings are now found.
+
+But it is mainly by the remains of plants and animals imbedded in the
+rocks that the geologist is guided in unravelling the chronological
+succession of geological changes. He has found that a certain order of
+appearance characterizes these organic remains, that each great group of
+rocks is marked by its own special types of life, and that these types
+can be recognized, and the rocks in which they occur can be correlated
+even in distant countries, and where no other means of comparison would
+be possible. At one moment he has to deal with the bones of some large
+mammal scattered through a deposit of superficial gravel, at another
+time with the minute foraminifers and ostracods of an upraised
+sea-bottom. Corals and crinoids crowded and crushed into a massive
+limestone where they lived and died, ferns and terrestrial plants matted
+together into a bed of coal where they originally grew, the scattered
+shells of a submarine sand-bank, the snails and lizards which lived and
+died within a hollow-tree, the insects which have been imprisoned within
+the exuding resin of old forests, the footprints of birds and
+quadrupeds, the trails of worms left upon former shores--these, and
+innumerable other pieces of evidence, enable the geologist to realize in
+some measure what the faunas and floras of successive periods have been,
+and what geographical changes the site of every land has undergone.
+
+It is evident that to deal successfully with these varied materials, a
+considerable acquaintance with different branches of science is needful.
+Especially necessary is a tolerably wide knowledge of the processes now
+at work in changing the surface of the earth, and of at least those
+forms of plant and animal life whose remains are apt to be preserved in
+geological deposits, or which in their structure and habitat enable us
+to realize what their forerunners were. It has often been insisted that
+the present is the key to the past; and in a wide sense this assertion
+is eminently true. Only in proportion as we understand the present,
+where everything is open on all sides to the fullest investigation, can
+we expect to decipher the past, where so much is obscure, imperfectly
+preserved or not preserved at all. A study of the existing economy of
+nature ought thus to be the foundation of the geologist's training.
+
+While, however, the present condition of things is thus employed, we
+must obviously be on our guard against the danger of unconsciously
+assuming that the phase of nature's operations which we now witness has
+been the same in all past time, that geological changes have always or
+generally taken place in former ages in the manner and on the scale
+which we behold to-day, and that at the present time all the great
+geological processes, which have produced changes in the past eras of
+the earth's history, are still existent and active. As a working
+hypothesis we may suppose that the nature of geological processes has
+remained constant from the beginning; but we cannot postulate that the
+action of these processes has never varied in energy. The few centuries
+wherein man has been observing nature obviously form much too brief an
+interval by which to measure the intensity of geological action in all
+past time. For aught we can tell the present is an era of quietude and
+slow change, compared with some of the eras which have preceded it. Nor
+perhaps can we be quite sure that, when we have explored every
+geological process now in progress, we have exhausted all the causes of
+change which, even in comparatively recent times, have been at work.
+
+In dealing with the geological record, as the accessible solid part of
+the globe is called, we cannot too vividly realize that at the best it
+forms but an imperfect chronicle. Geological history cannot be compiled
+from a full and continuous series of documents. From the very nature of
+its origin the record is necessarily fragmentary, and it has been
+further mutilated and obscured by the revolutions of successive ages.
+And even where the chronicle of events is continuous, it is of very
+unequal value in different places. In one case, for example, it may
+present us with an unbroken succession of deposits many thousands of
+feet in thickness, from which, however, only a few meagre facts as to
+geological history can be gleaned. In another instance it brings before
+us, within the compass of a few yards, the evidence of a most varied and
+complicated series of changes in physical geography, as well as an
+abundant and interesting suite of organic remains. These and other
+characteristics of the geological record become more apparent and
+intelligible as we proceed in the study of the science.
+
+_Classification._--For systematic treatment the subject may be
+conveniently arranged in the following parts:--
+
+1. _The Historical Development of Geological Science._--Here a brief
+outline will be given of the gradual growth of geological conceptions
+from the days of the Greeks and Romans down to modern times, tracing the
+separate progress of the more important branches of inquiry and noting
+some of the stages which in each case have led up to the present
+condition of the science.
+
+2. _The Cosmical Aspects of Geology._--This section embraces the
+evidence supplied by astronomy and physics regarding the form and
+motions of the earth, the composition of the planets and sun, and the
+probable history of the solar system. The subjects dealt with under this
+head are chiefly treated in separate articles.
+
+3. _Geognosy._--An inquiry into the materials of the earth's substance.
+This division, which deals with the parts of the earth, its envelopes of
+air and water, its solid crust and the probable condition of its
+interior, especially treats of the more important minerals of the crust,
+and the chief rocks of which that crust is built up. Geognosy thus lays
+a foundation of knowledge regarding the nature of the materials
+constituting the mass of the globe, and prepares the way for an
+investigation of the processes by which these materials are produced and
+altered.
+
+4. _Dynamical Geology_ studies the nature and working of the various
+geological processes whereby the rocks of the earth's crust are formed
+and metamorphosed, and by which changes are effected upon the
+distribution of sea and land, and upon the forms of terrestrial
+surfaces. Such an inquiry necessitates a careful examination of the
+existing geological economy of nature, and forms a fitting introduction
+to an inquiry into the geological changes of former periods.
+
+5. _Geotectonic or Structural Geology_ has for its object the
+architecture of the earth's crust. It embraces an inquiry into the
+manner in which the various materials composing this crust have been
+arranged. It shows that some have been formed in beds or strata of
+sediment on the floor of the sea, that others have been built up by the
+slow aggregation of organic forms, that others have been poured out in a
+molten condition or in showers of loose dust from subterranean sources.
+It further reveals that, though originally laid down in almost
+horizontal beds, the rocks have subsequently been crumpled, contorted
+and dislocated, that they have been incessantly worn down, and have
+often been depressed and buried beneath later accumulations.
+
+6. _Palaeontological Geology._--This branch of the subject, starting
+from the evidence supplied by the organic forms which are found
+preserved in the crust of the earth, includes such questions as the
+relations between extinct and living types, the laws which appear to
+have governed the distribution of life in time and in space, the
+relative importance of different genera of animals in geological
+inquiry, the nature and use of the evidence from organic remains
+regarding former conditions of physical geography. Some of these
+problems belong also to zoology and botany, and are more fully discussed
+in the articles PALAEONTOLOGY and PALAEOBOTANY.
+
+7. _Stratigraphical Geology._--This section might be called geological
+history. It works out the chronological succession of the great
+formations of the earth's crust, and endeavours to trace the sequence of
+events of which they contain the record. More particularly, it
+determines the order of succession of the various plants and animals
+which in past time have peopled the earth, and thus ascertains what has
+been the grand march of life upon this planet.
+
+8. _Physiographical Geology_, proceeding from the basis of fact laid
+down by stratigraphical geology regarding former geographical changes,
+embraces an inquiry into the origin and history of the features of the
+earth's surface--continental ridges and ocean basins, plains, valleys
+and mountains. It explains the causes on which local differences of
+scenery depend, and shows under what very different circumstances, and
+at what widely separated intervals, the hills and mountains, even of a
+single country, have been produced.
+
+Most of the detail embraced in these several sections is relegated to
+separate articles, to which references are here inserted. The following
+pages thus deal mainly with the general principles and historical
+development of the science:--
+
+
+  PART I.--HISTORICAL DEVELOPMENT
+
+  _Geological Ideas among the Greeks and Romans._--Many geological
+  phenomena present themselves in so striking a form that they could
+  hardly fail to impress the imagination of the earliest and rudest
+  races of mankind. Such incidents as earthquakes and volcanic
+  eruptions, destructive storms on land and sea, disastrous floods and
+  landslips suddenly strewing valleys with ruin, must have awakened the
+  terror of those who witnessed them. Prominent features of landscape,
+  such as mountain-chains with their snows, clouds and thunderstorms,
+  dark river-chasms that seem purposely cleft open in order to give
+  passage to the torrents that rush through them, crags with their
+  impressive array of pinnacles and recesses must have appealed of old,
+  as they still do, to the awe and wonder of those who for the first
+  time behold them. Again, banks of sea-shells in far inland districts
+  would, in course of time, arrest the attention of the more intelligent
+  and reflective observers, and raise in their minds some kind of
+  surmise as to how such shells could ever have come there. These and
+  other conspicuous geological problems found their earliest solution in
+  legends and myths, wherein the more striking terrestrial features and
+  the elemental forces of nature were represented to be the
+  manifestation of the power of unseen supernatural beings.
+
+  The basin of the Mediterranean Sea was especially well adapted, from
+  its physical conditions, to be the birth-place of such fables. It is a
+  region frequently shaken by earthquakes, and contains two distinct
+  centres of volcanic activity, one in the Aegean Sea and one in Italy.
+  It is bounded on the north by a long succession of lofty snow-capped
+  mountain-ranges, whence copious rivers, often swollen by heavy rains
+  or melted snows, carry the drainage into the sea. On the south it
+  boasts the Nile, once so full of mystery; likewise wide tracts of arid
+  desert with their dreaded dust storms. The Mediterranean itself,
+  though an inland sea, is subject to gales, which, on exposed coasts,
+  raise breakers quite large enough to give a vivid impression of the
+  power of ocean waves. The countries that surround this great sheet of
+  water display in many places widely-spread deposits full of sea
+  shells, like those that still live in the neighbouring bays and gulfs.
+  Such a region was not only well fitted to supply subjects for
+  mythology, but also to furnish, on every side, materials which, in
+  their interest and suggestiveness, would appeal to the reason of
+  observant men.
+
+  It was natural, therefore, that the early philosophers of Greece
+  should have noted some of these geological features, and should have
+  sought for other explanations of them than those to be found in the
+  popular myths. The opinions entertained in antiquity on these subjects
+  may be conveniently grouped under two heads: (1) Geological processes
+  now in operation, and (2) geological changes in the past.
+
+
+    Earthquakes and volcanoes.
+
+  1. _Contemporary Processes._--The geological processes of the present
+  time are partly at work underground and partly on the surface of the
+  earth. The former, from their frequently disastrous character,
+  received much attention from Greek and Roman authors. Aristotle, in
+  his _Meteorics_, cites the speculations of several of his predecessors
+  which he rejects in favour of his own opinion to the effect that
+  earthquakes are due to the generation of wind within the earth, under
+  the influence of the warmth of the sun and the internal heat. Wind,
+  being the lightest and most rapidly moving body, is the cause of
+  motion in other bodies, and fire, united with wind, becomes flame,
+  which is endowed with great rapidity of motion. Aristotle looked upon
+  earthquakes and volcanic eruptions as closely connected with each
+  other, the discharge of hot materials to the surface being the result
+  of a severe earthquake, when finally the wind rushes out with
+  violence, and sometimes buries the surrounding country under sparks
+  and cinders, as had happened at Lipari. These crude conceptions of
+  the nature of volcanic action, and the cause of earthquakes, continued
+  to prevail for many centuries. They are repeated by Lucretius, who,
+  however, following Anaximenes, includes as one of the causes of
+  earthquakes the fall of mountainous masses of rock undermined by time,
+  and the consequent propagation of gigantic tremors far and wide
+  through the earth. Strabo, having travelled through the volcanic
+  districts of Italy, was able to recognize that Vesuvius had once been
+  an active volcano, although no eruption had taken place from it within
+  human memory. He continued to hold the belief that volcanic energy
+  arose from the movement of subterranean wind. He believed that the
+  district around the Strait of Messina, which had formerly suffered
+  from destructive earthquakes, was seldom visited by them after the
+  volcanic vents of that region had been opened, so as to provide an
+  escape for the subterranean fire, wind, water and burning masses. He
+  cites in his _Geography_ a number of examples of widespread as well as
+  local sinkings of land, and alludes also to the uprise of the
+  sea-bottom. He likewise regards some islands as having been thrown up
+  by volcanic agency, and others as torn from the mainland by such
+  convulsions as earthquakes.
+
+  The most detailed account of earthquake phenomena which has come down
+  to us from antiquity is that of Seneca in his _Quaestiones Naturales_.
+  This philosopher had been much interested in the accounts given him by
+  survivors and witnesses of the earthquake which convulsed the district
+  of Naples in February A.D. 63. He distinguished several distinct
+  movements of the ground: 1st, the up and down motion (_succussio_);
+  2nd, the oscillatory motion (_inclinatio_); and probably a third, that
+  of trembling or vibration. While admitting that some earthquakes may
+  arise from the collapse of the walls of subterranean cavities, he
+  adhered to the old idea, held by the most numerous and important
+  previous writers, that these commotions are caused mainly by the
+  movements of wind imprisoned within the earth. As to the origin of
+  volcanic outbursts he supposed that the subterranean wind in
+  struggling for an outlet, and whirling through the chasms and
+  passages, meets with great store of sulphur and other combustible
+  substances, which by mere friction are set on fire. The elder Pliny
+  reiterates the commonly accepted opinion as to the efficacy of wind
+  underground. In discussing the phenomena of earthquakes he remarks
+  that towns with many culverts and houses with cellars suffer less than
+  others, and that at Naples those houses are most shaken which stand on
+  hard ground. It thus appears that with regard to subterranean
+  geological operations, no advance was made during the time of the
+  Greeks and Romans as to the theoretical explanation of these
+  phenomena; but a considerable body of facts was collected, especially
+  as to the effects of earthquakes and the occurrence of volcanic
+  eruptions.
+
+
+    Action of rivers.
+
+  The superficial processes of geology, being much less striking than
+  those of subterranean energy, naturally attracted less attention in
+  antiquity. The operations of rivers, however, which so intimately
+  affect a human population, were watched with more or less care.
+  Herodotus, struck by the amount of alluvial silt brought down annually
+  by the Nile and spread over the flat inundated land, inferred that
+  "Egypt is the gift of the river." Aristotle, in discussing some of the
+  features of rivers, displays considerable acquaintance with the
+  various drainage-systems on the north side of the Mediterranean basin.
+  He refers to the mountains as condensers of the atmospheric moisture,
+  and shows that the largest rivers rise among the loftiest high
+  grounds. He shows how sensibly the alluvial deposits carried down to
+  the sea increase the breadth of the land, and cites some parts of the
+  shores of the Black Sea, where, in sixty years, the rivers had brought
+  down such a quantity of material that the vessels then in use required
+  to be of much smaller draught than previously, the water shallowing so
+  much that the marshy ground would, in course of time, become dry land.
+  Strabo supplies further interesting information as to the work of
+  rivers in making their alluvial plains and in pushing their deltas
+  seaward. He remarks that these deltas are prevented from advancing
+  farther outward by the ebb and flow of the tides.
+
+
+    Occurrences of fossils.
+
+  2. _Past Processes._--The abundant well-preserved marine shells
+  exposed among the upraised Tertiary and post-Tertiary deposits in the
+  countries bordering the Mediterranean are not infrequently alluded to
+  in Greek and Latin literature. Xenophanes of Colophon (614 B.C.)
+  noticed the occurrence of shells and other marine productions inland
+  among the mountains, and inferred from them that the land had risen
+  out of the sea. A similar conclusion was drawn by Xanthus the Lydian
+  (464 B.C.) from shells like scallops and cockles, which were found far
+  from the sea in Armenia and Lower Phrygia. Herodotus, Eratosthenes,
+  Strato and Strabo noted the vast quantities of fossil shells in
+  different parts of Egypt, together with beds of salt, as evidence that
+  the sea had once spread over the country. But by far the most
+  philosophical opinions on the past mutations of the earth's surface
+  are those expressed by Aristotle in the treatise already cited.
+  Reviewing the evidence of these changes, he recognized that the sea
+  now covers tracts that were once dry land, and that land will one day
+  reappear where there is now sea. These alternations are to be regarded
+  as following each other in a certain order and periodicity. But they
+  are apt to escape our notice because they require successive periods
+  of time, which, compared with our brief existence, are of enormous
+  duration, and because they are brought about so imperceptibly that we
+  fail to detect them in progress. In a celebrated passage in his
+  _Metamorphoses_, Ovid puts into the mouth of the philosopher
+  Pythagoras an account of what was probably regarded as the Pythagorean
+  view of the subject in the Augustan age. It affirms the interchange of
+  land and sea, the erosion of valleys by descending rivers, the washing
+  down of mountains into the sea, the disappearance of the rivers and
+  the submergence of land by earthquake movements, the separation of
+  some islands from, and the union of others with, the mainland, the
+  uprise of hills by volcanic action, the rise and extinction of burning
+  mountains. There was a time before Etna began to glow, and the time is
+  coming when the mountain will cease to burn.
+
+  From this brief sketch it will be seen that while the ancients had
+  accumulated a good deal of information regarding the occurrence of
+  geological changes, their interpretations of the phenomena were to a
+  considerable extent mere fanciful speculation. They had acquired only
+  a most imperfect conception of the nature and operation of the
+  geological processes; and though many writers realized that the
+  surface of the earth has not always been, and will not always remain,
+  as it is now, they had no glimpse of the vast succession of changes of
+  that surface which have been revealed by geology. They built
+  hypotheses on the slenderest basis of fact, and did not realize the
+  necessity of testing or verifying them.
+
+  _Progress of Geological Conceptions in the Middle Ages._--During the
+  centuries that succeeded the fall of the Western empire little
+  progress was made in natural science. The schoolmen in the monasteries
+  and other seminaries were content to take their science from the
+  literature of Greece and Rome. The Arabs, however, not only collected
+  and translated that literature, but in some departments made original
+  observations themselves. To one of the most illustrious of their
+  number, Avicenna, the translator of Aristotle, a treatise has been
+  ascribed, in which singularly modern ideas are expressed regarding
+  mountains, some of which are there stated to have been produced by an
+  uplifting of the ground, while others have been left prominent, owing
+  to the wearing away of the softer rocks around them. In either case,
+  it is confessed that the process would demand long tracts of time for
+  its completion.
+
+  After the revival of learning the ancient problem presented by fossil
+  shells imbedded in the rocks of the interior of many countries
+  received renewed attention. But the conditions for its solution were
+  no longer what they had been in the days of the philosophers of
+  antiquity. Men were not now free to adopt and teach any doctrine they
+  pleased on the subject. The Christian church had meanwhile arisen to
+  power all over Europe, and adjudged as heretics all who ventured to
+  impugn any of her dogmas. She taught that the land and the sea had
+  been separated on the third day of creation, before the appearance of
+  any animal life, which was not created until the fifth day. To assert
+  that the dry land is made up in great part of rocks that were formed
+  in the sea, and are crowded with the remains of animals, was plainly
+  to impugn the veracity of the Bible. Again, it had come to be the
+  orthodox belief that only somewhere about 6000 years had elapsed since
+  the time of Adam and Eve. If any thoughtful observer, impressed with
+  the overwhelming force of the evidence that the fossiliferous
+  formations of the earth's crust must have taken long periods of time
+  for their accumulation, ventured to give public expression to his
+  conviction, he ran considerable risk of being proceeded against as a
+  heretic. It was needful, therefore, to find some explanation of the
+  facts of nature, which would not run counter to the ecclesiastical
+  system of the day. Various such interpretations were proposed,
+  doubtless in an honest endeavour at reconciliation. Three of these
+  deserve special notice: (1) Many able observers and diligent
+  collectors of fossils persuaded themselves that these objects never
+  belonged to organisms of any kind, but should be regarded as mere
+  "freaks of nature," having no more connexion with any once living
+  creature than the frost patterns on a window. They were styled
+  "formed" or "figured" stones, "lapides sui generis," and were asserted
+  to be due to some inorganic imitative process within the earth or to
+  the influence of the stars. (2) Observers who could not resist the
+  evidence of their senses that the fossil shells once belonged to
+  living animals, and who, at the same time, felt the necessity of
+  accounting for the presence of marine organisms in the rocks of which
+  the dry land is largely built up, sought a way out of the difficulty
+  by invoking the Deluge of Noah. Here was a catastrophe which, they
+  said, extended over the whole globe, and by which the entire dry land
+  was submerged even up to the tops of the high hills. True, it only
+  lasted one hundred and fifty days, but so little were the facts then
+  appreciated that no difficulty seems to have been generally felt in
+  crowding the accumulation of the thousands of feet of fossiliferous
+  formations into that brief space of time. (3) Some more intelligent
+  men in Italy, recognizing that these interpretations could not be
+  upheld, fell back upon the idea that the rocks in which fossil shells
+  are imbedded might have been heaped up by repeated and vigorous
+  eruptions from volcanic centres. Certain modern eruptions in the
+  Aegean Sea and in the Bay of Naples had drawn attention to the
+  rapidity with which hills of considerable size could be piled around
+  an active crater. It was argued that if Monte Nuovo near Naples could
+  have been accumulated to a height of nearly 500 ft. in two days, there
+  seemed to be no reason against believing that, during the time of the
+  Flood, and in the course of the centuries that have elapsed since
+  that event, the whole of the fossiliferous rocks might have been
+  deposited. Unfortunately for this hypothesis it ignored the fact that
+  these rocks do not consist of volcanic materials.
+
+
+    Leonardo da Vinci; Fracastorio; Falloppio.
+
+  So long as the fundamental question remained in dispute as to the true
+  character and history of the stratified portion of the earth's crust
+  containing organic remains, geology as a science could not begin its
+  existence. The diluvialists (those who relied on the hypothesis of the
+  Flood) held the field during the 16th, 17th and a great part of the
+  18th century. They were looked on as the champions of orthodoxy; and,
+  on that account, they doubtless wielded much more influence than would
+  have been gained by them from the force of their arguments. Yet during
+  those ages there were not wanting occasional observers who did good
+  service in combating the prevalent misconceptions, and in preparing
+  the way for the ultimate triumph of truth. It was more especially in
+  Italy, where many of the more striking phenomena of geology are
+  conspicuously displayed, that the early pioneers of the science arose,
+  and that for several generations the most marked progress was made
+  towards placing the investigations of the past history of the earth
+  upon a basis of careful observation and scientific deduction. One of
+  the first of these leaders was Leonardo da Vinci (1452-1519), who,
+  besides his achievements in painting, sculpture, architecture and
+  engineering, contributed some notable observations regarding the great
+  problem of the origin of fossil shells. He ridiculed the notion that
+  these objects could have been formed by the influence of the stars,
+  and maintained that they had once belonged to living organisms, and
+  therefore that what is now land was formerly covered by the sea.
+  Girolamo Fracastorio (1483-1553) claimed that the shells could never
+  have been left by the Flood, which was a mere temporary inundation,
+  but that they proved the mountains, in which they occur, to have been
+  successively uplifted out of the sea. On the other hand, even an
+  accomplished anatomist like Gabriello Falloppio (1523-1562) found it
+  easier to believe that the bones of elephants, teeth of sharks, shells
+  and other fossils were mere earthy inorganic concretions, than that
+  the waters of Noah's Flood could ever nave reached as far as Italy.
+
+
+    Nicolas Steno.
+
+  By much the most important member of this early band of Italian
+  writers was undoubtedly Nicolas Steno (1631-1687), who, though born in
+  Copenhagen, ultimately settled in Florence. Having made a European
+  reputation as an anatomist, his attention was drawn to geological
+  problems by finding that the rocks of the north of Italy contained
+  what appeared to be sharks' teeth closely resembling those of a
+  dog-fish, of which he had published the anatomy. Cautiously at first,
+  for fear of offending orthodox opinions, but afterwards more boldly,
+  he proclaimed his conviction that those objects had once been part of
+  living animals, and that they threw light on some of the past history
+  of the earth. He published in 1669 a small tract, _De solido intra
+  solidum naturaliter contento_, in which he developed the ideas he had
+  formed of this history from an attentive study of the rocks. He showed
+  that the stratified formations of the hills and valleys consist of
+  such materials as would be laid down in the form of sediment in turbid
+  water; that where they contain marine productions this water is proved
+  to have been the sea; that diversities in their composition point to
+  commingling of currents, carrying different kinds of sediment of which
+  the heaviest would first sink to the bottom. He made original and
+  important observations on stratification, and laid down some of the
+  fundamental axioms in stratigraphy. He reasoned that as the original
+  position of strata was approximately horizontal, when they are found
+  to be steeply inclined or vertical, or bent into arches, they have
+  been disrupted by subterranean exhalations, or by the falling in of
+  the roofs of underground cavernous spaces. It is to this alteration of
+  the original position of the strata that the inequalities of the
+  earth's surface, such as mountains, are to be ascribed, though some
+  have been formed by the outburst of fire, ashes and stones from inside
+  the earth. Another effect of the dislocation has been to provide
+  fissures, which serve as outlets for springs. Steno's anatomical
+  training peculiarly fitted him for dealing authoritatively with the
+  question of the nature and origin of the fossils contained in the
+  rocks. He had no hesitation in affirming that, even if no shells had
+  ever been found living in the sea, the internal structure of these
+  fossils would demonstrate that they once formed parts of living
+  animals. And not only shells, but teeth, bones and skeletons of many
+  kinds of fishes had been quarried out of the rocks, while some of the
+  strata had skulls, horns and teeth of land-animals. Illustrating his
+  general principles by a sketch of what he supposed to have been the
+  past history of Tuscany, he added a series of diagrams which show how
+  clearly he had conceived the essential elements of stratigraphy. He
+  thought he could perceive the records of six successive phases in the
+  evolution of the framework of that country, and was inclined to
+  believe that a similar chronological sequence would be found all over
+  the world. He anticipated the objections that would be brought against
+  his views on account of the insuperable difficulty in granting the
+  length of time that would be required for all the geographical
+  vicissitudes which his interpretation required. He thought that many
+  of the fossils must be as old as the time of the general deluge, but
+  he was careful not to indulge in any speculation as to the antiquity
+  of the earth.
+
+
+    Lazzaro Moro.
+
+  To the Italian school, as especially typified in Steno, must be
+  assigned the honour of having thus begun to lay firmly and truly the
+  first foundation stones of the modern science of geology. The same
+  school included Antonio Vallisneri (1661-1730), who surpassed his
+  predecessors in his wider and more exact knowledge of the
+  fossiliferous rocks that form the backbone of the Italian peninsula,
+  which he contended were formed during a wide and prolonged submergence
+  of the region, altogether different from the brief deluge of Noah.
+  There was likewise Lazzaro Moro (1687-1740), who did good service
+  against the diluvialists, but the fundamental feature of his system of
+  nature lay in the preponderant part which, unaware of the great
+  difference between volcanic materials and ordinary sediment, he
+  assigned to volcanic action in the production of the sedimentary rocks
+  of the earth's crust. He supposed that in the beginning the globe was
+  completely surrounded with water, beneath which the solid earth lay as
+  a smooth ball. On the third day of creation, however, vast fires were
+  kindled inside the globe, whereby the smooth surface of stone was
+  broken up, and portions of it, appearing above the water, formed the
+  earliest land. From that time onward, volcanic eruptions succeeded
+  each other, not only on the emerged land, but on the sea-floor, over
+  which the ejected material spread in an ever augmenting thickness of
+  sedimentary strata. In this way Moro carried the history of the
+  stratified rocks beyond the time of the Flood back to the Creation,
+  which was supposed to have been some 1600 years earlier; and he
+  brought it down to the present day, when fresh sedimentary deposits
+  are continually accumulating. He thus incurred no censure from the
+  ecclesiastical guardians of the faith, and he succeeded in attracting
+  increased public attention to the problems of geology. The influence
+  of his teaching, however, was subsequently in great part due to the
+  Carmelite friar Generelli, who published an eloquent exposition of
+  Moro's views.
+
+  _The Cosmogonists and Theories of the Earth._--While in Italy
+  substantial progress was made in collecting information regarding the
+  fossiliferous formations of that country, and in forming conclusions
+  concerning them based upon more or less accurate observations, the
+  tendency to mere fanciful speculation, which could not be wholly
+  repressed in any country, reached a remarkable extravagance in
+  England. In proportion as materials were yet lacking from which to
+  construct a history of the evolution of our planet in accordance with
+  the teaching of the church, imagination supplied the place of
+  ascertained fact, and there appeared during the last twenty years of
+  the 18th century a group of English cosmogonists, who, by the
+  sensational character of their speculations, aroused general attention
+  both in Britain and on the continent. It may be doubted, however,
+  whether the effect of their writings was not to hinder the advance of
+  true science by diverting men from the observation of nature into
+  barren controversy over unrealities. It is not needful here to do more
+  than mention the names of Thomas Burnet, whose _Sacred Theory of the
+  Earth_ appeared in 1681, and William Whiston, whose New Theory of the
+  Earth was published in 1696. Hardly less fanciful than these writers,
+  though his practical acquaintance with rocks and fossils was
+  infinitely greater, was John Woodward, whose _Essay towards a Natural
+  History of the Earth_ dates from 1695. More important as a
+  contribution to science was the catalogue of the large collection of
+  fossils, which he had made from the rocks of England and which he
+  bequeathed to the university of Cambridge. This catalogue appeared in
+  1728-1729 with the title of _An attempt towards a Natural History of
+  the Fossils of England_.
+
+
+    Descartes.
+
+  A striking contrast to these cosmogonists is furnished by another
+  group, which arose in France and Germany, and gave to the world the
+  first rational ideas concerning the probable primeval evolution of our
+  globe. The earliest of these pioneers was the illustrious philosopher
+  René Descartes (1596-1650). He propounded a scheme of cosmical
+  development in which he represented the earth, like the other planets,
+  to have been originally a mass of glowing material like the sun, and
+  to have gradually cooled on the outside, while still retaining an
+  incandescent, self-luminous nucleus. Yet with this noble conception,
+  which modern science has accepted, Descartes could not shake himself
+  free from the time-honoured error in regard to the origin of volcanic
+  action. He thought that certain exhalations within the earth condense
+  into oil, which, when in violent motion, enters into the subterranean
+  cavities, where it passes into a kind of smoke. This smoke is from
+  time to time ignited by a spark of fire and, pressing violently
+  against its containing walls, gives rise to earthquakes. If the flame
+  breaks through to the surface at the top of a mountain, it may escape
+  with enormous energy, hurling forth much earth mingled with sulphur or
+  bitumen, and thus producing a volcano. The mountain might burn for a
+  long time until at last its store of fuel in the shape of sulphur or
+  bitumen would be exhausted. Not only did the philosopher refrain from
+  availing himself of the high internal temperature of the globe as the
+  source of volcanic energy, he even did not make use of it as the cause
+  of the ignition of his supposed internal fuel, but speculated on the
+  kindling of the subterranean fires by the spirits or gases setting
+  fire to the exhalations, or by the fall of masses of rock and the
+  sparks produced by their friction or percussion.
+
+
+    Leibnitz.
+
+  The ideas of Descartes regarding planetary evolution were enlarged and
+  made more definite by Wilhelm Gottfried Leibnitz (1646-1716), whose
+  teaching has largely influenced all subsequent speculation on the
+  subject. In his great tract, the _Protogaea_ (published in 1749,
+  thirty-three years after his death), he traced the probable passage of
+  our earth from an original condition of incandescent vapour into that
+  of a smooth molten globe, which, by continuous cooling, acquired an
+  external solid crust and rugose surface. He thought that the more
+  ancient rocks, such as granite and gneiss, might be portions of the
+  earliest outer crust; and that as the external solidification
+  advanced, immense subterranean cavities were left which were filled
+  with air and water. By the collapse of the roofs of these caverns,
+  valleys might be originated at the surface, while the solid
+  intervening walls would remain in place and form mountains. By the
+  disruption of the crust, enormous bodies of water were launched over
+  the surface of the earth, which swept vast quantities of sediment
+  together, and thus gave rise to sedimentary deposits. After many
+  vicissitudes of this kind, the terrestrial forces calmed down, and a
+  more stable condition of things was established.
+
+  An important feature in the cosmogony of Leibnitz is the prominent
+  place which he assigned to organic remains in the stratified rocks of
+  the crust. Ridiculing the foolish attempts to account for the presence
+  of these objects by calling them "sports of nature," he showed that
+  they are to be regarded as historical monuments; and he adduced a
+  number of instances wherein successive platforms of strata, containing
+  organic remains, bear witness to a series of advances and retreats of
+  the sea. He recognized that some of the fossils appeared to have
+  nothing like them in the living world of to-day, but some analogous
+  forms might yet be found, he thought, in still unexplored parts of the
+  earth; and even if no living representatives should ever be
+  discovered, many types of animals might have undergone transformation
+  during the great changes which had affected the surface of the earth.
+  In spite of his clear realization of the vast store of potential
+  energy residing within the highly heated interior of the earth,
+  Leibnitz continued to regard volcanic action as due to the combustion
+  of inflammable substances enclosed within the terrestrial crust, such
+  as stone-coal, naphtha and sulphur.
+
+
+    Buffon.
+
+  Appealing to a much wider public than Descartes or Leibnitz, and
+  basing his speculations on a wider acquaintance with the organic and
+  inorganic realms of nature, G.L.L. de Buffon (1707-1788) was
+  undoubtedly one of the most influential forces that in Europe guided
+  the growth of geological ideas during the 18th century. He published
+  in 1749 a _Theory of the Earth_, in which he adopted views similar to
+  those of Descartes and Leibnitz as to planetary evolution; but though
+  he realized the importance of fossils as records of former conditions
+  of the earth's surface, he accounted for them by supposing that they
+  had been deposited from a universal ocean, a large part of which had
+  subsequently been engulfed into caverns in the interior of the globe.
+  Thirty years later, after having laboured with skill and enthusiasm in
+  all branches of natural history, he published another work, his famous
+  _Époques de la nature_ (1778), which is specially remarkable as the
+  first attempt to deal with the history of the earth in a chronological
+  manner, and to compute, on a basis of experiment, the antiquity of the
+  several stages of this history. His experiments were made with globes
+  of cast iron, and could not have yielded results of any value for his
+  purpose; but in so far as his calculations were not mere random
+  guesses but had some kind of foundation on experiment, they deserve
+  respectful recognition. He divided the history of our earth into six
+  periods of unequal duration, the whole comprising a period of some
+  70,000 or 75,000 years. He supposed that the stage of incandescence,
+  before the globe had consolidated to the centre, lasted 2936 years,
+  and that about 35,000 years elapsed before the surface had cooled
+  sufficiently to be touched, and therefore to be capable of supporting
+  living things. Terrestrial animal life, however, was not introduced
+  until 55,000 or 60,000 years after the beginning of the world or about
+  15,000 years before our time. Looking into the future, he foresaw
+  that, by continued refrigeration, our globe will eventually become
+  colder than ice, and this fair face of nature, with its manifold
+  varieties of plant and animal life, will perish after having existed
+  for 132,000 years.
+
+  Buffon's conception of the operation of the geological agents did not
+  become broader or more accurate in the interval between the appearance
+  of his two treatises. He still continued to believe in the lowering of
+  the ocean by subsidence into vast subterranean cavities, with a
+  consequent emergence of land. He still looked on volcanoes as due to
+  the burning of "pyritous and combustible stones," though he now called
+  in the co-operation of electricity. He calculated that the first
+  volcanoes could not arise until some 50,000 years after the beginning
+  of the world, by which time a sufficient extent of dense vegetation
+  had been buried in the earth to supply them with fuel. He appears to
+  have had but an imperfect acquaintance with the literature of his own
+  time. At least there can be little doubt that had he availed himself
+  of the labours of his own countryman, Jean Etienne Guettard
+  (1715-1786), of Giovanni Arduķno (1714-1795) in Italy, and of Johann
+  Gottlob Lehmann (d. 1767) and George Christian Füchsel (1722-1773) in
+  Germany, he would have been able to give to his "epochs" a more
+  definite succession of events and a greater correspondence with the
+  facts of nature.
+
+
+    James Hutton.
+
+  Among the writers of the 18th century, who formed philosophical
+  conceptions of the system of processes by which the life of our earth
+  as a habitable globe is carried on, a foremost place must be assigned
+  to James Hutton (1726-1797). Educated for the medical profession, he
+  studied at Edinburgh and at Paris, and took his doctor's degree at
+  Leiden. But having inherited a small landed property in Berwickshire,
+  he took to agriculture, and after putting his land into excellent
+  order, let his farm and betook himself to Edinburgh, there to gratify
+  the scientific tastes which he had developed early in life. He had
+  been more especially led to study minerals and rocks, and to meditate
+  on the problems which they suggest as to the constitution and history
+  of the earth. His journeys in Britain and on the continent of Europe
+  had furnished him with material for reflection; and he had gradually
+  evolved a system or theory in which all the scattered facts could be
+  arranged so as to show their mutual dependence and their place in the
+  orderly mechanism of the world. He used to discuss his views with one
+  or two of his friends, but refrained from publishing them to the world
+  until, on the foundation of the Royal Society of Edinburgh, he
+  communicated an outline of his doctrine to that learned body in 1785.
+  Some years later he expanded this first essay into a larger work in
+  two volumes, which were published in 1795 with the title of _Theory of
+  the Earth, with Proofs and Illustrations_.
+
+
+    John Playfair.
+
+  Hutton's teaching has exercised a profound influence on modern
+  geology. This influence, however, has arisen less from his own
+  writings than from the account of his doctrines given by his friend
+  John Playfair in the classic work entitled _Illustrations of the
+  Huttonian Theory_, published in 1802. Hutton wrote in so prolix and
+  obscure a style as rather to repel than attract readers. Playfair, on
+  the other hand, expressed himself in such clear and graceful language
+  as to command general attention, and to gain wide acceptance for his
+  master's views. Unlike the older cosmogonists, Hutton refrained from
+  trying to explain the origin of things, and from speculations as to
+  what might possibly have been the early history of our globe. He
+  determined from the outset to interpret the past by what can be seen
+  to be the present order of nature; and he refused to admit the
+  operation of causes which cannot be shown to be part of the actual
+  terrestrial system. Like other observers who had preceded him, he
+  recognized in the various rocks composing the dry land evidence of
+  former geographical conditions very different from those which now
+  prevail. He saw that the vast majority of rocks consist of hardened
+  sediments and must have been deposited in the sea. He could
+  distinguish among them an older or Primary series, and a younger or
+  Secondary series; and did not dispute the existence of a Tertiary
+  series claimed by Peter Simon Pallas (1741-1811). He believed that
+  these various aqueous accumulations had been consolidated by
+  subterranean heat, that the oldest and lowest rocks had suffered most
+  from this action, that into these more deep-seated masses subsequent
+  veins and larger bodies of molten matter were injected from below, and
+  thus that what was originally loose detritus eventually became changed
+  in such crystalline schists as are now found in mountain-chains. In
+  the course of these terrestrial revolutions sedimentary strata,
+  originally more or less nearly horizontal, have been pushed upward,
+  dislocated, crumpled, placed on end, and even elevated to form ranges
+  of lofty mountains. Hutton looked upon these disturbances as due to
+  the expansive power of subterranean heat; but he did not attempt to
+  sketch the mechanism of the process, and he expressly declined to
+  offer any conjecture as to how the land so elevated remains in that
+  position. He thought that the interior of our planet may "be a fluid
+  mass, melted, but unchanged by the action of heat"; and, far from
+  connecting volcanoes with the combustion of inflammable substances, as
+  had been the prevalent belief for so many centuries, he looked upon
+  them as a beneficent provision of "spiracles to the subterranean
+  furnace, in order to prevent the unnecessary elevation of land and
+  fatal effects of earthquakes."
+
+  A distinguishing feature of the Huttonian philosophy is to be seen in
+  the breadth of its conceptions regarding the geological operations
+  continually in progress on the surface of the globe. Hutton saw that
+  the land is undergoing a ceaseless process of degradation, through the
+  influence of the air, frost, rain, rivers and the sea, and that in
+  course of time, if no countervailing agency should intervene, the
+  whole of the dry land will be washed away into the sea. But he also
+  perceived that this universal erosion is not everywhere carried on at
+  the same rate; that it is specially active along the channels of
+  torrents and rivers, and that, owing to this difference these channels
+  are gradually deepened and widened, until the complicated
+  valley-system of a country is carved out. He recognized that the
+  detritus worn away from the land must be spread out over the floor of
+  the sea, so as to form there strata similar to those that compose most
+  of the dry land. As he could detect in the structure of land
+  convincing evidence that former sea floors had been elevated to form
+  the continents and islands of to-day, he could look forward to future
+  ages, when the same subterranean agency which had raised up the
+  present land would again be employed to uplift the bed of the existing
+  ocean, thus to renew the surface of our earth as a habitable globe,
+  and to start a fresh cycle of erosion and deposition.
+
+
+    Lamarck.
+
+  Though Hutton was not unaware that organic remains abound in many of
+  the stratified rocks, he left them out of consideration in the
+  elaboration of his theory. It was otherwise with one of his French
+  contemporaries, the illustrious J.B. Lamarck (1744-1829), who, after
+  having attained great eminence as a botanist, turned to zoology when
+  he was nearly fifty years of age, and before long rose to even greater
+  distinction in that department of science. His share in the
+  classification and description of the mollusca and in founding
+  invertebrate palaeontology, his theory of organic evolution and his
+  philosophical treatment of many biological questions have been tardily
+  recognized, but his contributions to geology have been less generally
+  acknowledged. When he accepted the "professorship of zoology; of
+  insects, of worms and of microscopic animals" at the Museum of Natural
+  History, Paris, in 1793, he at once entered with characteristic ardour
+  and capacity into the new field of research then opened to him. In
+  dealing with the mollusca he considered not merely the living but also
+  the extinct forms, especially the abundant, varied and well-preserved
+  genera and species furnished by the Tertiary deposits of the Paris
+  basin, of which he published descriptions and plates that proved of
+  essential service in the stratigraphical work of Cuvier and Alexandre
+  Brongniart (1770-1847). His labours among these relics of ancient seas
+  and lakes led him to ponder over the past history of the globe, and as
+  he was seldom dilatory in making known the opinions he had formed, he
+  communicated some of his conclusions to the National Institute in
+  1799. These, including a further elaboration of his views, he
+  published in 1802 in a small volume entitled Hydrogéologie.
+
+  This treatise, though it did not reach a second edition and has never
+  been reprinted, deserves an honourable place in geological literature.
+  Its object, the author states, was to present some important and novel
+  considerations, which he thought should form the basis of a true
+  theory of the earth. He entirely agreed with the doctrine of the
+  subaerial degradation of the land and the erosion of valleys by
+  running water. Not even Playfair could have stated this doctrine more
+  emphatically, and it is worthy of notice that Playfair's
+  _Illustrations of the Huttonian Theory_ appeared in the same year with
+  Lamarck's book. The French naturalist, however, carried his
+  conclusions so far as to take no account of any great movements of the
+  terrestrial crust, which might have produced or modified the main
+  physical features of the surface of the globe. He thought that all
+  mountains, except such as were thrown up by volcanic agency or local
+  accidents, have been cut out of plains, the original surfaces of which
+  are indicated by the crests and summits of these elevations.
+
+  Lamarck, in reflecting upon the wide diffusion of fossil shells and
+  the great height above the sea at which they are found, conceived the
+  extraordinary idea that the ocean basin has been scoured out by the
+  sea, and that, by an impulse communicated to the waters through the
+  influence chiefly of the moon, the sea is slowly eating away the
+  eastern margins of the continents, and throwing up detritus on their
+  western coasts, and is thus gradually shifting its basin round the
+  globe. He would not admit the operation of cataclysms; but insisted as
+  strongly as Hutton on the continuity of natural processes, and on the
+  necessity of explaining former changes of the earth's surface by
+  causes which can still be seen to be in operation. As might be
+  anticipated from his previous studies, he brought living things and
+  their remains into the forefront of his theory of the earth. He looked
+  upon fossils as one of the chief means of comprehending the
+  revolutions which the surface of the earth has undergone; and in his
+  little volume he again and again dwells on the vast antiquity to which
+  these revolutions bear witness. He acutely argues, from the condition
+  of fossil shells, that they must have lived and died where their
+  remains are now found.
+
+  In the last part of his treatise Lamarck advances some peculiar
+  opinions in physics and chemistry, which he had broached eighteen
+  years before, but which had met with no acceptance among the
+  scientific men of his time. He believed that the tendency of all
+  compound substances is to decay, and thereby to be resolved into their
+  component constituents. Yet he saw that the visible crust of the earth
+  consists almost wholly of compound bodies. He therefore set himself to
+  solve the problem thus presented. Perceiving that the biological
+  action of living organisms is constantly forming combinations of
+  matter, which would never have otherwise come into existence, he
+  proceeded to draw the extraordinary conclusion that the action of
+  plant and animal life (the _Pouvoir de la vie_) upon the inorganic
+  world is so universal and so potent, that the rocks and minerals which
+  form the outer part of the earth's crust are all, without exception,
+  the result of the operations of once living bodies. Though this
+  sweeping deduction must be allowed to detract from the value of
+  Lamarck's work, there can be no doubt that he realized, more fully
+  than any one had done before him, the efficacy of plants and animals
+  as agents of geological change.
+
+
+    Cuvier.
+
+  The last notable contributor to the cosmological literature of geology
+  was another illustrious Frenchman, the comparative anatomist Cuvier
+  (1769-1832). He was contemporary with Lamarck, but of a very different
+  type of mind. The brilliance of his speculations, and the charm with
+  which he expounded them, early gained for him a prominent place in the
+  society of Paris. He too was drawn by his zoological studies to
+  investigate fossil organic remains, and to consider the former
+  conditions of the earth's surface, of which they are memorials. It was
+  among the vertebrate organisms of the Paris basin that he found his
+  chief material, and from them that he prepared the memoirs which led
+  to him being regarded as the founder of vertebrate palaeontology. But
+  beyond their biological interest, they awakened in him a keen desire
+  to ascertain the character and sequence of the geographical
+  revolutions to which they bear witness. He approached the subject from
+  an opposite and less philosophical point of view than that of Lamarck,
+  coming to it with certain preconceived notions, which affected all
+  his subsequent writings. While Lamarck was by instinct an
+  evolutionist, who sought to trace in the history of the past the
+  operation of the same natural processes as are still at work, Cuvier,
+  on the other hand, was a catastrophist, who invoked a succession of
+  vast cataclysms to account for the interruptions in the continuity of
+  the geological record.
+
+  In a preliminary _Discourse_ prefixed to his _Recherches sur les
+  ossemens fossiles_ (1821) Cuvier gave an outline of what he conceived
+  to have been the past history of our globe, so far as he had been able
+  to comprehend it from his investigations of the Tertiary formations of
+  France. He believed that in that history evidence can be recognized of
+  the occurrence of many sudden and disastrous revolutions, which, to
+  judge from their effects on the animal life of the time, must have
+  exceeded in violence anything we can conceive at the present day, and
+  must have been brought about by other agencies than those which are
+  now in operation. Yet, in spite of these catastrophes, he saw that
+  there has been an upward progress in the animal forms inhabiting the
+  globe, until the series ended in the advent of man. He could not,
+  however, find any evidence that one species has been developed from
+  another, for in that case there should have been traces of
+  intermediate forms among the stratified formations, where he affirmed
+  that they had never been found. A prominent position in the
+  _Discourse_ is given to a strenuous argument to disprove the alleged
+  antiquity of some nations, and to show that the last great catastrophe
+  occurred not more than some 5000 or 6000 years ago. Cuvier thus linked
+  himself with those who in previous generations had contended for the
+  efficacy of the Deluge. But his researches among fossil animals had
+  given him a far wider outlook into the geological past, and had opened
+  up to him a succession of deeply interesting problems in the history
+  of life upon the earth, which, though he had not himself material for
+  their solution, he could foresee would be cleared up in the future.
+
+  _Gradual Shaping of Geology into a Distinct Branch of Science._--It
+  will be seen from the foregoing historical sketch that it was only
+  after the lapse of long centuries, and from the labours of many
+  successive generations of observers and writers, that what we now know
+  as the science of geology came to be recognized as a distinct
+  department of natural knowledge, founded upon careful and extended
+  study of the structure of the earth, and upon observation of the
+  natural processes, which are now at work in changing the earth's
+  surface. The term "geology,"[1] descriptive of this branch of the
+  investigation of nature, was not proposed until the last quarter of
+  the 18th century by Jean André De Luc (1727-1817) and Horace Benedict
+  De Saussure (1740-1749). But the science was then in a markedly
+  half-formed condition, theoretical speculation still in large part
+  supplying the place of deductions from a detailed examination of
+  actual fact. In 1807 a few enterprising spirits founded the Geological
+  Society of London for the special purpose of counteracting the
+  prevalent tendency and confining their intention "to investigate the
+  mineral structure of the earth." The cosmogonists and framers of
+  Theories of the Earth were succeeded by other schools of thought. The
+  Catastrophists saw in the composition of the crust of the earth
+  distinct evidence that the forces of nature were once much more
+  stupendous in their operation than they now are, and that they had
+  from time to time devastated the earth's surface; extirpating the
+  races of plants and animals, and preparing the ground for new
+  creations of organized life. Then came the Uniformitarians, who,
+  pushing the doctrines of Hutton to an extreme which he did not
+  propose, saw no evidence that the activity of the various geological
+  causes has ever seriously differed from what it is at present. They
+  were inclined to disbelieve that the stratified formations of the
+  earth's crust furnish conclusive evidence of a gradual progression,
+  from simple types of life in the oldest strata to the most highly
+  developed forms in the youngest; and saw no reason why remains of the
+  higher vertebrates should not be met with among the Palaeozoic
+  formations. Sir Charles Lyell (1797-1875) was the great leader of this
+  school. His admirably clear and philosophical presentations of
+  geological facts which, with unwearied industry, he collected from the
+  writings of observers in all parts of the world, impressed his views
+  upon the whole English-speaking world, and gave to geological science
+  a coherence and interest which largely accelerated its progress. In
+  his later years, however, he frankly accepted the views of Darwin in
+  regard to the progressive character of the geological record.
+
+  The youngest of the schools of geological thought is that of the
+  Evolutionists. Pointing to the whole body of evidence from inorganic
+  and organic nature, they maintain that the history of our planet has
+  been one of continual and unbroken development from the earliest
+  cosmical beginnings down to the present time, and that the crust of
+  the earth contains an abundant, though incomplete, record of the
+  successive stages through which the plant and animal kingdoms have
+  reached their existing organization. The publication of Darwin's
+  _Origin of Species_ in 1859, in which evolution was made the key to
+  the history of the animal and vegetable kingdoms, produced an
+  extraordinary revolution in geological opinion. The older schools of
+  thought rapidly died out, and evolution became the recognized creed of
+  geologists all over the world.
+
+
+    Werner.
+
+  _Development of Opinion regarding Igneous Rocks._--So long as the idea
+  prevailed that volcanoes are caused by the combustion of inflammable
+  substances underground, there could be no rational conception of
+  volcanic action and its products. Even so late as the middle of the
+  18th century, as above remarked, such a good observer as Lazzaro Moro
+  drew so little distinction between volcanic and other rocks that he
+  could believe the fossiliferous formations to have been mainly formed
+  of materials ejected from eruptive vents. After his time the notion
+  continued to prevail that all the rocks which form the dry land were
+  laid down under water. Even streams of lava, which were seen to flow
+  from an active crater, were regarded only as portions of sedimentary
+  or other rocks, which had been melted by the fervent heat of the
+  burning inflammable materials that had been kindled underground. In
+  spite of the speculations of Descartes and Leibnitz, it was not yet
+  generally comprehended that there exists beneath the terrestrial crust
+  a molten magma, which, from time to time, has been injected into that
+  crust, and has pierced through it, so as to escape at the surface with
+  all the energy of an active volcano. What we now recognize to be
+  memorials of these former injections and propulsions were all
+  confounded with the rocks of unquestionably aqueous origin. The last
+  great teacher by whom these antiquated doctrines were formulated into
+  a system and promulgated to the world was Abraham Gottlob Werner
+  (1749-1815), the most illustrious German mineralogist and geognost of
+  the second half of the 18th century. While still under twenty-six
+  years of age, he was appointed teacher of mining and mineralogy at the
+  Mining Academy of Freiberg in Saxony--a post which he continued to
+  fill up to the end of his life. Possessed of great enthusiasm for his
+  subject, clear, methodical and eloquent in his exposition of it, he
+  soon drew around him men from all parts of the world, who repaired to
+  study under the great oracle of what he called geognosy (Gr. [Greek:
+  gź], the earth, [Greek: gnōsis], knowledge) or earth-knowledge.
+  Reviving doctrines that had been current long before his time, he
+  taught that the globe was once completely surrounded with an ocean,
+  from which the rocks of the earth's crust were deposited as chemical
+  precipitates, in a certain definite order over the whole planet. Among
+  these "universal formations" of aqueous origin were included many
+  rocks, which have long been recognized to have been once molten, and
+  to have risen from below into the upper parts of the terrestrial
+  crust. Werner, following the old tradition, looked upon volcanoes as
+  modern features in the history of the planet, which could not have
+  come into existence until a sufficient amount of vegetation had been
+  buried to furnish fuel for their maintenance. Hence he attached but
+  little importance to them, and did not include in his system of rocks
+  any division of volcanic or igneous materials. From the predominant
+  part assigned by him to the sea in the accumulation of the materials
+  of the visible part of the earth, Werner and his school were known as
+  "Neptunists."
+
+
+    Origin of basalt.
+
+  But many years before the Saxon professor began to teach, clear
+  evidence had been produced from central France that basalt, one of the
+  rocks claimed by him as a chemical precipitate and a universal
+  formation, is a lava which has been poured out in a molten state at
+  various widely separated periods of time and at many different places.
+  So far back as 1752 J.E. Guettard (1715-1786) had shown that the
+  basaltic rocks of Auvergne are true lavas, which have flowed out in
+  streams from groups of once active cones. Eleven years later the
+  observation was confirmed and greatly extended by Nicholas Desmarest
+  (1725-1815), who, during a long course of years, worked out and mapped
+  the complicated volcanic records of that interesting region, and
+  demonstrated to all who were willing impartially to examine the
+  evidence the true volcanic nature of basalt. These views found
+  acceptance from some observers, but they were vehemently opposed by
+  the followers of Werner, who, by the force of his genius, made his
+  theoretical conceptions predominate all over Europe. The controversy
+  as to the origin of basalt was waged with great vigour during the
+  later decades of the 18th century. Desmarest took no part in it. He
+  had accumulated such conclusive proof of the correctness of his
+  deductions, and had so fully expounded the clearness of the evidence
+  in their favour furnished by the region of Auvergne, that, when any
+  one came to consult him on the subject, he contented himself with
+  giving the advice to "go and see." While the debate was in progress on
+  the continent, the subject was approached from a new and independent
+  point of view by Hutton in Scotland. This illustrious philosopher, as
+  already stated, realized the importance of the internal heat of the
+  globe in consolidating the sedimentary rocks, and believed that molten
+  material from the earth's interior has been protruded from below into
+  the overlying crust. Some of the material thus injected could be
+  recognized, he thought, in granite and in the various dark massive
+  rocks which, known in Scotland under the name of "whinstone," were
+  afterwards called "Trap," and are now grouped under various names,
+  such as basalt, dolerite and diorite. So important a share did Hutton
+  thus assign to the internal heat in the geological evolution of the
+  planet, that he and those who adopted the same opinions were styled
+  "Plutonists," or, especially where they concerned themselves with the
+  volcanic origin of basalt, "Vulcanists." The geological world was thus
+  divided into two hostile camps, that of the Neptunists or Wernerians,
+  and that of the Plutonists, Vulcanists or Huttonians.
+
+  After many years of futile controversy the first serious weakening of
+  the position of the dominant Neptunist school arose from the defection
+  of some of the most prominent of Werner's pupils. In particular Jean
+  Franēois D'Aubuisson de Voisins (1769-1819), who had written a
+  treatise on the aqueous origin of the basalts of Saxony, went
+  afterwards to Auvergne, where he was speedily a convert to the views
+  expounded by Desmarest as to the volcanic nature of basalt. Having
+  thus to relinquish one of the fundamental articles of the Freiberg
+  faith, he was subsequently led to modify his adherence to others
+  until, as he himself confessed, his views came almost wholly to agree
+  with those of Hutton. Not less complete, and even more important, was
+  the conversion of the great Leopold von Buch (1774-1853). He, too, was
+  trained by Werner himself, and proved to be the most illustrious pupil
+  of the Saxon professor. Full of admiration for the Neptunism in which
+  he had been reared, he, in his earliest separate work, maintained the
+  aqueous origin of basalt, and contrasted the wide field opened up to
+  the spirit of observation by his master's teaching with the narrower
+  outlook offered by "the volcanic theory." But a little further
+  acquaintance with the facts of nature led Von Buch also to abandon his
+  earlier prepossessions. It was a personal visit to the volcanic region
+  of Auvergne that first opened his eyes, and led him to recant what he
+  had believed and written about basalt. But the abandonment of so
+  essential a portion of the Wernerian creed prepared the way for
+  further relinquishments. When a few years later he went to Norway and
+  found to his astonishment that granite, which he had been taught to
+  regard as the oldest chemical precipitate from the universal ocean,
+  could there be seen to have broken through and metamorphosed
+  fossiliferous limestones, and to have sent veins into them, his faith
+  in Werner's order of the succession of the rocks in the earth's crust
+  received a further momentous shock. While one after another of the
+  Freiberg doctrines crumbled away before him, he was now able to
+  interrogate nature on a wider field than the narrow limits of Saxony,
+  and he was thus gradually led to embrace the tenets of the opposite
+  school. His commanding position, as the most accomplished geologist on
+  the continent, gave great importance to his recantation of the
+  Neptunist creed. His defection indeed was the severest blow that this
+  creed had yet sustained. It may be said to have rung the knell of
+  Wernerianism, which thereafter rapidly declined in influence, while
+  Plutonism came steadily to the front, where it has ever since
+  remained.
+
+  Although Desmarest had traced in Auvergne a long succession of
+  volcanic eruptions, of which the oldest went back to a remote period
+  of time, and although he had shown that this succession, coupled with
+  the records of contemporaneous denudation, might be used in defining
+  epochs of geological history, it was not until many years after his
+  day that volcanic action came to be recognized as a normal part of the
+  mechanism of our globe, which had been in operation from the remotest
+  past, and which had left numerous records among the rocks of the
+  terrestrial crust. During the progress of the controversy between the
+  two great opposing factions in the later portion of the 18th and the
+  first three decades of the 19th century, those who espoused the
+  Vulcanist cause were intent on proving that certain rocks, which are
+  intercalated among the stratified formations and which were claimed by
+  the Neptunists as obviously formed by water, are nevertheless of truly
+  igneous origin. These observers fixed their eyes on the evidence that
+  the material of such rocks, instead of having been deposited from
+  aqueous solution, had once been actually molten, and had in that
+  condition been thrust between the strata, had enveloped portions of
+  them, and had indurated or otherwise altered them. They spoke of these
+  masses as "unerupted lavas"; and undoubtedly in innumerable instances
+  they were right. But their zeal to establish an intrusive origin led
+  them to overlook the proofs that some intercalated sheets of igneous
+  material had not been injected into the strata, but had been poured
+  out at the surface as truly volcanic discharges, and therefore
+  belonged to the ancient periods represented by the strata between
+  which they are interposed. It may readily be supposed that any proofs
+  of the contemporaneous intercalation of such sheets would be eagerly
+  seized upon by the Neptunists in favour of their aqueous theory. The
+  influence of the ancient belief that "burning mountains" could only
+  rise from the combustion of subterranean inflammable materials
+  extended even into the ranks of the Vulcanists, so far at least as to
+  lead to a general acquiescence in the assumption that volcanoes
+  appeared to belong to a late phase in the history of the planet. It
+  was not until after considerable progress had been made in determining
+  the palaeontological distinctions and order of succession of the
+  stratified formations of the earth's crust that it became possible to
+  trace among these formations a succession of volcanic episodes which
+  were contemporaneous with them. In no part of the world has an ampler
+  record of such episodes been preserved than in the British Isles. It
+  was natural, therefore, that the subject should there receive most
+  attention. As far back as 1820 Ami Boué (1794-1881) showed that the
+  Old Red Sandstone of Scotland includes a great series of volcanic
+  rocks, and that other rocks of volcanic origin are associated with the
+  Carboniferous formations. H.T. de la Beche (1796-1855) afterwards
+  traced proofs of contemporaneous eruptions among the Devonian rocks of
+  the south-west of England. Adam Sedgwick (1785-1873) showed, first in
+  the Lake District, and afterwards in North Wales, the presence of
+  abundant volcanic sheets among the oldest divisions of the Palaeozoic
+  series; while Roderick Impey Murchison (1792-1871) made similar
+  discoveries among the Lower Silurian rocks. From the time of these
+  pioneers the volcanic history of the country has been worked out by
+  many observers until it is now known with a fulness as yet unattained
+  in any other region.
+
+  _Growth of Opinion regarding Earthquakes._--We have seen how crude
+  were the conceptions of the ancients regarding the causes of volcanic
+  action, and that they connected volcanoes and earthquakes as results
+  of the commotion of wind imprisoned within subterranean caverns and
+  passages. One of the earliest treatises, in which the phenomena of
+  terrestrial movements were discussed in the spirit of modern science,
+  was the posthumous collection of papers by Robert Hooke (1635-1703),
+  entitled _Lectures and Discourses of Earthquakes and Subterranean
+  Eruptions_, where the probable agency of earthquakes in upheaving and
+  depressing land is fully considered, but without any definite
+  pronouncement as to the author's conception of its origin. Hooke still
+  associated earthquakes with volcanic action, and connected both with
+  what he called "the general congregation of sulphurous subterraneous
+  vapours." He conceived that some kind of "fermentation" takes place
+  within the earth, and that the materials which catch fire and give
+  rise to eruptions or earthquakes are analogous to those that
+  constitute gunpowder. The first essay wherein earthquakes are treated
+  from the modern point of view as the results of a shock that sends
+  waves through the crust of the earth was written by the Rev. John
+  Michell, and communicated to the Royal Society in the year 1760. Still
+  under the old misconception that volcanoes are due to the combustion
+  of inflammable materials, which he thought might be set on fire by the
+  spontaneous combustion of pyritous strata, he supposed that, by the
+  sudden access of large bodies of water to these subterranean fires,
+  vapour is produced in such quantity and with such force as to give
+  rise to the shock. From the centre of origin of this shock waves, he
+  thought, are propagated through the earth, which are largest at the
+  start and gradually diminish as they travel outwards. By drawing lines
+  at different places in the direction of the track of these waves, he
+  believed that the place of common intersection of these lines would be
+  nearly the centre of the disturbance. In this way he showed that the
+  great Lisbon earthquake of 1755 had its focus under the Atlantic,
+  somewhere between the latitudes of Lisbon and Oporto, and he estimated
+  that the depth at which it originated could not be much less than 1
+  m., and probably did not exceed 3 m. Michell, however, misconceived
+  the character of the waves which he described, seeing that he believed
+  them to be due to the actual propagation of the vapour itself
+  underneath the surface of the earth. A century had almost passed after
+  the date of his essay before modern scientific methods of observation
+  and the use of recording instruments began to be applied to the study
+  of earthquake phenomena. In 1846 Robert Mallet (1810-1881) published
+  an important paper "On the Dynamics of Earthquakes" in the
+  _Transactions of the Royal Irish Academy_. From that time onward he
+  continued to devote his energies to the investigation, studying the
+  effects of the Calabrian earthquake of 1857, experimenting on the
+  transmission of waves of shock through various materials, caused by
+  exploding charges of gunpowder, and collecting all the information to
+  be obtained on the subject. His writings, and especially his work in
+  two volumes on _The First Principles of Observational Seismology_,
+  must be regarded as having laid the foundations of this branch of
+  modern geology (see EARTHQUAKE; SEISMOMETER).
+
+  _History of the Evolution of Stratigraphical Geology._--Men had long
+  been familiar with the evidence that the present dry land once lay
+  under the sea, before they began to realize that the rocks, of which
+  the land consists, contain a record of many alternations of land and
+  sea, and relics of a long succession of plants and animals from early
+  and simple types up to the manifold and complex forms of to-day. In
+  countries where coal-mining had been prosecuted for generations, it
+  had been recognized that the rocks consist of strata superposed on
+  each other in a definite order, which was found to extend over the
+  whole of a district. As far back as 1719 John Strachey drew attention
+  to this fact in a communication published in the _Philosophical
+  Transactions_. John Michell (1760), in the paper on earthquakes
+  already cited, showed that he had acquired a clear understanding of
+  the order of succession among stratified formations, and perceived
+  that to disturbances of the terrestrial crust must be ascribed the
+  fact that the lower or older and more inclined strata form the
+  mountains, while the younger and more horizontal strata are spread
+  over the plains.
+
+  In Italy G. Arduķno (1713-1795) classified the rocks in the north of
+  the peninsula as Primitive, Secondary, Tertiary and Volcanic. A
+  similar threefold order was announced for the Harz and Erzgebirge by
+  J.G. Lehmann in 1756. He recognized in that region an ancient series
+  of rocks in inclined or vertical strata, which rise to the tops of the
+  hills and descend to an unknown depth into the interior. These masses,
+  he thought, were contemporaneous with the making of the world. Next
+  came the Flötzgebirge, consisting of younger sediments, disposed in
+  flat or gently inclined sheets which overlie the first and more
+  disturbed series, and are full of petrified remains of plants and
+  animals. Lastly he included the mountains which have from time to time
+  been formed by local accidents. Still more advanced were the
+  conceptions of G.C. Füchsel, who in the year 1762 published in Latin
+  _A History of the Earth and the Sea, based on a History of the
+  Mountains of Thuringia_; and in 1773, in German, a _Sketch of the most
+  Ancient History of the Earth and Man_. In these works he described the
+  stratigraphical relations and general characters of the various
+  geological formations in his little principality; and taking them as
+  indicative of a general order of succession, he traced what he
+  believed to have been a series of revolutions through which the earth
+  has passed. In interpreting this geological history, he laid great
+  stress on the evidence of the fossils contained in the rocks. He
+  recognized that the various formations differ from each other in their
+  enclosed organic remains, and that from these differences the
+  existence of former sea-bottoms and land surfaces can be determined.
+
+  The labours of these pioneers paved the way for the advent of Werner.
+  Though the system evolved by this teacher claimed to discard theory
+  and to be established on a basis of observed facts, it rested on a
+  succession of hypotheses, for which no better foundation could be
+  shown than the belief of their author in their validity. Starting from
+  the extremely limited stratigraphical range displayed in the
+  geological structure of Saxony, he took it as a type for the rest of
+  the globe, persuading himself and impressing upon his followers that
+  the rocks of that small kingdom were to be taken as examples of his
+  "universal formations." The oldest portion of the series, classed by
+  him as "Primitive," consisted of rocks which he maintained had been
+  deposited from chemical solution. Yet they included granite, gneiss,
+  basalt, porphyry and serpentine, which, even in his own day, were by
+  many observers correctly regarded as of igneous origin. A later group
+  of rocks, to which he gave the name of "Transition," comprised, in his
+  belief, partly chemical, partly mechanical sediments, and contained
+  the earliest fossil organic remains. A third group, for which he
+  reserved Lehmann's name "Flötz," was made up chiefly of mechanical
+  detritus, while youngest of all came the "Alluvial" series of loams,
+  clays, sands, gravels and peat. It was by the gradual subsidence of
+  the ocean that, as he believed, the general mass of the dry land
+  emerged, the first-formed rocks being left standing up, sometimes on
+  end, to form the mountains, while those of later date, less steeply
+  inclined, occupied successively lower levels down to the flat alluvial
+  accumulations of the plains. Neither Werner, nor any of his followers,
+  ventured to account for what became of the water as the sea-level
+  subsided, though, in despite of their antipathy to anything like
+  speculation, they could not help suggesting, as an answer to the
+  cogent arguments of their opponents, that "one of the celestial bodies
+  which sometimes approach near to the earth may have been able to
+  withdraw a portion of our atmosphere and of our ocean." Nor was any
+  attempt made to explain the extraordinary nature of the supposed
+  chemical precipitates of the universal ocean. The progress of inquiry
+  even in Werner's lifetime disproved some of the fundamental portions
+  of his system. Many of the chemical precipitates were shown to be
+  masses that had been erupted in a molten state from below. His order
+  of succession was found not to hold good; and though he tried to
+  readjust his sequence and to introduce into it modifications to suit
+  new facts, its inherent artificiality led to its speedy decline after
+  his death. It must be conceded, however, that the stress which he laid
+  upon the fact that the rocks of the earth's crust were deposited in a
+  definite order had an important influence in directing attention to
+  this subject, and in preparing the way for a more natural system,
+  based not on mere mineralogical characters, but having regard to the
+  organic remains, which were now being gathered in ever-increasing
+  numbers and variety from stratified formations of many different ages
+  and from all parts of the globe.
+
+  It was in France and in England that the foundations of stratigraphy,
+  based upon a knowledge of organic remains, were first successfully
+  laid. Abbé J.L. Giraud-Soulavie (1752-1813), in his _Histoire
+  naturelle de la France méridionale_, which appeared in seven volumes,
+  subdivided the limestones of Vivarais into five ages, each marked by a
+  distinct assemblage of shells. In the lowest strata, representing the
+  first age, none of the fossils were believed by him to have any living
+  representatives, and he called these rocks "Primordial." In the next
+  group a mingling of living with extinct forms was observable. The
+  third age was marked by the presence of shells of still existing
+  species. The strata of the fourth series were characterized by
+  carbonaceous shales or slates, containing remains of primordial
+  vegetation, and perhaps equivalents of the first three calcareous
+  series. The fifth age was marked by recent deposits containing remains
+  of terrestrial vegetation and of land animals. It is remarkable that
+  these sagacious conclusions should have been formed and published at a
+  time when the geologists of the Continent were engaged in the
+  controversy about the origin of basalt, or in disputes about the
+  character and stratigraphical position of the supposed universal
+  formations, and when the interest and importance of fossil organic
+  remains still remained unrecognized by the vast majority of the
+  combatants.
+
+  The rocks of the Paris basin display so clearly an orderly
+  arrangement, and are so distinguished for the variety and perfect
+  preservation of their enclosed organic remains, that they could not
+  fail to attract the early notice of observers. J. É. Guettard, G.F.
+  Rouelle (1703-1770), N. Desmarest, A.L. Lavoisier (1743-1794) and
+  others made observations in this interesting district. But it was
+  reserved for Cuvier (1769-1832) and A. Brongniart (1770-1847) to work
+  out the detailed succession of the Tertiary formations, and to show
+  how each of these is characterized by its own peculiar assemblage of
+  organic remains. The later progress of investigation has slightly
+  corrected and greatly amplified the tabular arrangement established by
+  these authors in 1808, but the broad outlines of the Tertiary
+  stratigraphy of the Paris basin remain still as Cuvier and Brongniart
+  left them. The most important subsequent change in the classification
+  of the Tertiary formations was made by Sir Charles Lyell, who,
+  conceiving in 1828 the idea of a classification of these rocks by
+  reference to their relative proportions of living and extinct species
+  of shells, established, in collaboration with G.P. Deshayes, the now
+  universally accepted divisions Eocene, Miocene and Pliocene.
+
+  Long before Cuvier and Brongniart published an account of their
+  researches, another observer had been at work among the Secondary
+  formations of the west of England, and had independently discovered
+  that the component members of these formations were each distinguished
+  by a peculiar group of organic remains; and that this distinction
+  could be used to discriminate them over all the region through which
+  he had traced them. The remarkable man who arrived at this
+  far-reaching generalization was William Smith (1769-1839), a land
+  surveyor who, in the prosecution of his professional business, found
+  opportunities of traversing a great part of England, and of putting
+  his deductions to the test. As the result of these journeys he
+  accumulated materials enough to enable him to produce a geological map
+  of the country, on which the distribution and succession of the rocks
+  were for the first time delineated. Smith's labours laid the
+  foundation of stratigraphical geology in England and he was styled
+  even in his lifetime the "Father of English geology." From his day
+  onward the significance of fossil organic remains gained rapidly
+  increasing recognition. Thus in England the outlines traced by him
+  among the Secondary and Tertiary formations were admirably filled in
+  by Thomas Webster (1773-1844); while the Cretaceous series was worked
+  out in still greater detail in the classic memoirs of William Henry
+  Fitton (1780-1861).
+
+  There was one stratigraphical domain, however, into which William
+  Smith did not enter. He traced his sequence of rocks down into the
+  Coal Measures, but contented himself with only a vague reference to
+  what lay underneath that formation. Though some of these underlying
+  rocks had in various countries yielded abundant fossils, they had
+  generally suffered so much from terrestrial disturbances, and their
+  order of succession was consequently often so much obscured throughout
+  western Europe, that they remained but little known for many years
+  after the stratigraphy of the Secondary and Tertiary series had been
+  established. At last in 1831 Murchison began to attack this _terra
+  incognita_ on the borders of South Wales, working into it from the Old
+  Red Sandstone, the stratigraphical position of which was well known.
+  In a few years he succeeded in demonstrating the existence of a
+  succession of formations, each distinguished by its own peculiar
+  assemblage of organic remains which were distinct from those in any of
+  the overlying strata. To these formations he gave the name of Silurian
+  (q.v.). From the key which his researches supplied, it was possible to
+  recognize in other countries the same order of formations and the same
+  sequence of fossils, so that, in the course of a few years,
+  representatives of the Silurian system were found far and wide over
+  the globe. While Murchison was thus engaged, Sedgwick devoted himself
+  to the more difficult task of unravelling the complicated structure of
+  North Wales. He eventually made out the order of the several
+  formations there, with their vast intercalations of volcanic material.
+  He named them the Cambrian system (q.v.), and found them to contain
+  fossils, which, however, lay for some time unexamined by him. He at
+  first believed, as Murchison also did, that his rocks were all older
+  than any part of the Silurian series. It was eventually discovered
+  that a portion of them was equivalent to the lower part of that
+  series. The oldest of Sedgwick's groups, containing distinctive
+  fossils, retain the name Cambrian, and are of high interest, as they
+  enclose the remains of the earliest faunas which are yet well known.
+  Sedgwick and Murchison rendered yet another signal service to
+  stratigraphical geology by establishing, in 1839, on a basis of
+  palaeontological evidence supplied by W. Lonsdale, the independence of
+  the Devonian system (q.v.).
+
+  For many years the rocks below the oldest fossiliferous deposits
+  received comparatively little attention. They were vaguely described
+  as the "crystalline schists" and were often referred to as parts of
+  the primeval crust in which no chronology was to be looked for. W.E.
+  Logan (1798-1875) led the way, in Canada, by establishing there
+  several vast series of rocks, partly of crystalline schists and
+  gneisses (Laurentian) and partly of slates and conglomerates
+  (Huronian). Later observers, both in Canada and the United States,
+  have greatly increased our knowledge of these rocks, and have shown
+  their structure to be much more complex than was at first supposed
+  (see ARCHEAN SYSTEM).
+
+  During the latter half of the 19th century the most important
+  development of stratigraphical geology was the detailed working out
+  and application of the principle of zonal classification to the
+  fossiliferous formations--that is, the determination of the sequence
+  and distribution of organic remains in these formations, and the
+  arrangement of the strata into zones, each of which is distinguished
+  by a peculiar assemblage of fossil species (see under Part VI.). The
+  zones are usually named after one especially characteristic species.
+  This system of classification was begun in Germany with reference to
+  the members of the Jurassic system (q.v.) by A. Oppel (1856-1858) and
+  F.A. von Quenstedt (1858), and it has since been extended through the
+  other Mesozoic formations. It has even been found to be applicable to
+  the Palaeozoic rocks, which are now subdivided into palaeontological
+  zones. In the Silurian system, for example, the graptolites have been
+  shown by C. Lapworth to furnish a useful basis for zonal subdivisions.
+  The lowest fossiliferous horizon in the Cambrian rocks of Europe and
+  North America is known as the _Olenellus_ zone, from the prominence in
+  it of that genus of trilobite.
+
+  Another conspicuous feature in the progress of stratigraphy during the
+  second half of the 19th century was displayed by the rise and rapid
+  development of what is known as Glacial geology. The various deposits
+  of "drift" spread over northern Europe, and the boulders scattered
+  across the surface of the plains had long attracted notice, and had
+  even found a place in popular legend and superstition. When men began
+  to examine them with a view to ascertain their origin, they were
+  naturally regarded as evidences of the Noachian deluge. The first
+  observer who drew attention to the smoothed and striated surfaces of
+  rock that underlie the Drifts was Hutton's friend, Sir James Hall, who
+  studied them in the lowlands of Scotland and referred them to the
+  action of great debacles of water, which, in the course of some
+  ancient terrestrial convulsion, had been launched across the face of
+  the country. Playfair, however, pointed out that the most potent
+  geological agents for the transportation of large blocks of stone are
+  the glaciers. But no one was then bold enough to connect the travelled
+  boulders with glaciers on the plains of Germany and of Britain. Yet
+  the transporting agency of ice was invoked in explanation of their
+  diffusion. It came to be the prevalent belief among the geologists of
+  the first half of the 19th century, that the fall of temperature,
+  indicated by the gradual increase in the number of northern species of
+  shells in the English Crag deposits, reached its climax during the
+  time of the Drift, and that much of the north and centre of Europe was
+  then submerged beneath a sea, across which floating icebergs and floes
+  transported the materials of the Drift and dropped the scattered
+  boulders. As the phenomena are well developed around the Alps, it was
+  necessary to suppose that the submergence involved the lowlands of the
+  Continent up to the foot of that mountain chain--a geographical change
+  so stupendous as to demand much more evidence than was adduced in its
+  support. At last Louis Agassiz (1807-1873), who had varied his
+  palaeontological studies at Neuchātel by excursions into the Alps, was
+  so much struck by the proofs of the former far greater extension of
+  the Swiss glaciers, that he pursued the investigation and satisfied
+  himself that the ice had formerly extended from the Alpine valleys
+  right across the great plain of Switzerland, and had transported huge
+  boulders from the central mountains to the flanks of the Jura. In the
+  year 1840 he visited Britain and soon found evidence of similar
+  conditions there. He showed that it was not by submergence in a sea
+  cumbered with floating ice, but by the former presence of vast
+  glaciers or sheets of ice that the Drift and erratic blocks had been
+  distributed. The idea thus propounded by him did not at once command
+  complete approval, though traces of ancient glaciers in Scotland and
+  Wales were soon detected by native geologists, particularly by W.
+  Buckland, Lyell, J.D. Forbes and Charles Maclaren. Robert Chambers
+  (1802-1871) did good service in gathering additional evidence from
+  Scotland and Norway in favour of Agassiz's views, which steadily
+  gained adherents until, after some quarter of a century, they were
+  adopted by the great majority of geologists in Britain, and
+  subsequently in other countries. Since that time the literature of
+  geology has been swollen by a vast number of contributions in which
+  the history of the Glacial period, and its records both in the Old and
+  New World, have been fully discussed.
+
+  _Rise and Progress of Palaeontological Geology._--As this branch of
+  the science deals with the evidence furnished by fossil organic
+  remains as to former geographical conditions, it early attracted
+  observers who, in the superficial beds of marine shells found at some
+  distance from the coast, saw proofs of the former submergence of the
+  land under the sea. But the occurrence of fossils embedded in the
+  heart of the solid rocks of the mountains offered much greater
+  difficulties of explanation, and further progress was consequently
+  slow. Especially baneful was the belief that these objects were mere
+  sports of nature, and had no connexion with any once living organisms.
+  So long as the true organic origin of the fossil plants and animals
+  contained in the rocks was in dispute, it was hardly possible that
+  much advance could be made in their systematic study, or in the
+  geological deductions to be drawn from them. One good result of the
+  controversy, however, was to be seen in the large collections of these
+  "formed stones" that were gathered together in the cabinets and
+  museums of the 17th and 18th centuries. The accumulation and
+  comparison of these objects naturally led to the production of
+  treatises in which they were described and not unfrequently
+  illustrated by good engravings. Switzerland was more particularly
+  noted for the number and merit of its works of this kind, such as
+  that of K.N. Lang (_Historia lapidum figuratorum Helvetiae_, 1708) and
+  those of Johann Jacob Scheuchzer (1672-1733). In England, also,
+  illustrated treatises were published both by men who looked on fossils
+  as mere freaks of nature, and by those who regarded them as proofs of
+  Noah's flood. Of the former type were the works of Martin Lister
+  (1638-1712) and Robert Plot (_Natural History of Oxfordshire_, 1677).
+  The Celtic scholar Edward Llwyd (1660-1709) wrote a Latin treatise
+  containing good plates of a thousand fossils in the Ashmolean Museum,
+  Oxford, and J. Woodward, in 1728-1729, published his _Natural History
+  of the Fossils of England_, already mentioned, wherein he described
+  his own extensive collection, which he bequeathed to the University of
+  Cambridge, where it is still carefully preserved. The most voluminous
+  and important of all these works, however, appeared at a later date at
+  Nuremberg. It was begun by G.W. Knorr (1705-1761), who himself
+  engraved for it a series of plates, which for beauty and accuracy have
+  seldom been surpassed. After his death the work was continued by
+  J.E.I. Walch (1725-1778), and ultimately consisted of four massive
+  folio volumes and nearly 300 plates under the title of _Lapides
+  diluvii universalis testes_. Although the authors supposed their
+  fossils to be relics of Noah's flood, their work must be acknowledged
+  to mark a distinct onward stage in the palaeontological department of
+  geology.
+
+  It was in France that palaeontological geology began to be cultivated
+  in a scientific spirit. The potter Bernard Palissy, as far back as
+  1580, had dwelt on the importance of fossil shells as monuments of
+  revolutions of the earth's surface; but the observer who first
+  undertook the detailed study of the subject was Jean Etienne Guettard,
+  who began in 1751 to publish his descriptions of fossils in the form
+  of memoirs presented to the Academy of Sciences of Paris. To him they
+  were not only of deep interest as monuments of former types of
+  existence, but they had an especial value as records of the changes
+  which the country had undergone from sea to land and from land to sea.
+  More especially noteworthy was a monograph by him which appeared in
+  1765 bearing the title "On the accidents that have befallen Fossil
+  Shells compared with those which are found to happen to shells now
+  living in the Sea." In this treatise he showed that the fossils have
+  been encrusted with barnacles and serpulae, have been bored into by
+  other organisms, and have often been rounded or broken before final
+  entombment; and he inferred that these fossils must have lived and
+  died on the sea-floor under similar conditions to those which obtain
+  on the sea-floor to-day. His argument was the most triumphant that had
+  ever been brought against the doctrine of _lusus naturae_, and that of
+  the efficacy of Noah's flood--doctrines which still held their ground
+  in Guettard's day. When Soulavie, Cuvier and Brongniart in France, and
+  William Smith in England, showed that the rock formations of the
+  earth's crust could be arranged in chronological order, and could be
+  recognized far and wide by means of their enclosed organic remains,
+  the vast significance of these remains in geological research was
+  speedily realized, and palaeontological geology at once entered on a
+  new and enlarged phase of development. But apart from their value as
+  chronological monuments, and as witnesses of former conditions of
+  geography, fossils presented in themselves a wide field of
+  investigation as types of life that had formerly existed, but had now
+  passed away. It was in France that this subject first took definite
+  shape as an important branch of science. The mollusca of the Tertiary
+  deposits of the Paris basin became, in the hands of Lamarck, the basis
+  on which invertebrate palaeontology was founded. The same series of
+  strata furnished to Cuvier the remains of extinct land animals, of
+  which, by critical study of their fragmentary bones and skeletons, he
+  worked out restorations that may be looked on as the starting-point of
+  vertebrate palaeontology. These brilliant researches, rousing
+  widespread interest in such studies, showed how great a flood of light
+  could be thrown on the past history of the earth and its inhabitants.
+  But the full significance of these extinct types of life could not be
+  understood so long as the doctrine of the immutability of species, so
+  strenuously upheld by Cuvier, maintained its sway among naturalists.
+  Lamarck, as far back as the year 1800, had begun to propound his
+  theory of evolution and the transformation of species; but his views,
+  strongly opposed by Cuvier and the great body of naturalists of the
+  day, fell into neglect. Not until after the publication in 1859 of the
+  _Origin of Species_ by Charles Darwin were the barriers of old
+  prejudice in this matter finally broken down. The possibility of
+  tracing the ancestry of living forms back into the remotest ages was
+  then perceived; the time-honoured fiction that the stratified
+  formations record a series of catastrophes and re-creations was
+  finally dissipated; and the earth's crust was seen to contain a noble,
+  though imperfect, record of the grand evolution of organic types of
+  which our planet has been the theatre.
+
+  _Development of Petrographical Geology._--Theophrastus, the favourite
+  pupil of Aristotle, wrote a treatise _On Stones_, which has come down
+  to our own day, and may be regarded as the earliest work on
+  petrography. At a subsequent period Pliny, in his _Natural History_,
+  collected all that was known in his day regarding the occurrence and
+  uses of minerals and rocks. But neither of these works is of great
+  scientific importance, though containing much interesting information.
+  Minerals from their beauty and value attracted notice before much
+  attention was paid to rocks, and their study gave rise to the science
+  of mineralogy long before geology came into existence. When rocks
+  began to be more particularly scrutinized, it was chiefly from the
+  side of their usefulness for building and other economic purposes. The
+  occurrence of marine shells in many of them had early attracted
+  attention to them. But their varieties of composition and origin did
+  not become the subject of serious study until after Linnaeus and J.G.
+  Wallerius in the 18th century had made a beginning. The first
+  important contribution to this department of the science was that of
+  Werner, who in 1786 published a classification and description of
+  rocks in which he arranged them in two divisions, simple and compound,
+  and further distinguished them by various external characters and by
+  their relative age. The publication of this scheme may be said to mark
+  the beginning of scientific petrography. Werner's system, however, had
+  the serious defect that the chronological order in which he grouped
+  the rocks, and the hypothesis by which he accounted for them as
+  chemical precipitates from the original ocean, were both alike
+  contrary to nature. It was hardly possible indeed that much progress
+  could be made in this branch of geology until chemistry and mineralogy
+  had made greater advances; and especially until it was possible to
+  ascertain the intimate chemical and mineralogical composition, and the
+  minute structure of rocks. The study, however, continued to be pursued
+  in Germany, where the influence of Werner's enthusiasm still led men
+  to enter the petrographical rather than the palaeontological domain.
+  The resources of modern chemistry were pressed into the service, and
+  analyses were made and multiplied to such a degree that it seemed as
+  if the ultimate chemical constitution of every type of rock had now
+  been thoroughly revealed. The condition of the science in the middle
+  of the 19th century was well shown by J.L.A. Roth, who in 1861
+  collected about 1000 trustworthy analyses which up to that time had
+  been made. But though the chemical elements of the rocks had been
+  fairly well determined, the manner in which they were combined in the
+  compound rocks could for the most part be only more or less plausibly
+  conjectured. As far back as 1831 an account was published of a process
+  devised by William Nicol of Edinburgh, whereby sections of fossil wood
+  could be cut, mounted on glass, and reduced to such a degree of
+  transparency as to be easily examined under a microscope. Henry Sorby,
+  of Sheffield, having seen Nicol's preparations, perceived how
+  admirably adapted the process was for the study of the minute
+  structure and composition of rocks. In 1858 he published in the
+  _Quarterly Journal of the Geological Society_ a paper "On the
+  Microscopical Structure of Crystals." This essay led to a complete
+  revolution of petrographical methods and gave a vast impetus to the
+  study of rocks. Petrology entered upon a new and wider field of
+  investigation. Not only were the mineralogical constituents of the
+  rocks detected, but minute structures were revealed which shed new
+  light on the origin and history of these mineral masses, and opened up
+  new paths in theoretical geology. In the hands of H. Vogelsang, F.
+  Zirkel, H. Rosenbusch, and a host of other workers in all civilized
+  countries, the literature of this department of the science has grown
+  to a remarkable extent. Armed with the powerful aid of modern optical
+  instruments, geologists are now able with far more prospect of success
+  to resume the experiments begun a century before by de Saussure and
+  Hall. G.A. Daubrée, C. Friedel, E. Sarasin, F. Fouqué and A. Michel
+  Lévy in France, C. Doelter y Cisterich and E. Hussak of Gratz, J.
+  Morozewicz of Warsaw and others, have greatly advanced our knowledge
+  by their synthetical analyses, and there is every reason to hope that
+  further advances will be made in this field of research.
+
+  _Rise of Physiographical Geology._--Until stratigraphical geology had
+  advanced so far as to show of what a vast succession of rocks the
+  crust of the earth is built up, by what a long and complicated series
+  of revolutions these rocks have come to assume their present
+  positions, and how enormous has been the lapse of time which all these
+  changes represent, it was not possible to make a scientific study of
+  the surface features of our globe. From ancient times it had been
+  known that many parts of the land had once been under the sea; but
+  down even to the beginning of the 19th century the vaguest conceptions
+  continued to prevail as to the operations concerned in the submergence
+  and elevation of land, and as to the processes whereby the present
+  outlines of terrestrial topography were determined. We have seen, for
+  instance, that according to the teaching of Werner the oldest rocks
+  were first precipitated from solution in the universal ocean to form
+  the mountains, that the vertical position of their strata was
+  original, that as the waters subsided successive formations were
+  deposited and laid bare, and that finally the superfluous portion of
+  the ocean was whisked away into space by some unexplained co-operation
+  of another planetary body. Desmarest, in his investigation of the
+  volcanic history of Auvergne, was the first observer to perceive by
+  what a long process of sculpture the present configuration of the land
+  has been brought about. He showed conclusively that the valleys have
+  been carved out by the streams that flow in them, and that while they
+  have sunk deeper and deeper into the framework of the land, the spaces
+  of ground between them have been left as intervening ridges and hills.
+  De Saussure learnt a similar lesson from his studies of the Alps, and
+  Hutton and Playfair made it a cardinal feature in their theory of the
+  earth. Nevertheless the idea encountered so much opposition that it
+  made but little way until after the middle of the 19th century.
+  Geologists preferred to believe in convulsions of nature, whereby
+  valleys were opened and mountains were upheaved. That the main
+  features of the land, such as the great mountain-chains, had been
+  produced by gigantic plication of the terrestrial crust was now
+  generally admitted, and also that minor fractures and folds had
+  probably initiated many of the valleys. But those who realized most
+  vividly the momentous results achieved by ages of subaerial denudation
+  perceived that, as Hutton showed, even without the aid of underground
+  agency, the mere flow of water in streams across a mass of land must
+  in course of time carve out just such a system of valleys as may
+  anywhere be seen. It was J.B. Jukes who, in 1862, first revived the
+  Huttonian doctrine, and showed how completely it explained the
+  drainage-lines in the south of Ireland. Other writers followed in
+  quick succession until, in a few years, the doctrine came to be widely
+  recognized as one of the established principles of modern geology.
+  Much help was derived from the admirable illustrations of
+  land-sculpture and river-erosion supplied from the Western Territories
+  and States of the American Union.
+
+  Another branch of physiographical geology which could only come into
+  existence after most of the other departments of the science had made
+  large progress, deals with the evolution of the framework of each
+  country and of the several continents and oceans of the globe. It is
+  now possible, with more or less confidence, to trace backward the
+  history of every terrestrial area, to see how sea and land have there
+  succeeded each other, how rivers and lakes have come and gone, how the
+  crust of the earth has been ridged up at widely separated intervals,
+  each movement determining some line of mountains or plains, how the
+  boundaries of the oceans have shifted again and again in the past, and
+  thus how, after so prolonged a series of revolutions, the present
+  topography of each country, and of the globe as a whole, has been
+  produced. In the prosecution of this subject maps have been
+  constructed to show what is conjectured to have been the distribution
+  of sea and land during the various geological periods in different
+  parts of the world, and thus to indicate the successive stages through
+  which the architecture of the land has been gradually evolved. The
+  most noteworthy contribution to this department of the science is the
+  _Antlitz der Erde_ of Professor Suess of Vienna. This important and
+  suggestive work has been translated into French and English.
+
+
+PART II.--COSMICAL ASPECTS
+
+Before geology had attained to the position of an inductive science, it
+was customary to begin investigations into the history of the earth by
+propounding or adopting some more or less fanciful hypothesis in
+explanation of the origin of our planet, or even of the universe. Such
+preliminary notions were looked upon as essential to a right
+understanding of the manner in which the materials of the globe had been
+put together. One of the distinguishing features of Hutton's Theory of
+the Earth consisted in his protest that it is no part of the province of
+geology to discuss the origin of things. He taught that in the materials
+from which geological evidence is to be compiled there can be found "no
+traces of a beginning, no prospect of an end." In England, mainly to the
+influence of the school which he founded, and to the subsequent rise of
+the Geological Society of London, which resolved to collect facts
+instead of fighting over hypotheses, is due the disappearance of the
+crude and unscientific cosmologies by which the writings of the earlier
+geologists were distinguished.
+
+But there can now be little doubt that in the reaction against those
+visionary and often grotesque speculations, geologists were carried too
+far in an opposite direction. In allowing themselves to believe that
+geology had nothing to do with questions of cosmogony, they gradually
+grew up in the conviction that such questions could never be other than
+mere speculation, interesting or amusing as a theme for the employment
+of the fancy, but hardly coming within the domain of sober and inductive
+science. Nor would they soon have been awakened out of this belief by
+anything in their own science. It is still true that in the data with
+which they are accustomed to deal, as comprising the sum of geological
+evidence, there can be found no trace of a beginning, though the
+evidence furnished by the terrestrial crust shows a general evolution of
+organic forms from some starting-point which cannot be seen. The oldest
+rocks which have been discovered on any part of the globe have probably
+been derived from other rocks older than themselves. Geology by itself
+has not yet revealed, and is little likely ever to reveal, a trace of
+the first solid crust of our globe. If, then, geological history is to
+be compiled from direct evidence furnished by the rocks of the earth, it
+cannot begin at the beginning of things, but must be content to date
+its first chapter from the earliest period of which any record has been
+preserved among the rocks.
+
+Nevertheless, though geology in its usual restricted sense has been, and
+must ever be, unable to reveal the earliest history of our planet, it no
+longer ignores, as mere speculation, what is attempted in this subject
+by its sister sciences. Astronomy, physics and chemistry have in late
+years all contributed to cast light on the earlier stages of the earth's
+existence, previous to the beginning of what is commonly regarded as
+geological history. But whatever extends our knowledge of the former
+conditions of our globe may be legitimately claimed as part of the
+domain of geology. If this branch of inquiry, therefore, is to continue
+worthy of its name as the science of the earth, it must take cognizance
+of these recent contributions from other sciences. It must no longer be
+content to begin its annals with the records of the oldest rocks, but
+must endeavour to grope its way through the ages which preceded the
+formation of any rocks. Thanks to the results achieved with the
+telescope, the spectroscope and the chemical laboratory, the story of
+these earliest ages of our earth is every year becoming more definite
+and intelligible.
+
+Up to the present time no definite light has been thrown by physics on
+the origin and earliest condition of our globe. The famous nebular
+theory (q.v.) of Kant and Laplace sketched the supposed evolution of the
+solar system from a gaseous nebula, slowly rotating round a more
+condensed central portion of its mass, which eventually became the sun.
+As a consequence of increased rapidity of rotation resulting from
+cooling and contraction, the nebula acquired a more and more lenticular
+form, until at last it threw off from its equatorial protuberance a ring
+of matter. Subsequently the same process was repeated, and other similar
+rings successively separated from the parent mass. Each ring went
+through a corresponding series of changes until it ultimately became a
+planet, with or without one or more attendant satellites. The intimate
+relationship of our earth to the sun and the other planets was, in this
+way, shown. But there are some serious physical difficulties in the way
+of the acceptance of the nebular hypothesis. Another explanation is
+given by the meteoritic hypothesis, according to which, out of the
+swarms of meteorites with which the regions of space are crowded, the
+sun and planets have been formed by gradual accretion.
+
+According to these theoretical views we should expect to find a general
+uniformity of composition in the constituent matter of the solar system.
+For many years the only available evidence on this point was derived
+from the meteorites (q.v.) which so constantly fall from outer space
+upon the surface of the earth. These bodies were found to consist of
+elements, all of which had been recognized as entering into the
+constitution of the earth. But the discoveries of spectroscopic research
+have made known a far more widely serviceable method of investigation,
+which can be applied even to the luminous stars and nebulae that lie far
+beyond the bounds of the solar system. By this method information has
+been obtained regarding the constitution of the sun, and many of our
+terrestrial metals, such as iron, nickel and magnesium, have been
+ascertained to exist in the form of incandescent vapour in the solar
+atmosphere. The present condition of the sun probably represents one of
+the phases through which stars and planets pass in their progress
+towards becoming cool and dark bodies in space. If our globe was at
+first, like its parent sun, an incandescent mass of probably gaseous
+matter, occupying much more space than it now fills, we can conceive
+that it has ever since been cooling and contracting until it has reached
+its present form and dimensions, and that it still retains a high
+internal temperature. Its oblately spheroidal form is such as would be
+assumed by a rotating mass of matter in the transition from a vaporous
+and self-luminous or liquid condition to one of cool and dark solidity.
+But it has been claimed that even a solid spherical globe might develop,
+under the influence of protracted rotation, such a shape as the earth at
+present possesses.
+
+The observed increase of temperature downwards in our planet has
+hitherto been generally accepted as a relic and proof of an original
+high temperature and mobility of substance. Recently, however, the
+validity of this proof has been challenged on the ground that the
+ascertained amount of radium in the rocks of the outer crust is more
+than sufficient to account for the observed downward increase of
+temperature. Too little, however, is known of the history and properties
+of what is called radium to afford a satisfactory ground on which to
+discard what has been, and still remains, the prevalent belief on this
+subject.
+
+An important epoch in the geological history of the earth was marked by
+the separation of the moon from its mass (see TIDE). Whether the
+severance arose from the rupture of a surrounding ring or the gradual
+condensation of matter in such a ring, or from the ejection of a single
+mass of matter from the rapidly rotating planet, it has been shown that
+our satellite was only a few thousand miles from the earth's surface,
+since when it has retreated to its present distance of 240,000 m. Hence
+the influence of the moon's attraction, and all the geological effects
+to which it gives rise, attained their maximum far back in the
+development of the globe, and have been slowly diminishing throughout
+geological history.
+
+The sun by virtue of its vast size has not yet passed out of the
+condition of glowing gas, and still continues to radiate heat beyond the
+farthest planet of the solar system. The earth, however, being so small
+a body in comparison, would cool down much more quickly. Underneath its
+hot atmosphere a crust would conceivably begin to form over its molten
+surface, though the interior might still possess a high temperature and,
+owing to the feeble conducting power of rocks, would remain intensely
+hot for a protracted series of ages.
+
+Full information regarding the form and size of the earth, and its
+relations to the other planetary members of the solar system, will be
+found in the articles PLANET and SOLAR SYSTEM. For the purposes of
+geological inquiry the reader will bear in mind that the equatorial
+diameter of our globe is estimated to be about 7925 m., and the polar
+diameter about 7899 m.; the difference between these two sums
+representing the amount of flattening at the poles (about 26½ m.). The
+planet has been compared in shape to an orange, but it resembles an
+orange which has been somewhat squeezed, for its equatorial
+circumference is not a regular circle but an ellipse, of which the major
+axis lies in long. 8° 15' W.--on a meridian which cuts the north-west
+corner of America, passing through Portugal and Ireland, and the
+north-east corner of Asia in the opposite hemisphere.
+
+The rotation of the earth on its axis exerts an important influence on
+the movements of the atmosphere, and thereby affects the geological
+operations connected with these movements. The influence of rotation is
+most marked in the great aerial circulation between the poles and the
+equator. Currents of air, which set out in a meridional direction from
+high latitudes towards the equator, come from regions where the velocity
+due to rotation is small to where it is greater, and they consequently
+fall behind. Thus, in the northern hemisphere a north wind, as it moves
+away from its northern source of origin, is gradually deflected more and
+more towards the west and becomes a north-east current; while in the
+opposite hemisphere a wind making from high southern latitudes towards
+the equator becomes, from the same cause, a south-east current. Where,
+on the other hand, the air moves from the equatorial to the polar
+regions its higher velocity of rotation carries it eastward, so that on
+the south side of the equator it becomes a north-west current and on the
+north side a south-west current. It is to this cause that the easting
+and westing of the great atmospheric currents are to be attributed, as
+is familiarly exemplified in the trade winds.
+
+The atmospheric circulation thus deflected influences the circulation of
+the ocean. The winds which persistently blow from the north-east on the
+north side of the equator, and from the south-east on the south side,
+drive the superficial waters onwards, and give rise to converging
+oceanic currents which unite to form the great westerly equatorial
+current.
+
+A more direct effect of terrestrial rotation has been claimed in the
+case of rivers which flow in a meridional direction. It has been
+asserted that those, which in the northern hemisphere flow from north to
+south, like the Volga, by continually passing into regions where the
+velocity of rotation is increasingly greater, are thrown more against
+their western than their eastern banks, while those whose general course
+is in an opposite direction, like the Irtisch and Yenesei, press more
+upon their eastern sides. There cannot be any doubt that the tendency of
+the streams must be in the directions indicated. But when the
+comparatively slow current and constantly meandering course of most
+rivers are taken into consideration, it may be doubted whether the
+influence of rotation is of much practical account so far as
+river-erosion is concerned.
+
+One of the cosmical relations of our planet which has been more
+especially prominent in geological speculations relates to the position
+of the earth's axis of rotation. Abundant evidence has now been obtained
+to prove that at a comparatively late geological period a rich flora,
+resembling that of warm climates at the present day, existed in high
+latitudes even within less than 9° of the north pole, where, with an
+extremely low temperature and darkness lasting for half of the year, no
+such vegetation could possibly now exist. It has accordingly been
+maintained by many geologists that the axis of rotation must have
+shifted, and that when the remarkable Arctic assemblage of fossil plants
+lived the region of their growth must have lain in latitudes much nearer
+to the equator of the time.
+
+The possibility of any serious displacement of the rotational axis since
+a very early period in the earth's history has been strenuously denied
+by astronomers, and their arguments have been generally, but somewhat
+reluctantly, accepted by geologists, who find themselves confronted with
+a problem which has hitherto seemed insoluble. That the axis is not
+rigidly stable, however, has been postulated by some physicists, and has
+now been demonstrated by actual observation and measurement. It is
+admitted that by the movement of large bodies of water the air over the
+surface of the globe, and more particularly by the accumulation of vast
+masses of snow and ice in different regions, the position of the axis
+might be to some extent shifted; more serious effects might follow from
+widespread upheavals or depressions of the surface of the lithosphere.
+On the assumption of the extreme rigidity of the earth's interior,
+however, the general result of mathematical calculation is to negative
+the supposition that in any of these ways within the period represented
+by what is known as the "geological record," that is, since the time of
+the oldest known sedimentary formations, the rotational axis has ever
+been so seriously displaced as to account for such stupendous geological
+events as the spread of a luxuriant vegetation far up into polar
+latitudes. If, however, the inside of the globe possesses a great
+plasticity than has been allowed, the shifting of the axis might not be
+impossible, even to such an extent as would satisfy the geological
+requirements. This question is one on which the last word has not been
+said, and regarding which judgment must remain in suspense.
+
+In recent years fresh information bearing on the minor devagations of
+the pole has been obtained from a series of several thousand careful
+observations made in Europe and North America. It has thus been
+ascertained that the pole wanders with a curiously irregular but
+somewhat spiral movement, within an amplitude of between 40 and 50 ft.,
+and completes its erratic circuit in about 428 days. It was not supposed
+that its movement had any geological interest, but Dr John Milne has
+recently pointed out that the times of sharpest curvature in the path of
+the pole coincide with the occurrence of large earthquakes, and has
+suggested that, although it can hardly be assumed that this coincidence
+shows any direct connexion between earthquake frequency and changes in
+the position of the earth's axis, both effects may not improbably arise
+from the same redistribution of surface material by ocean currents and
+meteorological causes.
+
+If for any reason the earth's centre of gravity were sensibly displaced,
+momentous geological changes would necessarily ensue. That the centre of
+gravity does not coincide with the centre of figure of the globe, but
+lies to the south of it, has long been known. This greater aggregation
+of dense material in the southern hemisphere probably dates from the
+early ages of the earth's consolidation, and it is difficult to believe
+that any readjustment of the distribution of this material in the
+earth's interior is now possible. But certain rearrangements of the
+hydrosphere on the surface of the globe may, from time to time, cause a
+shifting of the centre of gravity, which will affect the level of the
+ocean. The accumulation of enormous masses of ice around the pole will
+give rise to such a displacement, and will thus increase the body of
+oceanic water in the glaciated hemisphere. Various calculations have
+been made of the effect of the transference of the ice-cap from one pole
+to the other, a revolution which may possibly have occurred more than
+once in the past history of the globe. James Croll estimated that if the
+mass of ice in the southern hemisphere be assumed to be 1000 ft. thick
+down to lat. 60°, its removal to the opposite hemisphere would raise the
+level of the sea 80 ft. at the north pole, while the Rev. Osmond Fisher
+made the rise as much as 409 ft. The melting of the ice would still
+further raise the sea-level by the addition of so large a volume of
+water to the ocean. To what extent superficial changes of this kind have
+operated in geological history remains an unsolved problem, but their
+probable occurrence in the past has to be recognized as one of the
+factors that must be considered in tracing the revolutions of the
+earth's surface.
+
+_The Age of the Earth._--Intimately connected with the relations of our
+globe to the sun and the other members of the solar system is the
+question of the planet's antiquity--a subject of great geological
+importance, regarding which much discussion has taken place since the
+middle of the 19th century. Though an account of this discussion
+necessarily involves allusion to departments of geology which are more
+appropriately referred to in later parts of this article, it may perhaps
+be most conveniently included here.
+
+Geologists were for many years in the habit of believing that no limit
+could be assigned to the antiquity of the planet, and that they were at
+liberty to make unlimited drafts on the ages of the past. In 1862 and
+subsequent years, however, Lord Kelvin (then Sir William Thomson)
+pointed out that these demands were opposed to known physical facts, and
+that the amount of time required for geological history was not only
+limited, but must have been comprised within a comparatively narrow
+compass. His argument rested on three kinds of evidence: (1) the
+internal heat and rate of cooling of the earth; (2) the tidal
+retardation of the earth's rotation; and (3) the origin and age of the
+sun's heat.
+
+1. Applying Fourier's theory of thermal conductivity, Lord Kelvin
+contended that in the known rate of increase of temperature downward and
+beneath the surface, and the rate of loss of heat from the earth, we
+have a limit to the antiquity of the planet. He showed, from the data
+available at the time, that the superficial consolidation of the globe
+could not have occurred less than 20 million years ago, or the
+underground heat would have been greater than it is; nor more than 400
+million years ago, otherwise the underground temperature would have
+shown no sensible increase downwards. He admitted that very wide limits
+were necessary. In subsequently discussing the subject, he inclined
+rather towards the lower than the higher antiquity, but concluded that
+the limit, from a consideration of all the evidence, must be placed
+within some such period of past time as 100 millions of years.
+
+2. The argument from tidal retardation proceeds on the admitted fact
+that, owing to the friction of the tide-wave, the rotation of the earth
+is retarded, and is, therefore, much slower now than it must have been
+at one time. Lord Kelvin affirmed that had the globe become solid some
+10,000 million years ago, or indeed any high antiquity beyond 100
+million years, the centrifugal force due to the more rapid rotation must
+have given the planet a very much greater polar flattening than it
+actually possesses. He admitted, however, that, though 100 million years
+ago that force must have been about 3% greater than now, yet "nothing
+we know regarding the figure of the earth, and the disposition of land
+and water, would justify us in saying that a body consolidated when
+there was more centrifugal force by 3% than now, might not now be in all
+respects like the earth, so far as we know it at present."
+
+3. The third argument, based upon the age of the sun's heat, is
+confessedly less to be relied on than the two previous ones. It proceeds
+upon calculations as to the amount of heat which would be available by
+the falling together of masses from space, which gave rise by their
+impact to our sun. The vagueness of the data on which this argument
+rests may be inferred from the fact that in one passage P.G. Tait placed
+the limit of time during which the sun has been illuminating the earth
+as, "on the very highest computation, not more than about 15 or 20
+millions of years"; while, in another sentence of the same volume, he
+admitted that, "by calculations in which there is no possibility of
+large error, this hypothesis [of the origin of the sun's heat by the
+falling together of masses of matter] is thoroughly competent to explain
+100 millions of years' solar radiation at the present rate, perhaps
+more." In more recently reviewing his argument, Lord Kelvin expressed
+himself in favour of more strictly limiting geological time than he had
+at first been disposed to do. He insists that the time "was more than 20
+and less than 40 millions of years and probably much nearer 20 than 40."
+Geologists appear to have reluctantly brought themselves to believe that
+perhaps, after all, 100 millions of years might suffice for the
+evolution of geological history. But when the time was cut down to 15 or
+20 millions they protested that such a restricted period was
+insufficient for that evolution, and though they did not offer any
+effective criticism of the arguments of the physicists they felt
+convinced that there must be some flaw in the premises on which these
+arguments were based.
+
+By degrees, however, there have arisen among the physicists themselves
+grave doubts as to the validity of the physical evidence on which the
+limitation of the earth's age has been founded, and at the same time
+greater appreciation has been shown of the signification and strength of
+the geological proofs of the high antiquity of our planet. In an address
+from the chair of the Mathematical Section of the British Association in
+1886, Professor (afterwards Sir) George Darwin reviewed the controversy,
+and pronounced the following deliberate judgment in regard to it: "In
+considering these three arguments I have adduced some reasons against
+the validity of the first [tidal friction], and have endeavoured to show
+that there are elements of uncertainty surrounding the second [secular
+cooling of the earth]; nevertheless, they undoubtedly constitute a
+contribution of the first importance to physical geology. Whilst, then,
+we may protest against the precision with which Professor Tait seeks to
+deduce results from them, we are fully justified in following Sir
+William Thomson, who says that 'the existing state of things on the
+earth, life on the earth--all geological history showing continuity of
+life--must be limited within some such period of past time as 100
+million years'." Lord Kelvin has never dealt with the geological and
+palaeontological objections against the limitation of geological time to
+a few millions of years. But Professor Darwin, in the address just
+cited, uttered the memorable warning: "At present our knowledge of a
+definite limit to geological time has so little precision that we should
+do wrong summarily to reject theories which appear to demand longer
+periods of time than those which now appear allowable." In his
+presidential address to the British Association at Cape Town in 1905 he
+returned to the subject, remarking that the argument derived from the
+increase of underground temperature "seems to be entirely destroyed" by
+the discovery of the properties of radium. He thinks that "it does not
+seem extravagant to suppose that 500 to 1000 million years may have
+elapsed since the birth of the moon." He has "always believed that the
+geologists were more nearly correct than the physicists, notwithstanding
+the fact that appearances were so strongly against them," and he
+concludes thus: "It appears, then, that the physical argument is not
+susceptible of a greater degree of certainty than that of the
+geologists, and the scale of geological time remains in great measure
+unknown" (see also Tide, chap. viii.).
+
+In an address to the mathematical section of the American Association
+for the Advancement of Science in 1889, the vice-president of the
+section, R.S. Woodward, thus expressed himself with regard to the
+physical arguments brought forward by Lord Kelvin and Professor Tait in
+limitation of geological time: "Having been at some pains to look into
+this matter, I feel bound to state that, although the hypothesis appears
+to be the best which can be formulated at present, the odds are against
+its correctness. Its weak links are the unverified assumptions of an
+initial uniform temperature and a constant diffusivity. Very likely
+these are approximations, but of what order we cannot decide.
+Furthermore, if we accept the hypothesis, the odds appear to be against
+the present attainment of trustworthy numerical results, since the data
+for calculation, obtained mostly from observations on continental areas,
+are far too meagre to give satisfactory average values for the entire
+mass of the earth."
+
+Still more emphatic is the protest made from the physical side by
+Professor John Perry. He has attacked each of the three lines of
+argument of Lord Kelvin, and has impugned the validity of the
+conclusions drawn from them. The argument from tidal retardation he
+dismisses as fallacious, following in this contention the previous
+criticism of the Rev. Maxwell Close and Sir George Darwin. In dealing
+with the argument based on the secular cooling of the earth, he holds it
+to be perfectly allowable to assume a much higher conductivity for the
+interior of the globe, and that such a reasonable assumption would
+enable us greatly to increase our estimate of the earth's antiquity. As
+for the third argument, from the age of the sun's heat, he points out
+that the sun may have been repeatedly fed by a supply of meteorites from
+outside, while the earth may have been protected from radiation, and
+been able to retain much of its heat by being enveloped in a dense
+atmosphere. Remarking that "almost anything is possible as to the
+present internal state of the earth," he concludes thus: "To sum up, we
+can find no published record of any lower maximum age of life on the
+earth, as calculated by physicists, than 400 millions of years. From the
+three physical arguments Lord Kelvin's higher limits are 1000, 400 and
+500 million years. I have shown that we have reasons for believing that
+the age, from all these, may be very considerably underestimated. It is
+to be observed that if we exclude everything but the arguments from mere
+physics, the _probable_ age of life on the earth is much less than any
+of the above estimates; but if the palaeontologists have good reasons
+for demanding much greater times, I see nothing from the physicists'
+point of view which denies them four times the greatest of these
+estimates."
+
+A fresh line of argument against Lord Kelvin's limitation of the
+antiquity of our globe has recently been started by the remarkable
+discoveries in radio-activity. From the ascertained properties of radium
+it appears to be possible that our estimates of solar heat, as derived
+from the theory of gravitation, may have to be augmented ten or twenty
+times; that stores of radium and similar bodies within the earth may
+have indefinitely deferred the establishment of the present temperature
+gradient from the surface inward; that consequently the earth may have
+remained for long ages at a temperature not greatly different from that
+which it now possesses, and hence that the times during which our globe
+has supported animal and vegetable life may be very much longer than
+that allowed in the estimates previously made by physicists from other
+data (see RADIOACTIVITY).
+
+The arguments from the geological side against the physical contention
+that would limit the age of our globe to some 10 or 20 millions of years
+are mainly based on the observed rates of geological and biological
+changes at the present time upon land and sea, and on the nature,
+physical history and organic contents of the stratified crust of the
+earth. Unfortunately, actual numerical data are not obtainable in many
+departments of geological activity, and even where they can be procured
+they do not yet rest on a sufficiently wide collection of accurate and
+co-ordinated observations. But in some branches of dynamical geology,
+material exists for, at least, a preliminary computation of the rate of
+change. This is more especially the case in respect of the wide domain
+of denudation. The observational records of the action of the sea, of
+springs, rivers and glaciers are becoming gradually fuller and more
+trustworthy. A method of making use of these records for estimating the
+rate of denudation of the land has been devised. Taking the Mississippi
+as a general type of river action, it has been shown that the amount of
+material conveyed by this stream into the sea in one year is equivalent
+to the lowering of the general surface of the drainage basin of the
+river by 1/6000 of a foot. This would amount to one foot in 6000 years
+and 1000 ft. in 6 million years. So that at the present rate of waste in
+the Mississippi basin a whole continent might be worn away in a few
+millions of years.
+
+It is evident that as deposition and denudation are simultaneous
+processes, the ascertainment of the rate at which solid material is
+removed from the surface of the land supplies some necessary information
+for estimating the rate at which new sedimentary formations are being
+accumulated on the floor of the sea, and for a computation of the length
+of time that would be required at the present rate of change for the
+deposition of all the stratified rocks that enter into the composition
+of the crust of our globe. If the thickness of these rocks be assumed to
+be 100,000 ft., and if we could suppose them to have been laid down over
+as wide an area as that of the drainage basins from the waste of which
+they were derived, then at the present rate of denudation their
+accumulation would require some 600 millions of years. But, as Dr A.R.
+Wallace has justly pointed out, the tract of sea-floor over which the
+material derived from the waste of the terrestrial surface is laid down
+is at present much less than that from which this material is worn away.
+We have no means, however, of determining what may have been the ratio
+between the two areas in past time. Certainly ancient marine sedimentary
+rocks cover at the present day a much more extensive area than that in
+which they are now being elaborated. If we take the ratio postulated by
+Dr Wallace--1 to 19--the 100,000 ft. of sedimentary strata would require
+31 millions of years for their accumulation. It is quite possible,
+however, that this ratio may be much too high. There are reasons for
+believing that the proportion of coast-line to land area has been
+diminishing during geological time; in other words, that in early times
+the land was more insular and is now more continental. So that the 31
+millions of years may be much less than the period that would be
+required, even on the supposition of continuous uninterrupted denudation
+and sedimentation, during the whole of the time represented by the
+stratified formations.
+
+But no one who has made himself familiar with the actual composition of
+these formations and the detailed structure of the terrestrial crust can
+fail to recognize how vague, imperfect and misleading are the data on
+which such computations are founded. It requires no prolonged
+acquaintance with the earth's crust to impress upon the mind that one
+all-important element is omitted, and indeed can hardly be allowed for
+from want of sufficiently precise data, but the neglect of which must
+needs seriously impair the value of all numerical calculations made
+without it. The assumption that the stratified formations can be treated
+as if they consisted of a continuous unbroken sequence of sediments,
+indicating a vast and uninterrupted process of waste and deposition, is
+one that is belied on every hand by the actual structure of these
+formations. It can only give us a minimum of the time required; for,
+instead of an unbroken series, the sedimentary formations are full of
+"unconformabilities"--gaps in the sequence of the chronological
+records--as if whole chapters and groups of chapters had been torn out
+of a historical work. It can often be shown that these breaks of
+continuity must have been of vast duration, and actually exceeded in
+chronological importance thick groups of strata lying below and above
+them (see Part VI.). Moreover, even among the uninterrupted strata,
+where no such unconformabilities exist, but where the sediments follow
+each other in apparently uninterrupted sequence, and might be thought to
+have been deposited continuously at the same general rate, and without
+the intervention of any pause, it can be demonstrated that sometimes an
+inch or two of sediment might, on certain horizons, represent the
+deposit of an enormously longer period than a hundred or a thousand
+times the same amount of sediment on other horizons. A prolonged study
+of these questions leads to a profound conviction that in many parts of
+the geological record the time represented by sedimentary deposits may
+be vastly less than the time which is not so represented.
+
+It has often been objected that the present rate of geological change
+ought not to be taken as a measure of the rate in past time, because the
+total sum of terrestrial energy has been steadily diminishing, and
+geological processes must consequently have been more vigorous in former
+ages than they are now. Geologists do not pretend to assert that there
+has been no variation or diminution in the activities of the various
+processes which they have to study. What they do insist on is that the
+present rate of change is the only one which we can watch and measure,
+and which will thus supply a statistical basis for any computations on
+the subject. But it has been dogmatically affirmed that because
+terrestrial energy has been diminishing therefore all kinds of
+geological work must have been more vigorously and more rapidly carried
+on in former times than now; that there were far more abundant and more
+stupendous volcanoes, more frequent and more destructive earthquakes,
+more gigantic upheavals and subsidences, more powerful oceanic waves and
+tides, more violent atmospheric disturbances with heavier rainfall and
+more active denudation.
+
+It is easy to make these assertions, and they look plausible; but, after
+all, they rest on nothing stronger than assumption. They can be tested
+by an appeal to the crust of the earth, in which the geological history
+of our planet has been so fully recorded. Had such portentous
+manifestations of geological activity ever been the normal condition of
+things since the beginning of that history, there ought to be a record
+of them in the rocks. But no evidence for them has been found there,
+though it has been diligently sought for in all quarters of the globe.
+We may confidently assert that while geological changes may quite
+possibly have taken place on a gigantic scale in the earliest ages of
+the earth's existence, of which no geological record remains, there is
+no proof that they have ever done so since the time when the very oldest
+of the stratified formations were deposited. There is no need to
+maintain that they have always been conducted precisely on the same
+scale as now, or to deny that they may have gradually become less
+vigorous as the general sum of terrestrial energy has diminished. But we
+may unhesitatingly affirm that no actual evidence of any such
+progressive diminution of activity has been adduced from the geological
+record in the crust of the earth: that, on the contrary, no appearances
+have been detected there which necessarily demand the assumption of
+those more powerful operations postulated by physicists, or which are
+not satisfactorily explicable by reference to the existing scale of
+nature's processes.
+
+That this conclusion is warranted even with regard to the innate energy
+of the globe itself will be seen if we institute a comparison between
+the more ancient and the more recent manifestations of that energy.
+Take, for example, the proofs of gigantic plication, fracture and
+displacement within the terrestrial crust. These, as they have affected
+the most ancient rocks of Europe, have been worked out in great detail
+in the north-west of Scotland. But they are not essentially different
+from or on a greater scale than those which have been proved to have
+affected the Alps, and to have involved strata of so recent a date as
+the older Tertiary formations. On the contrary, it may be doubted
+whether any denuded core of an ancient mountain-chain reveals traces of
+such stupendous disturbances of the crust as those which have given rise
+to the younger mountain-chains of the globe. It may, indeed, quite well
+have been the rule that instead of diminishing in intensity of effect,
+the consequences of terrestrial contraction have increased in magnitude,
+the augmenting thickness of the crust offering greater resistance to
+the stresses, and giving rise to vaster plications, faults,
+thrust-planes and metamorphism, as this growing resistance had to be
+overcome.
+
+The assertion that volcanic action must have been more violent and more
+persistent in ancient times than it is now has assuredly no geological
+evidence in its support. It is quite true that there are vastly more
+remains of former volcanoes scattered over the surface of the globe than
+there are active craters now, and that traces of copious eruptions of
+volcanic material can be followed back into some of the oldest parts of
+the geological record. But we have no proof that ever at any one time in
+geological history there have been more or larger or more vigorous
+volcanoes than those of recent periods. It may be said that the absence
+of such proof ought not to invalidate the assertion until a far wider
+area of the earth's surface has been geologically studied. But most
+assuredly, as far as geological investigation has yet gone, there is an
+overwhelming body of evidence to show that from the earliest epochs in
+geological history, as registered in the stratified rocks, volcanic
+action has manifested itself very much as it does now, but on a less
+rather than on a greater scale. Nowhere can this subject be more
+exhaustively studied than in the British Isles, where a remarkably
+complete series of volcanic eruptions has been chronicled ranging from
+the earliest Palaeozoic down to older Tertiary time. The result of a
+prolonged study of British volcanic geology has demonstrated that, even
+to minute points of detail, there has been a singular uniformity in the
+phenomena from beginning to end. The oldest lavas and ashes differ in no
+essential respect from the youngest. Nor have they been erupted more
+copiously or more frequently. Many successive volcanic periods have
+followed each other after prolonged intervals of repose, each displaying
+the same general sequence of phenomena and similar evidence of gradual
+diminution and extinction. The youngest, instead of being the feeblest,
+were the most extensive outbursts in the whole of this prolonged series.
+
+If now we turn for evidence of the alleged greater activity of all the
+epigene or superficial forces, and especially for proofs of more rapid
+denudation and deposition on the earth's surface, we search for it in
+vain among the stratified formations of the terrestrial crust. Had the
+oldest of these rocks been accumulated in a time of great atmospheric
+perturbation, of torrential rains, colossal tides and violent storms, we
+might surely expect to find among the sediments some proof of such
+disturbed meteorological and geographical conditions. We should look, on
+the one hand, for tumultuous accumulations of coarse unworn detritus,
+rapidly swept by rains, floods and waves from land to sea, and on the
+other hand, for an absence of any evidence of the tranquil and
+continuous deposit of such fine laminated silt as could only settle in
+quiet water. But an appeal to the geological record is made in vain for
+any such proofs. The oldest sediments, like the youngest, reveal the
+operation only of such agents and such rates of activity as are still to
+be witnessed in the accumulation of the same kind of deposits. If, for
+instance, we search the most ancient thick sedimentary formation in
+Britain--the Torridon Sandstone of north-west Scotland, which is older
+than the oldest fossiliferous deposits--we meet with nothing which might
+not be found in any Palaeozoic, Mesozoic or Cainozoic group of similar
+sediments. We see an accumulation, at least 8000 or 10,000 ft. thick, of
+consolidated sand, gravel and mud, such as may be gathering now on the
+floor of any large mountain-girdled lake. The conglomerates of this
+ancient series are not pell-mell heaps of angular detritus, violently
+swept away from the land and huddled promiscuously on the sea-floor.
+They are, in general, built up of pebbles that have been worn smooth,
+rounded and polished by prolonged attrition in running water, and they
+follow each other on successive platforms with intervening layers of
+finer sediment. The sandstones are composed of well water-worn sand,
+some of which has been laid down so tranquilly that its component grains
+have been separated out in layers according to their specific gravity,
+in such manner that they now present dark laminae in which particles of
+magnetic iron, zircon and other heavy minerals have been sifted out
+together, just as iron-sand may be seen gathered into thin sheets on
+sandy beaches at the present day. Again, the same series of primeval
+sediments includes intercalations of fine silt, which has been deposited
+as regularly and intermittently there as it has been among the most
+recent formations. These bands of shale have been diligently searched
+for fossils, as yet without success; but they may eventually disclose
+organic remains older than any hitherto found in Europe.
+
+We now come to the consideration of the palaeontological evidence as to
+the value of geological time. Here the conclusions derived from a study
+of the structure of the sedimentary formations are vastly strengthened
+and extended. In the first place, the organization of the most ancient
+plants and animals furnishes no indication that they had to contend with
+any greater violence of storm, flood, wave or ocean-current than is
+familiar to their modern descendants. The oldest trees, shrubs, ferns
+and club-mosses display no special structures that suggest a difference
+in the general conditions of their environment. The most ancient
+crinoids, sponges, crustaceans, arachnids and molluscs were as
+delicately constructed as those of to-day, and their remains are often
+found in such perfect preservation as to show that neither during their
+lifetime nor after their death were they subject to any greater violence
+of the elements than their living representatives now experience. Of
+much more cogency, however, is the evidence supplied by the grand upward
+succession of organic forms, from the most ancient stratified rocks up
+to the present day. No biologist now doubts for a moment that this
+marvellous succession is the result of a gradual process of evolution
+from lower to higher types of organization. There may be differences of
+opinion as to the causes which have governed this process and the order
+of the steps through which it has advanced, but no one who is conversant
+with the facts will now venture to deny that it has taken place, and
+that, on any possible explanation of its progress, it must have demanded
+an enormous lapse of time. In the Cambrian or oldest fossiliferous
+formations there is already a large and varied fauna, in which the
+leading groups of invertebrate life are represented. On no tenable
+hypothesis can these be regarded as the first organisms that came into
+being on our planet. They must have had a long ancestry, and as Darwin
+first maintained, the time required for their evolution may have been
+"as long as, or probably far longer than, the whole interval from the
+Silurian [Cambrian] age to the present day." The records of these
+earliest eras of organic development have unfortunately not survived the
+geological revolutions of the past; at least, they have not yet been
+recovered. But it cannot be doubted that they once existed and
+registered their testimony to the prodigious lapse of time prior to the
+deposition of the most ancient fossiliferous formations which have
+escaped destruction.
+
+The impressive character of the evidence furnished by the sequence of
+organic forms throughout the great series of fossiliferous strata can
+hardly be fully realized without a detailed and careful study of the
+subject. Professor E.B. Poulton, in an address to the zoological section
+of the British Association at the Liverpool Meeting in 1896, showed how
+overwhelming are the demands which this evidence makes for long periods
+of time, and how impossible it is of comprehension unless these demands
+be conceded. The history of life upon the earth, though it will probably
+always be surrounded with great and even insuperable difficulties,
+becomes broadly comprehensible in its general progress when sufficient
+time is granted for the evolution which it records; but it remains
+unintelligible on any other conditions.
+
+Taken then as a whole, the body of evidence, geological and
+palaeontological, in favour of the high antiquity of our globe is so
+great, so manifold, and based on such an ever-increasing breadth of
+observation and reflection, that it may be confidently appealed to in
+answer to the physical arguments which would seek to limit that
+antiquity to ten or twenty millions of years. In the present state of
+science it is out of our power to state positively what must be the
+lowest limit of the age of the earth. But we cannot assume it to be much
+less, and it may possibly have been much more, than the 100 millions of
+years which Lord Kelvin was at one time willing to concede.[2]
+
+
+PART III.--GEOGNOSY. THE INVESTIGATION OF THE NATURE AND COMPOSITION OF
+THE MATERIALS OF WHICH THE EARTH CONSISTS
+
+This division of the science is devoted to a description of the parts of
+the earth--of the atmosphere and ocean that surround the planet, and
+more especially of the solid materials that underlie these envelopes and
+extend downwards to an unknown distance into the interior. These various
+constituents of the globe are here considered as forms of matter capable
+of being analysed, and arranged according to their composition and the
+place they take in the general composition of the globe.
+
+Viewed in the simplest way the earth may be regarded as made up of three
+distinct parts, each of which ever since an early period of planetary
+history has been the theatre of important geological operations. (1) An
+envelope of air, termed the _atmosphere_, which surrounds the whole
+globe; (2) A lower and less extensive envelope of water, known as the
+_hydrosphere_ (Gr. [Greek: hydōr], water) which, constituting the oceans
+and seas, covers nearly three-fourths of the underlying solid surface of
+the planet; (3) A globe, called the _lithosphere_ (Gr. [Greek: lithos],
+stone), the external part of which, consisting of solid stone, forms the
+_crust_, while underneath, and forming the vast mass of the interior,
+lies the _nucleus_, regarding the true constitution of which we are
+still ignorant.
+
+1. _The Atmosphere._--The general characters of the atmosphere are
+described in separate articles (see especially ATMOSPHERE; METEOROLOGY).
+Only its relations to geology have here to be considered. As this
+gaseous envelope encircles the whole globe it is the most universally
+present and active of all the agents of geological change. Its efficacy
+in this respect arises partly from its composition, and the chemical
+reactions which it effects upon the surface of the land, partly from its
+great variations in temperature and moisture, and partly from its
+movements.
+
+  Many speculations have been made regarding the chemical composition of
+  the atmosphere during former geological periods. There can indeed be
+  little doubt that it must originally have differed greatly from its
+  present condition. If the whole mass of the planet originally existed
+  in a gaseous state, there would be practically no atmosphere. The
+  present outer envelope of air may be considered to be the surviving
+  relic of this condition, after all the other constituents have been
+  incorporated into the hydrosphere and lithosphere. The oxygen, which
+  now forms fully a half of the outer crust of the earth, was doubtless
+  originally, whether free or in combination, part of the atmosphere.
+  So, too, the vast beds of coal found all over the world, in geological
+  formations of many different ages, represent so much carbonic acid
+  once present in the air. The chlorides and other salts in the sea may
+  likewise partly represent materials carried down out of the atmosphere
+  in the primitive condensation of the aqueous vapour, though they have
+  been continually increased ever since by contributions from the
+  drainage of the land. It has often been suggested that, during the
+  Carboniferous period, the atmosphere must have been warmer and more
+  charged with aqueous vapour and carbon dioxide than at the present
+  day, to admit of so luxuriant a flora as that from which the
+  coal-seams were formed. There seems, however, to be at present no
+  method of arriving at any certainty on this subject. Lastly, the
+  amount of carbonic acid absorbed in the weathering of rocks at the
+  surface, and the consequent production of carbonates, represents an
+  enormous abstraction of this gas.
+
+  As at present constituted, the atmosphere is regarded as a mechanical
+  mixture of nearly four volumes of nitrogen and one of oxygen, together
+  with an average of 3.5 parts of carbon dioxide in every 10,000 parts
+  of air, and minute quantities of various other gases and solid
+  particles. Of the vapours contained in it by far the most important is
+  that of water which, although always present, varies greatly in amount
+  according to variations in temperature. By condensation the water
+  vapour appears in visible form as dew, mist, cloud, rain, hail, snow
+  and ice, and in these forms includes and carries down some of the
+  other vapours, gases and solid particles present in the air. The
+  circulation of water from the atmosphere to the land, from the land to
+  the sea, and again from the sea to the land, forms the great
+  geological process whereby the habitable condition of the planet is
+  maintained and the surface of the land is sculptured (Part IV.).
+
+2. _The Hydrosphere._--The water envelope covers nearly three-fourths of
+the surface of the earth, and forms the various oceans and seas which,
+though for convenience of reference distinguished by separate names, are
+all linked together in one great body. The physical characters of this
+vast envelope are discussed in separate articles (see OCEAN and
+OCEANOGRAPHY). Viewed from the geological standpoint, the features of
+the sea that specially deserve attention are first the composition of
+its waters, and secondly its movements.
+
+  Sea-water is distinguished from that of ordinary lakes and rivers by
+  its greater specific gravity and its saline taste. Its average density
+  is about 1.026, but it varies even within the same ocean, being least
+  where large quantities of fresh water are added from rain or melting
+  snow and ice, and greatest where evaporation is most active. That
+  sea-water is heavier than fresh arises from the greater proportion of
+  salts which it contains in solution. These salts constitute about
+  three and a half parts in every hundred of water. They consist mainly
+  of chlorides of sodium and magnesium, the sulphates of magnesium,
+  calcium and potassium, with minuter quantities of magnesium bromide
+  and calcium carbonate. Still smaller proportions of other substances
+  have been detected, gold for example having been found in the
+  proportion of 1 part in 15,180,000.
+
+  That many of the salts have existed in the sea from the time of its
+  first condensation out of the primeval atmosphere appears to be
+  probable. It is manifest, however, that, whatever may have been the
+  original composition of the oceans, they have for a vast section of
+  geological time been constantly receiving mineral matter in solution
+  from the land. Every spring, brook and river removes various salts
+  from the rocks over which it moves, and these substances, thus
+  dissolved, eventually find their way into the sea. Consequently
+  sea-water ought to contain more or less traceable proportions of every
+  substance which the terrestrial waters can remove from the land, in
+  short, of probably every element present in the outer shell of the
+  globe, for there seems to be no constituent of this earth which may
+  not, under certain circumstances, be held in solution in water.
+  Moreover, unless there be some counteracting process to remove these
+  mineral ingredients, the ocean water ought to be growing, insensibly
+  perhaps, but still assuredly, saltier, for the supply of saline matter
+  from the land is incessant.
+
+  To the geologist the presence of mineral solutions in sea-water is a
+  fact of much importance, for it explains the origin of a considerable
+  part of the stratified rocks of the earth's crust. By evaporation the
+  water has given rise to deposits of rock-salt, gypsum and other
+  materials. The lime contained in solution, whether as sulphate or
+  carbonate, has been extracted by many tribes of marine animals, which
+  have thus built up out of their remains vast masses of solid
+  limestone, of which many mountain-chains largely consist.
+
+  Another important geological feature of the sea is to be seen in the
+  fact that its basins form the great receptacles for the detritus worn
+  away from the land. Besides the limestones, the visible parts of the
+  terrestrial crust are, in large measure, composed of sedimentary rocks
+  which were originally laid down on the sea-bottom. Moreover, by its
+  various movements, the sea occupies a prominent place among the
+  epigene or superficial agents which produce geological changes on the
+  surface of the globe.
+
+3. _The Lithosphere._--Beneath the gaseous and liquid envelopes lies the
+solid part of the planet, which is conveniently regarded as consisting
+of two parts,--(a) the crust, and (b) the interior or nucleus.
+
+
+  The crust.
+
+It was for a long time a prevalent belief that the interior of the globe
+is a molten mass round which an outer shell has gradually formed through
+cooling. Hence the term "crust" was applied to this external solid
+envelope, which was variously computed to be 10, 20, or more miles in
+thickness. The portion of this crust accessible to human observation was
+seen to afford abundant evidence of vast plications and corrugations of
+its substance, which were regarded as only explicable on the supposition
+of a thin solid collapsible shell floating on a denser liquid interior.
+When, however, physical arguments were adduced to show the great
+rigidity of the earth as a whole, the idea of a thin crust enclosing a
+molten nucleus was reluctantly abandoned by geologists, who found the
+problem of the earth's interior to be incapable of solution by any
+evidence which their science could produce. They continued, however, to
+use the term "crust" as a convenient word to denote the cool outer layer
+of the earth's mass, the structure and history of which form the main
+subjects of geological investigation. More recently, however, various
+lines of research have concurred in suggesting that, whatever may be the
+condition of the interior, its substance must differ greatly from that
+of the outer shell, and that there may be more reason than appeared for
+the retention of the name of crust. Observations on earthquake motion by
+Dr John Milne and others, show that the rate and character of the waves
+transmitted through the interior of the earth differ in a marked degree
+from those propagated along the crust. This difference indicates that
+rocky material, such as we know at the surface, may extend inwards for
+some 30 m., below which the earth's interior rapidly becomes fairly
+homogeneous and possesses a high rigidity. From measurements of the
+force of gravity in India by Colonel S.G. Burrard, it has been inferred
+that the variations in density of the outer parts of the earth do not
+descend farther than 30 or 40 m., which might be assumed to be the limit
+of the thickness of the crust. Recent researches in regard to the
+radio-active substances present in rocks suggest that the crust is not
+more than 50 m. thick, and that the interior differs from it in
+possessing little or no radio-active material.
+
+
+  The interior.
+
+Though we cannot hope ever to have direct acquaintance with more than
+the mere outside skin of our planet, we may be led to infer the
+irregular distribution of materials within the crust from the present
+distribution of land and water, and the observed differences in the
+amount of deflection of the plumb-line near the sea and near
+mountain-chains. The fact that the southern hemisphere is almost wholly
+covered with water appears explicable only on the assumption of an
+excess of density in the mass of that portion of the planet. The
+existence of such a vast sheet of water as that of the Pacific Ocean is
+to be accounted for, as Archdeacon J.H. Pratt pointed out, by the
+presence of "some excess of matter in the solid parts of the earth
+between the Pacific Ocean and the earth's centre, which retains the
+water in its place, otherwise the ocean would flow away to the other
+parts of the earth." A deflection of the plumb-line towards the sea,
+which has in a number of cases been observed, indicates that "the
+density of the crust beneath the mountains must be less than that below
+the plains, and still less than that below the ocean-bed." Apart
+therefore from the depression of the earth's surface in which the oceans
+lie, we must regard the internal density, whether of crust or nucleus,
+to be somewhat irregularly arranged, there being an excess of heavy
+materials in the water hemisphere, and beneath the ocean-beds, as
+compared with the continental masses.
+
+In our ignorance regarding the chemical constitution of the nucleus of
+our planet, an argument has sometimes been based upon the known fact
+that the specific gravity of the globe as a whole is about double that
+of the crust. This has been held by some writers to prove that the
+interior must consist of much heavier material and is therefore probably
+metallic. But the effect of pressure ought to make the density of the
+nucleus much higher, even if the interior consisted of matter no heavier
+than the crust. That the total density of the planet does not greatly
+exceed its observed amount seems only explicable on the supposition that
+some antagonistic force counteracts the effects of pressure. The only
+force we can suppose capable of so acting is heat. But comparatively
+little is yet known regarding the compression of gases, liquids and
+solids under such vast pressures as must exist within the nucleus.
+
+That the interior of the earth possesses a high temperature is inferred
+from the evidence of various sources. (1) Volcanoes, which are openings
+that constantly, or intermittently, give out hot vapours and molten lava
+from reservoirs beneath the crust. Besides active volcanoes, it is known
+that former eruptive vents have been abundantly and widely distributed
+over the globe from the earliest geological periods down to our own day.
+(2) Hot springs are found in many parts of the globe, with temperatures
+varying up to the boiling point of water. (3) From mines, tunnels and
+deep borings into the earth it has been ascertained that in all quarters
+of the globe below the superficial zone of invariable temperature, there
+is a progressive increase of heat towards the interior. The rate of this
+increase varies, being influenced, among other causes, by the varying
+conductivity of the rocks. But the average appears to be about 1° Fahr.
+for every 50 or 60 ft. of descent, as far down as observations have
+extended. Though the increase may not advance in the same proportion at
+great depths, the inference has been confidently drawn that the
+temperature of the nucleus must be exceedingly high.
+
+The probable condition of the earth's interior has been a fruitful
+source of speculation ever since geology came into existence; but no
+general agreement has been arrived at on the subject. Three chief
+hypotheses have been propounded: (1) that the nucleus is a molten mass
+enclosed within a solid shell; (2) that, save in local vesicular spaces
+which may be filled with molten or gaseous material, the globe is solid
+and rigid to the centre; (3) that the great body of the nucleus consists
+of incandescent vapours and gases, especially vaporous iron, which under
+the gigantic pressure within the earth are so compressed as to confer
+practical rigidity on the globe as a whole, and that outside this main
+part of the nucleus the gases pass into a shell of molten magma, which,
+in turn, shades off outwards into the comparatively thin, cool
+solidified crust. Recent seismological observations have led to the
+inference that the outer crust, some 30 to 45 m. thick, must rapidly
+merge into a fairly homogeneous nucleus which, whatever be its
+constitution, transmits undulatory movements through its substance with
+uniform velocity and is believed to possess a high rigidity.
+
+The origin of the earth's high internal temperature has been variously
+accounted for. Most usually it has been assumed to be the residue of the
+original "tracts of fluent heat" out of which the planet shaped itself
+into a globe. According to another supposition the effects of the
+gradual gravitational compression of the earth's mass have been the main
+source of the high temperature. Recent researches in radio-activity, to
+which reference has already been made, have indicated another possible
+source of the internal heat in the presence of radium in the rocks of
+the crust. This substance has been detected in all igneous rocks,
+especially among the granites, in quantity sufficient, according to the
+Hon. R.J. Strutt, to account for the observed temperature-gradient in
+the crust, and to indicate that this crust cannot be more than 45 m.
+thick, otherwise the outflow of heat would be greater than the amount
+actually ascertained. Inside this external crust containing radio-active
+substances, it is supposed, as already stated, that the nucleus consists
+of some totally different matter containing little or no radium.
+
+  _Constitution of the Earth's Crust._--As the crust of the earth
+  contains the "geological record," or stony chronicle from which
+  geology interprets the history of our globe, it forms the main subject
+  of study to the geologist. The materials of which this crust consists
+  are known as minerals and rocks. From many chemical analyses, which
+  have been made of these materials, the general chemical constitution
+  of, at least, the accessible portion of the crust has been
+  satisfactorily ascertained. This information becomes of much
+  importance in speculations regarding the early history of the globe.
+  Of the elements known to the chemist the great majority form but a
+  small proportion of the composition of the crust, which is mainly
+  built up of about twenty of them. Of these by far the most important
+  are the non-metallic elements oxygen and silicon. The former forms
+  about 47% and the latter rather more than 28% of the original crust,
+  so that these two elements make up about three-fourths of the whole.
+  Next after them come the metals aluminium (8.16%), iron (4.64),
+  calcium (3.50), magnesium (2.62), sodium (2.63), and potassium (2.35).
+  The other twelve elements included in the twenty vary in amount from a
+  proportion of 0.41% in the case of titanium, to not more than 0.01% of
+  chlorine, fluorine, chromium, nickel and lithium. The other fifty or
+  more elements exist in such minute proportions in the crust that,
+  probably, not one of them amounts to as much as 0.01%, though they
+  include the useful metals, except iron. Taking the crust, and the
+  external envelopes of the ocean and the air, we thus perceive that
+  these outer parts of our planet consist of more than three-fourths of
+  non-metals and less than one-fourth of metals.
+
+  The combinations of the elements which are of most importance in the
+  constitution of the terrestrial crust consist of oxides. From the mean
+  of a large number of analyses of the rocks of the lower or primitive
+  portion of the crust, it has been ascertained that silica (SiO2) forms
+  almost 60% and alumina (Al2O3) upwards of 15% of the whole. The other
+  combinations in order of importance are lime (CaO) 4.90%, magnesia
+  (MgO) 4.36, soda (Na2O) 3.55, ferrous oxide (FeO) 3.52, potash (K2O)
+  2.80, ferric oxide (Fe2O3) 2.63, water (H2O) 1.52, titanium oxide
+  (TiO2) 0.60, phosphoric acid (P2O5) 0.22; the other combinations of
+  elements thus form less than 1% of the crust.
+
+  These different combinations of the elements enter into further
+  combinations with each other so as to produce the wide assortment of
+  simple minerals (see MINERALOGY). Thus, silica and alumina are
+  combined to form the aluminous silicates, which enter so largely into
+  the composition of the crust of the earth. The silicates of magnesia,
+  potash and soda constitute other important families of minerals. A
+  mass of material composed of one, but more usually of more than one
+  mineral, is known as a _rock_. Under this term geologists are
+  accustomed to class not only solid stone, such as granite and
+  limestone, but also less coherent materials such as clay, peat and
+  even loose sand. The accessible portion of the earth's crust consists
+  of various kinds of rocks, which differ from each other in structure,
+  composition and origin, and are therefore susceptible of diverse
+  classifications according to the point of view from which they are
+  considered. The details of this subject will be found in the article
+  PETROLOGY.
+
+  _Classification of Rocks._--Various systems of classification of rocks
+  have been proposed, but none of them is wholly satisfactory. The most
+  useful arrangement for most purposes of the geologist is one based on
+  the broad differences between them in regard to their mode of origin.
+  From this point of view they may be ranged in three divisions:
+
+  1. In the first place, a large number of rocks may be described as
+  original or underived, for it is not possible to trace them back to
+  any earlier source. They belong to the primitive constitution of the
+  planet, and, as they have all come up from below through the crust,
+  they serve to show the nature of the material which lies immediately
+  below the outer parts of that crust. They include the numerous
+  varieties of lava, which have been poured out in a molten state from
+  volcanic vents, also a great series of other rocks which, though they
+  may never have been erupted to the surface, have been forced upward in
+  a melted condition into the other rocks of the crust and have
+  solidified there. From their mode of origin this great class of rocks
+  has been called "igneous" or "eruptive." As they generally show no
+  definite internal structure save such as may result from joints, they
+  have been termed "massive" or "unstratified," to distinguish them from
+  those of the second division which are strongly marked out by the
+  presence of a stratified structure. The igneous rocks present a
+  considerable range of composition. For the most part they consist
+  mainly of aluminous silicates, some of them being highly acid
+  compounds with 75% or more of silica. But they also include highly
+  basic varieties wherein the proportion of silica sinks to 40%, and
+  where magnesia greatly predominates over alumina. The textures of
+  igneous rocks likewise comprise a wide series of varieties. On the one
+  hand, some are completely vitreous, like obsidian, which is a natural
+  glass. From this extreme every gradation may be traced through gradual
+  increase of the products of devitrification, until the mass may become
+  completely crystalline. Again, some crystalline igneous rocks are so
+  fine in grain as not to show their component crystals save under the
+  microscope, while in others the texture is so coarse as to present the
+  component minerals in separate crystals an inch or more in length.
+  These differences indicate that, at first, the materials of the rock
+  may have been as completely molten as artificial glass, and that the
+  crystalline condition has been subsequently developed by cooling, and
+  the separation of the chemical constituents into definite crystalline
+  minerals. Many of the characters of igneous rocks have been reproduced
+  experimentally by fusing together their minerals, or the constituents
+  of their minerals, in the proper proportion. But it has not yet been
+  found possible to imitate the structure of such rocks as granite.
+  Doubtless these rocks consolidated with extreme slowness at great
+  depths below the surface, under vast pressures and probably in the
+  presence of water or water-vapour--conditions which cannot be
+  adequately imitated in a laboratory.
+
+  Though the igneous rocks occupy extensive areas in some countries,
+  they nevertheless cover a much smaller part of the whole surface of
+  the land than is taken up by the second division or stratified rocks.
+  But they increase in quantity downwards and probably extend
+  continuously round the globe below the other rocks. This important
+  series brings before us the relations of the molten magma within the
+  earth to the overlying crust and to the outer surface. On the one
+  hand, it includes the oldest and most deep-seated extravasations of
+  that magma, which have been brought to light by ruptures and upheavals
+  of the crust and prolonged denudation. On the other, it presents to
+  our study the varied outpourings of molten and fragmentary materials
+  in the discharges of modern and ancient volcanoes. Between these two
+  extremes of position and age, we find that the crust has been, as it
+  were, riddled with injections of the magma from below. These features
+  will be further noticed in Part V. of this article.
+
+  2. The "sedimentary" or "stratified rocks" form by much the larger
+  part of the dry land of the globe, and they are prolonged to an
+  unknown distance from the shores under the bed of the sea. They
+  include those masses of mineral matter which, unlike the igneous
+  rocks, can be traced back to a definite origin on the surface of the
+  earth. Three distinct types may be recognized among them: (a) By far
+  the largest proportion of them consists of different kinds of sediment
+  derived from the disintegration of pre-existing rocks. In this
+  "fragmental" group are placed all the varieties of shingle, gravel,
+  sand, clay and mud, whether these materials remain in a loose
+  incoherent condition, or have been compacted into solid stone. (b)
+  Another group consists of materials that have been deposited by
+  chemical precipitation from solution in water. The white sinter laid
+  down by calcareous springs is a familiar example on a small scale.
+  Beds of rock-salt, gypsum and dolomite have, in some regions, been
+  accumulated to a thickness of many thousand feet, by successive
+  precipitations of the salt contained in the water of inland seas. (c)
+  An abundant and highly important series of sedimentary formations has
+  been formed from the remains of plants and animals. Such accumulations
+  may arise either from the transport and deposit of these remains, as
+  in the case of sheets of drift-wood, and banks of drifted sea-shells,
+  or from the growth and decay of the organisms on the spot, as happens
+  in peat bogs and in coral-reefs.
+
+  As the sedimentary rocks have for the most part been laid down under
+  water, and more especially on the sea-floor, they are often spoken of
+  as "aqueous," in contradistinction to the igneous rocks. Some of them,
+  however, are accumulated by the drifting action of wind upon loose
+  materials, and are known as "aeolian" formations. Familiar instances
+  of such wind-formed deposits are the sand-dunes along many parts of
+  the sea coast. Much more extensive in area are the sands of the great
+  deserts in the arid regions of the globe.
+
+  It is from the sedimentary rocks that the main portion of geological
+  history is derived. They have been deposited one over another in
+  successive strata from a remote period in the development of the globe
+  down to the present time. From this arrangement they have been termed
+  "stratified," in contrast to the unstratified or igneous series. They
+  have preserved memorials of the geographical revolutions which the
+  surface of the earth has undergone; and above all, in the abundant
+  fossils which they have enclosed, they furnish a momentous record of
+  the various tribes of plants and animals which have successively
+  flourished on land and sea. Their investigation is thus the most
+  important task which devolves upon the geologist.
+
+  3. In the third place comes a series of rocks which are not now in
+  their original condition, but have undergone such alteration as to
+  have acquired new characters that more or less conceal their first
+  structures. Some of them can be readily recognized as altered igneous
+  masses; others are as manifestly of sedimentary origin; while of many
+  it is difficult to decide what may have been their pristine character.
+  To this series the term "metamorphic" has been applied. Its members
+  are specially distinguished by a prevailing fissile, or schistose,
+  structure which they did not at first possess, and which differs from
+  anything found in unaltered igneous or sedimentary rocks. This
+  fissility is combined with a more or less pronounced crystalline
+  structure. These changes are believed to be the result of movements
+  within the crust of the earth, whereby the most solid rocks were
+  crushed and sheared, while, at the same time, under the influence of a
+  high temperature and the presence of water, they underwent internal
+  chemical reactions, which led to a rearrangement and recomposition of
+  their mineral constituents and the production of a crystalline
+  structure (see METAMORPHISM).
+
+  Among the less altered metamorphic rocks of sedimentary origin, the
+  successive laminae of deposit of the original sediment can be easily
+  observed; but they are also traversed by a new set of divisional
+  planes, along which they split across the original bedding. Together
+  with this superinduced cleavage there have been developed in them
+  minute hairs, scales and rudimentary crystals. Further stages of
+  alteration are marked by the increase of micaceous scales, garnets and
+  other minerals, especially along the planes of cleavage, until the
+  whole rock becomes crystalline, and displays its chief component
+  minerals in successive discontinuous folia which merge into each
+  other, and are often crumpled and puckered. Massive igneous rocks can
+  be observed to have undergone intense crushing and cleavage, and to
+  have ultimately assumed a crystalline foliated character. Rocks which
+  present this aspect are known as schists (q.v.). They range from the
+  finest silky slates, or phyllites, up to the coarsest gneisses, which
+  in hand-specimens can hardly be distinguished from granites. There is
+  indeed every reason to believe that such gneisses were probably
+  originally true granites, and that their foliation and
+  recrystallization have been the result of metamorphism.
+
+  The schists are more especially to be found in the heart of
+  mountain-chains, and in regions where the lowest and oldest parts of
+  the earth's crust have, in the course of geological revolutions, been
+  exposed to the light of day. They have been claimed by some writers
+  to be part of the original or primitive surface of our globe that
+  first consolidated on the molten nucleus. But the progress of
+  investigation all over the world has shown that this supposition
+  cannot be sustained. The oldest known rocks present none of the
+  characters of molten material that has cooled and hardened in the air,
+  like the various forms of recent lava. On the contrary, they possess
+  many of the features characteristic of bodies of eruptive material
+  that have been injected into the crust at some depth underground, and
+  are now visible at the surface, owing to the removal by denudation of
+  the rocks under which they consolidated. In their less foliated
+  portions they can be recognized as true eruptive rocks. In many places
+  gneisses that possess a thoroughly typical foliation have been found
+  to pierce ancient sedimentary formations as intrusive bosses and
+  veins.
+
+
+PART IV.--DYNAMICAL GEOLOGY
+
+This section of the science includes the investigation of those
+processes of change which are at present in progress upon the earth,
+whereby modifications are made on the structure and composition of the
+crust, on the relations between the interior and the surface, as shown
+by volcanoes, earthquakes and other terrestrial disturbances, on the
+distribution of oceans and continents, on the outlines of the land, on
+the form and depth of the sea-bottom, on climate, and on the races of
+plants and animals by which the earth is tenanted. It brings before us,
+in short, the whole range of activities which it is the province of
+geology to study, and leads us to precise notions regarding their
+relations to each other and the results which they achieve. A knowledge
+of this branch of the subject is thus the essential groundwork of a true
+and fruitful acquaintance with the principles of geology, seeing that it
+necessitates a study of the present order of nature, and thus provides a
+key for the interpretation of the past.
+
+The whole range of operations included within the scope of inquiry in
+this branch of the science may be regarded as a vast cycle of change,
+into which we may break at any point, and round which we may travel,
+only to find ourselves brought back to our starting-point. It is a
+matter of comparatively small moment at what part of the cycle we begin
+our inquiries. We shall always find that the changes we see in action
+have resulted from some that preceded, and give place to others which
+follow them.
+
+At an early time in the earth's history, anterior to any of the periods
+of which a record remains in the visible rocks, the chief sources of
+geological action probably lay within the earth itself. If, as is
+generally supposed, the planet still retained a great store of its
+initial heat, it was doubtless the theatre of great chemical changes,
+giving rise, perhaps, to manifestations of volcanic energy somewhat like
+those which have so marvellously roughened the surface of the moon. As
+the outer layers of the globe cooled, and the disturbances due to
+internal heat and chemical action became less marked, the conditions
+would arise in which the materials for geological history were
+accumulated. The influence of the sun, which must always have operated,
+would then stand out more clearly, giving rise to that wide circle of
+superficial changes wherein variations of temperature and the
+circulation of air and water over the surface of the earth come into
+play.
+
+In the pursuit of his inquiries into the past history and into the
+present _régime_ of the earth, the geologist must needs keep his mind
+ever open to the reception of evidence for kinds and especially for
+degrees of action which he had not before imagined. Human experience has
+been too short to allow him to assume that all the causes and modes of
+geological change have been definitively ascertained. On the earth
+itself there may remain for future discovery evidence of former
+operations by heat, magnetism, chemical change or otherwise, which may
+explain many of the phenomena with which geology has to deal. Of the
+influences, so many and profound, which the sun exerts upon our planet,
+we can as yet only perceive a little. Nor can we tell what other
+cosmical influences may have lent their aid in the evolution of
+geological changes.
+
+Much useful information regarding many geological processes has been
+obtained from experimental research in laboratories and elsewhere, and
+much more may be confidently looked for from future extensions of this
+method of inquiry. The early experiments of Sir James Hall, already
+noticed, formed the starting-point for numerous subsequent researches,
+which have elucidated many points in the origin and history of rocks. It
+is true that we cannot hope to imitate those operations of nature which
+demand enormous pressures and excessively high temperatures combined
+with a long lapse of time. But experience has shown that in regard to a
+large number of processes, it is possible to imitate nature's working
+with sufficient accuracy to enable us to understand them, and so to
+modify and control the results as to obtain a satisfactory solution of
+some geological problems.
+
+In the present state of our knowledge, all the geological energy upon
+and within the earth must ultimately be traced back to the primeval
+energy of the parent nebula or sun. There is, however, a certain
+propriety and convenience in distinguishing between that part of it
+which is due to the survival of some of the original energy of the
+planet and that part which arises from the present supply of energy
+received day by day from the sun. In the former case we have to deal
+with the interior of the earth, and its reaction upon the surface; in
+the latter, we deal with the surface of the earth and to some extent
+with its reaction on the interior. This distinction allows of a broad
+treatment of the subject under two divisions:
+
+I. Hypogene or Plutonic Action: The changes within the earth caused by
+internal heat, mechanical movement and chemical rearrangements.
+
+II. Epigene or Surface Action: The changes produced on the superficial
+parts of the earth, chiefly by the circulation of air and water set in
+motion by the sun's heat.
+
+
+_DIVISION I.--HYPOGENE OR PLUTONIC ACTION_
+
+In the discussion of this branch of the subject we must carry in our
+minds the conception of a globe still possessing a high internal
+temperature, radiating heat into space and consequently contracting in
+bulk. Portions of molten rocks from inside are from time to time poured
+out at the surface. Sudden shocks are generated by which destructive
+earthquakes are propagated through the diameter of the globe as well as
+to and along its surface. Wide geographical areas are pushed up or sink
+down. In the midst of these movements remarkable changes are produced
+upon the rocks of the crust; they are plicated, fractured, crushed,
+rendered crystalline and even fused.
+
+
+  (A) _Volcanoes and Volcanic Action._
+
+  This subject is discussed in the article VOLCANO, and only a general
+  view of its main features will be given here. Under the term volcanic
+  action (vulcanism, vulcanicity) are embraced all the phenomena
+  connected with the expulsion of heated materials from the interior of
+  the earth to the surface. A volcano may be defined as a conical hill
+  or mountain, built up wholly or mainly of materials which have been
+  ejected from below, and which have accumulated around the central vent
+  of eruption. As a rule its truncated summit presents a cup-shaped
+  cavity, termed the crater, at the bottom of which is the opening of
+  the main funnel or pipe whereby communication is maintained with the
+  heated interior. From time to time, however, in large volcanoes rents
+  are formed on the sides of the cone, whence steam and other hot
+  vapours and also streams of molten lava are poured forth. On such
+  rents smaller or parasitic cones are often formed, which imitate the
+  operations of the parent cone and, after repeated eruptions, may rise
+  to hills hundreds of feet in height. In course of centuries the result
+  of the constant outpouring of volcanic materials may be to build up a
+  large mountain like Etna, which towers above the sea to a height of
+  10,840 feet, and has some 200 minor cones along its flanks.
+
+  But all volcanic eruptions do not proceed from central orifices. In
+  Iceland it has been observed that, from fissures opened in the ground
+  and extending for long distances, molten material has issued in such
+  abundance as to be spread over the surrounding country for many miles,
+  while along the lines of fissure small cones or hillocks of
+  fragmentary material have accumulated round more active parts of the
+  rent. There is reason to believe that in the geological past this
+  fissure-type of eruption has repeatedly been developed, as well as the
+  more common form of central cones like Vesuvius or Etna.
+
+  In the operations of existing volcanoes only the superficial
+  manifestations of volcanic action are observable. But when the rocks
+  of the earth's crust are studied, they are found to enclose the relics
+  of former volcanic eruptions. The roots of ancient volcanoes have thus
+  been laid bare by geological revolutions; and some of the
+  subterranean phases of volcanic action are thereby revealed which are
+  wholly concealed in an active volcano. Hence to obtain as complete a
+  conception as possible of the nature and history of volcanic action,
+  regard must be had, not merely to modern volcanoes, but to the records
+  of ancient eruptions which have been preserved within the crust.
+
+  The substances discharged from volcanic vents consist of--(1) Gases
+  and vapours: which, dissolved in the molten magma of the interior,
+  take the chief share in volcanic activity. They include in greatest
+  abundance water-gas, which condenses into the clouds of steam so
+  conspicuous in volcanic eruptions. Hydrochloric acid and sulphuretted
+  hydrogen are likewise plentiful, together with many other substances
+  which, sublimed by the high internal temperature, take a solid form on
+  cooling at the surface. (2) Molten rock or lava: which ranges from the
+  extremely acid type of the obsidians and rhyolites with 70% or more of
+  silica, to the more basic and heavy varieties such as basalts and
+  leucite-lavas with much iron, and sometimes no more than 45% of
+  silica. The specific gravity of lavas varies between 2.37 and 3.22,
+  and the texture ranges from nearly pure glass, like obsidian, to a
+  coarse granitoid compound, as in some rhyolites. (3) Fragmentary
+  materials, which are sometimes discharged in enormous quantity and
+  dispersed over a wide extent of country, the finer particles being
+  transported by upper air-currents for hundreds of miles. These
+  materials arise either from the explosion of lava by the sudden
+  expansion of the dissolved vapours and gases, as the molten rock rises
+  to the surface, or from the breaking up and expulsion of portions of
+  the walls of the vent, or of the lava, which happens to have
+  solidified within these walls. They vary from the finest impalpable
+  dust and ashes, through increasing stages of coarseness up to huge
+  "bombs" torn from the upper surface of the molten rock in the vent,
+  and large blocks of already solidified lava, or of non-volcanic rock
+  detached from the sides of the pipe up which the eruptions take place.
+
+  Nothing is yet known as to the determining cause of any particular
+  volcanic eruption. Some vents, like that of Stromboli, in the
+  Mediterranean, are continually active, and have been so ever since man
+  has observed them. Others again have been only intermittently in
+  eruption, with intervals of centuries between their periods of
+  activity. We are equally in the dark as to what has determined the
+  sites on which volcanic action has manifested itself. There is reason,
+  indeed, to believe that extensive fractures of the terrestrial crust
+  have often provided passages up which the vapours, imprisoned in the
+  internal magma, have been able to make their way, accompanied by other
+  products. Where chains of volcanoes rise along definite lines, like
+  those of Sumatra, Java, and many other tracts both in the Old and the
+  New World, there appears to be little doubt that their linear
+  distribution should be attributed to this cause. But where a volcano
+  has appeared by itself, in a region previously exempt from volcanic
+  action, the existence of a contributing fissure cannot be so
+  confidently presumed. The study of certain ancient volcanoes, the
+  roots of which have been exposed by long denudation, has shown an
+  absence of any visible trace of their having availed themselves of
+  fractures in the crust. The inference has been drawn that volcanic
+  energy is capable of itself drilling an orifice through the crust,
+  probably at some weaker part, and ejecting its products at the
+  surface. The source of this energy is to be sought in the enormous
+  expansive force of the vapours and gases dissolved in the magma. They
+  are kept in solution by the enormous pressure within the earth; but as
+  the lava approaches the surface and this pressure is relieved these
+  dissolved vapours and gases rush out with explosive violence, blowing
+  the upper part of the lava column into dust, and allowing portions of
+  the liquid mass below to rise and escape, either from the crater or
+  from some fissure which the vigour of explosion has opened on the side
+  of the cone. So gigantic is the energy of these pent-up vapours, that,
+  after a long period of volcanic quiescence, they sometimes burst forth
+  with such violence as to blow off the whole of the upper part or even
+  one side of a large cone. The history of Vesuvius, and the great
+  eruptions of Krakatoa in 1883 and of Bandaizan in 1888 furnish
+  memorable examples of great volcanic convulsions. It has been observed
+  that such stupendous discharges of aeriform and fragmentary matter may
+  be attended with the emission of little or no lava. On the other hand,
+  some of the largest outflows of lava have been accompanied by
+  comparatively little fragmentary material. Thus, the great lava-floods
+  of Iceland in 1783 spread for 40 m. away from their parent fissure,
+  which was marked only by a line of little cones of slag.
+
+  The temperature of lava as it issues from underground has been
+  measured more or less satisfactorily, and affords an indication of
+  that existing within the earth. At Vesuvius it has been ascertained to
+  be more than 2000° Fahr. At first the molten rock glows with a white
+  light, which rapidly reddens, and disappears under the rugged brown
+  and black crust that forms on the surface. Underneath this badly
+  conducting crust, the lava cools so slowly that columns of steam have
+  been noticed rising from its surface more than 80 years after its
+  eruption.
+
+  Considerable alteration in the topography of volcanic regions may be
+  produced by successive eruptions. The fragmentary materials are
+  sometimes discharged in such abundance as to cover the ground for many
+  miles around with a deposit of loose ashes, cinders and slag. Such a
+  deposit accumulating to a depth of many feet may completely bury
+  valleys and water-courses, and thus greatly affect the drainage. The
+  coarsest materials accumulate nearest to the vent that emits them. The
+  finer dust is not infrequently hurled forth with such an impetus as to
+  be carried for thousands of feet into the tracks of upper
+  air-currents, whereby it may be borne for hundreds of miles away from
+  the vent so as ultimately to fall to the ground in countries far
+  removed from any active volcano. Outflows of lava, from their greater
+  solidity and durability, produce still more serious and lasting
+  changes in the external features of the ground over which they flow.
+  As they naturally seek the lowest levels, they find their way into the
+  channels of streams. If they keep along the channels, they seal them
+  up under a mass of compact stone which the running water, if not
+  wholly diverted elsewhere, will take many long centuries to cut
+  through. If, on the other hand, the lava crosses a stream, it forms a
+  massive dam, above which the water is ponded back so as to form a
+  lake.
+
+  As the result of prolonged activity a volcanic cone is gradually built
+  up by successive outflows of lava and showers of dust and stones.
+  These materials are arranged in beds, or sheets, inclined outwards
+  from the central vent. On surrounding level ground the alternating
+  beds are flat. In course of time, deep gullies are cut on the outer
+  slopes of the cone by rain, and by the heavy showers that arise from
+  the condensation of the copious discharges of steam during eruptions.
+  Along the sides of these ravines instructive sections may be studied
+  of the volcanic strata. The larger rivers of some volcanic regions
+  have likewise eroded vast gorges in the more horizontal lavas and
+  ashes of the flatter country, and have thus laid bare stupendous
+  cliffs, along which the successive volcanic sheets can be seen piled
+  above each other for many hundred feet. On a small scale, some of
+  these features are well displayed among the rivers that drain the
+  volcanic tracts of central France; on a great scale, they are
+  presented in the course of the Snake river, and other streams that
+  traverse the great volcanic country of western North America. Similar
+  volcanic scenery has been produced in western Europe by the action of
+  denudation in dissecting the flat Tertiary lavas of Scotland, the
+  Faeroe Isles and Iceland.
+
+  Of special interest to the geologist are those volcanoes which have
+  taken their rise on the sea-bottom; for the volcanic intercalations
+  among the stratified formations of the earth's crust are almost
+  entirely of submarine origin. Many active volcanoes situated on
+  islands have begun their eruptions below sea-level. Both Vesuvius and
+  Etna sprang up on the floor of the Mediterranean sea, and have
+  gradually built up their cones into conspicuous parts of the dry land.
+  Examples of a similar history are to be found among the volcanic
+  islands of the Pacific Ocean. In some of these cases a movement of
+  elevation has carried the submarine lavas, tuffs and agglomerates
+  above sea-level, and has furnished opportunities of comparing these
+  materials with those of recent subaerial origin, and also with the
+  ancient records of submarine eruptions which have been preserved among
+  the stratified formations. From the evidence thus supplied, it can be
+  shown that the materials ejected from modern submarine volcanic vents
+  closely resemble those accumulated by subaerial volcanoes; that the
+  dust, ashes and stones become intermingled or interstratified with
+  coral-mud, or other non-volcanic deposit of the sea-bottom, that
+  vesicular lavas may be intercalated among them as on land, and that
+  between the successive sheets of volcanic origin, layers of limestone
+  may be laid down which are composed chiefly, or wholly, of the remains
+  of calcareous marine organisms.
+
+  Though active volcanoes are widely distributed over the globe, and are
+  especially abundant around the vast basin of the Pacific Ocean, they
+  afford an incomplete picture of the extent to which volcanic action
+  has displayed itself on the surface of our planet. When the rocks of
+  the land are attentively studied they disclose proofs of that action
+  in many districts where there is now no outward sign of it. Not only
+  so, but they reveal that volcanoes have been in eruption in some of
+  these districts during many different periods of the past, back to the
+  beginnings of geological history. The British Islands furnish a
+  remarkable example of such a series of ancient eruptions. From the
+  Cambrian period all through Palaeozoic times there rose at intervals
+  in that country a succession of volcanic centres from some of which
+  thousands of feet of lavas and tuffs were discharged. Again in older
+  Tertiary times the same region witnessed a stupendous outpouring of
+  basalt, the surviving relics of which are more than 3000 ft. thick,
+  and cover many hundreds of square miles. Similar evidence is supplied
+  in other countries both in the Old and the New world. Hence it is
+  proved that, in the geological past, volcanic action has been vigorous
+  at long intervals on the same sites during a vast series of ages,
+  though no active vents are to be seen there now. The volcanoes now
+  active form but a small proportion of the total number which has
+  appeared on the surface of the earth.
+
+  With regard to the cause of volcanic action much has been speculated,
+  but little can be confidently affirmed. That water in the form of
+  occluded gas plays the chief part in forcing the lava column up a
+  volcanic chimney, and in the violent explosions that accompany the
+  rise of the molten material, is generally admitted. But opinions
+  differ as to the source of this water. According to some
+  investigators, it should be regarded as in large measure of meteoric
+  origin, derived from the descent of rain into the earth, and its
+  absorption by the molten magma in the interior. Others, contending
+  that the supply so furnished, even if it could reach and be dissolved
+  in the magma, would yet be insufficient to furnish the prodigious
+  quantity of aqueous vapour discharged during an eruption, maintain
+  that the water belongs to the magma itself. They point to the admitted
+  fact that many substances, particularly metals in a state of fusion,
+  can absorb large quantities of vapours and gases without chemical
+  combination, and on cooling discharge them with eruptive phenomena
+  somewhat like those of volcanoes. This question must be regarded as
+  one of the still unsolved problems of geology.
+
+
+  (B) _Movements of the Earth's Crust._
+
+  Among the hypogene forces in geological dynamics an important place
+  must be assigned to movements of the terrestrial crust. Though the
+  expression "the solid earth" has become proverbial, it appears
+  singularly inappropriate in the light of the results obtained in
+  recent years by the use of delicate instruments of observation. With
+  the facilities supplied by these instruments (see SEISMOMETER), it has
+  been ascertained that the ground beneath our feet is subject to
+  continual slight tremors, and feeble pulsations of longer duration,
+  some of which may be due to daily or seasonal variations of
+  temperature, atmospheric pressure or other meteorological causes. The
+  establishment of self-recording seismometers all over the world has
+  led to the detection of many otherwise imperceptible shocks, over and
+  above the appreciable earth-waves propagated from earthquake centres
+  of disturbance. Moreover, it has been ascertained that some parts of
+  the surface of the land are slowly rising, while others are falling
+  with reference to the sea-level. From time to time the surface suffers
+  calamitous devastation from earthquakes, when portions of the crust
+  under great strain suddenly give way. Lastly, at intervals, probably
+  separated from each other by vast periods of time, the terrestrial
+  crust undergoes intense plication and fracture, and is consequently
+  ridged up into mountain-chains. No event of this kind has been
+  witnessed since man began to record his experiences. But from the
+  structure of mountains, as laid open by prolonged denudation, it is
+  possible to form a vivid conception of the nature and effects of these
+  most stupendous of all geological revolutions.
+
+  In considering this department of geological inquiry it will be
+  convenient to treat it under the following heads: (1) Slow depression
+  and upheaval; (2) Earthquakes; (3) Mountain-making; (4) Metamorphism
+  of rocks.
+
+  1. _Slow Depression and Upheaval._--On the west side of Japan the land
+  is believed to be sinking below the sea, for fields are replaced by
+  beaches of sand or shingle, while the depth of the sea off shore has
+  perceptibly increased. A subsidence of the south of Sweden has taken
+  place in comparatively recent times, for streets and foundations of
+  houses at successive levels are found below high-water mark. The west
+  coast of Greenland over an extent of more than 600 m. is sinking, and
+  old settlements are now submerged. Proofs of submergence of land are
+  furnished by "submerged forests," and beds of terrestrial peat now
+  lying at various depths below the level of the sea, of which many
+  examples have been collected along the shores of the British Isles,
+  Holland and France. Interesting evidence that the west of Europe now
+  stands at a lower level than it did at a late geological period is
+  supplied in the charts of the North Sea and Atlantic, which show that
+  the valleys of the land are prolonged under the sea. These valleys
+  have been eroded out of the rocks by the streams which flow in them,
+  and the depth of their submerged portions below the sea level affords
+  an indication of the extent of the subsidence.
+
+  The uprise of land has been detected in various parts of the world.
+  One of the most celebrated instances is that of the shores of the Gulf
+  of Bothnia, where, at Stockholm, the elevation, between the years 1774
+  and 1875, appears to have been 48 centimetres (18½ in.) in a century.
+  But on the west side of Sweden, fronting the Skager Rak, the coast,
+  between the years 1820 and 1870, rose 30 centimetres, which is at the
+  rate of 60 centimetres, or nearly 2 ft. in a century. In the region of
+  the Great Lakes in the interior of Canada and the United States it has
+  been ascertained that the land is undergoing a slow tilt towards the
+  south-west, of which the mean rate appears to be rather less than 6
+  in. in a century. If this rate of change should continue the waters of
+  Lake Michigan, owing to the progress of the tilt, will, in some 500 or
+  600 years, submerge the city of Chicago, and eventually the drainage
+  of the lakes will be diverted into the basin of the Mississippi. Proof
+  of recent emergence of land is supplied by what are called "raised
+  beaches" or "strand-lines," that is, lines of former shores marked by
+  sheets of littoral deposits, or platforms cut by shore-waves in rock
+  and flanked by old sea-cliffs and lines of sea-worn caves. Admirable
+  examples of these features are to be seen along the west coast of
+  Europe from the south of England to the north of Norway. These lines
+  of old shores become fainter in proportion to their antiquity. In
+  Britain they occur at various heights, the platforms at 25, 50 and 100
+  ft. being well marked.
+
+  The cause of these slow upward and downward movements of the crust of
+  the earth is still imperfectly understood. Upheaval might conceivably
+  be produced by an ascent of the internal magma, and the consequent
+  expansion of the overlying crust by heat; while depression might
+  follow any subsidence of the magma, or its displacement to another
+  district. If, as is generally believed, the globe is still
+  contracting, the shrinkage of the surface may cause both these
+  movements. Subsidence will be in excess, but between subsiding tracts
+  lateral thrust may suffice to push upward intervening more solid and
+  stable ground; but no solution of the problem yet proposed is wholly
+  satisfactory.
+
+  2. _Earthquakes._--As this subject is discussed in a separate article
+  it will be sufficient here to take note of its more important
+  geological bearings. It was for many centuries taken for granted that
+  earthquakes and volcanoes are due to a common cause. We have seen that
+  in classical antiquity they were looked on as the results of the
+  movements of wind imprisoned within the earth. Long after this notion
+  was discarded, and a more scientific appreciation of volcanic action
+  was reached, it was still thought that earthquakes should be regarded
+  as manifestations of the same source of energy as that which displays
+  itself in volcanic eruptions. It is true that earthquakes are frequent
+  in districts of active volcanoes, and they may undoubtedly be often
+  due there to the explosions of the magma, or to the rupture of rocks
+  caused by its ascent towards the surface. But such shocks are
+  comparatively local in their range and feeble in their effects. There
+  is now a general agreement that between the great world-shaking
+  earthquakes and volcanic phenomena, no immediate and intimate
+  relationship can be traced, though they may be connected in ways which
+  are not yet perceived. Some of the more recent great earthquakes on
+  land have proved that the waves of shock are produced by the sudden
+  rupture or collapse of rocks under great strain, either along lines of
+  previous fracture or of new rents in the terrestrial crust; and that
+  such ruptures may occur at a remote distance from any volcano. Thus
+  the recent disastrous San Francisco earthquake has been recognized to
+  have resulted from a slipping of ground along the line of an old
+  fault, which has been traced for a long distance in California
+  generally parallel to the coast. The position of this fault at the
+  surface has long been clearly followed by its characteristic
+  topography. After the earthquake these superficial features were found
+  to have been removed by the same cause that had originated them. For
+  some 300 m. on the track of this old fault-line a renewed slipping was
+  seen to have taken place along one or both sides, and the ground at
+  the surface was ruptured as well as displaced horizontally. Obviously,
+  the jar occasioned by the sudden and simultaneous subsidence of a
+  portion of the earth's crust several hundred miles long, must be far
+  more serious than could be produced by an earthquake radiating from a
+  single local volcanic focus.
+
+  From their disastrous effects on buildings and human lives, an
+  exaggerated importance has been imputed to earthquakes as agents of
+  geological change. Experience shows that even after a severe shock
+  which may have destroyed numerous towns and villages, together with
+  thousands of their inhabitants, the face of the country has suffered
+  scarcely any perceptible change, and that, in the course of a year or
+  two, when the ruined houses and prostrate trees have been cleared
+  away, little or no obvious trace of the catastrophe may remain. Among
+  the more enduring records of a great earthquake may be enumerated (a)
+  landslips, which lay bare hillsides, and sometimes pond back the
+  drainage of valleys so as to give rise to lakes; (b) alterations of
+  the topography, as in fissuring of the ground, or in the production of
+  inequalities whereby the drainage is affected; new valleys and new
+  lakes may thus be formed, while previously existing lakes may be
+  emptied; (c) permanent changes of level, either in an upward or
+  downward direction.
+
+  3. _Mountain-making._--This subject may be referred to here for the
+  striking evidence which it supplies of the importance of movements of
+  the earth's crust among geological processes. The structure of a great
+  mountain-chain such as the Alps proves that the crust of the earth has
+  been intensely plicated, crumpled and fractured. Vast piles of
+  sedimentary strata have been folded to such an extent as to occupy now
+  only half of their original horizontal extent. This compression in the
+  case of the Alps has been computed to amount to as much as 120,000
+  metres or 74 English miles, so that two points on the opposite sides
+  of that chain have been brought by so much nearer to each other than
+  they were originally before the movements. Besides such intense
+  plication, extensive rupturing of the crust has taken place in the
+  same range of mountains. Not only have the most ancient rocks been
+  squeezed up into the central axis of the chain, but huge slices of
+  them have been torn away from the main body, and thrust forward for
+  many miles, so as now actually to form the summits of mountains, which
+  are almost entirely composed of much younger formations. If these
+  colossal disturbances occurred rapidly, they would give rise to
+  cataclysms of inconceivable magnitude over the surface of the globe.
+  No record has been discovered of such accompanying devastation. But
+  whether sudden and violent, or prolonged and gradual, such stupendous
+  upturnings of the crust did undoubtedly take place, as is clearly
+  revealed in innumerable natural sections, which have been laid open by
+  the denudation of the crests and sides of the mountains.
+
+  4. _Metamorphism of Rocks_ (see METAMORPHISM).--During the movements
+  to which the crust of the earth has been subject, not only have the
+  rocks been folded and fractured, but they have likewise, in many
+  regions, acquired new internal structures, and have thus undergone a
+  process of "regional metamorphism." This rearrangement of their
+  substance has been governed by conditions which are probably not yet
+  all recognized, but among them we should doubtless include a high
+  temperature, intense pressure, mechanical movement resulting in
+  crushing, shearing and foliation, and the presence of water in their
+  pores. It is among igneous rocks that the progressive stages of
+  metamorphism can be most easily traced. Their definite original
+  structure and mineral composition afford a starting-point from which
+  the investigation may be begun and pursued. Where an igneous rock has
+  been invaded by metamorphic changes, it may be observed to have been
+  first broken down into separate lenticles, the cores of which may
+  still retain, with little or no alteration, the original
+  characteristic minerals and crystalline structure of the rock. Between
+  these lenticles, the intervening portions have been crushed down into
+  a powder or paste, which seems to have been squeezed round and past
+  them, and shows a laminated arrangement that resembles the
+  flow-structure in lavas. As the degree of metamorphism increases, the
+  lenticles diminish in size, and the intervening crushed and foliated
+  matrix increases in amount, until at last it may form the entire mass
+  of the rock. While the original minerals are thus broken down, new
+  varieties make their appearance. Of these, among the earliest to
+  present themselves are usually the micas, that impart their
+  characteristic silvery sheen to the surfaces of the folia along which
+  they spread. Younger felspars, as well as mica, are developed, and
+  there arise also sillimanite, garnet, andalusite and many others. The
+  texture becomes more coarsely crystalline, and the segregation of the
+  constituent minerals more definite along the lines of foliation. From
+  the finest silky phyllites a graduation may be traced through
+  successively coarser mica-schists, until we reach the almost granitic
+  texture of the coarsest gneisses.
+
+  Regional metamorphism has arisen in the heart of mountain-chains, and
+  in any other district where the deformation of the crust has been
+  sufficiently intense. There is another type of alteration termed
+  "contact-metamorphism," which is developed around masses of igneous
+  rock, especially where these have been intruded in large bosses among
+  stratified formations. It is particularly displayed around masses of
+  granite, where sandstones are found altered into quartzite, shales and
+  grits into schistose compounds, and where sometimes fossils are still
+  recognizable among the metamorphic minerals.
+
+
+_DIVISION II.--EPIGENE OR SUPERFICIAL ACTION_
+
+It is on the surface of the globe, and by the operation of agents
+working there, that at present the chief amount of visible geological
+change is effected. In considering this branch of inquiry, we are not
+involved in a preliminary difficulty regarding the very nature of the
+agencies as is the case in the investigation of plutonic action. On the
+contrary, the surface agents are carrying on their work under our very
+eyes. We can watch it in all its stages, measure its progress, and mark
+in many ways how accurately it represents similar changes which, for
+long ages previously, must have been effected by the same means. But in
+the systematic treatment of this subject we encounter a difficulty of
+another kind. We discover that while the operations to be discussed are
+numerous and readily observable, they are so interwoven into one great
+network that any separation of them under different subdivisions is sure
+to be more or less artificial and to convey an erroneous impression.
+While, therefore, under the unavoidable necessity of making use of such
+a classification of subjects, we must always bear in mind that it is
+employed merely for convenience, and that in nature superficial
+geological action must be continually viewed as a whole, since the work
+of each agent has constant reference to that of the others, and is not
+properly intelligible unless that connexion be kept in view.
+
+The movements of the air; the evaporation from land and sea; the fall of
+rain, hail and snow; the flow of rivers and glaciers; the tides,
+currents and waves of the ocean; the growth and decay of organized
+existence, alike on land and in the depths of the sea;--in short, the
+whole circle of movement, which is continually in progress upon the
+surface of our planet, are the subjects now to be examined. It is
+desirable to adopt some general term to embrace the whole of this range
+of inquiry. For this end the word epigene (Gr. [Greek: epi], upon) has
+been suggested as a convenient term, and antithetical to hypogene (Gr.
+[Greek: hypo], under), or subterranean action.
+
+A simple arrangement of this part of Geological Dynamics is in three
+sections:
+
+A. _Air._--The influence of the atmosphere in destroying and forming
+rocks.
+
+B. _Water._--The geological functions of the circulation of water
+through the air and between sea and land, and the action of the sea.
+
+C. _Life._--The part taken by plants and animals in preserving,
+destroying or reproducing geological formations.
+
+The words destructive, reproductive and conservative, employed in
+describing the operations of the epigene agents, do not necessarily
+imply that anything useful to man is destroyed, reproduced or preserved.
+On the contrary, the destructive action of the atmosphere may turn
+barren rock into rich soil, while its reproductive effects sometimes
+turn rich land into barren desert. Again, the conservative influence of
+vegetation has sometimes for centuries retained as barren morass what
+might otherwise have become rich meadow or luxuriant woodland. The
+terms, therefore, are used in a strictly geological sense, to denote the
+removal and re-deposition of material, and its agency in preserving what
+lies beneath it.
+
+
+  (A) _The Air._
+
+  As a geological agent, the air brings about changes partly by its
+  component gases and partly by its movements. Its destructive action is
+  both chemical and mechanical. The chemical changes are probably
+  mainly, if not entirely, due to the moisture of the air, and
+  particularly to the gases, vapours and organic matter which the
+  moisture contains. Dry air seems to have little or no appreciable
+  influence in promoting these reactions. As the changes in question are
+  similar to those much more abundantly brought about by rain they are
+  described in the following section under the division on rain.
+
+  Among the more recognizable mechanical changes effected in the
+  atmosphere, one of considerable importance is to be seen in the result
+  of great and rapid changes of temperature. Heat expands rocks, while
+  cold contracts them. In countries with a great annual range of
+  temperature, considerable difficulty is sometimes experienced in
+  selecting building materials liable to be little affected by the
+  alternate expansion and contraction, which prevents the joints of
+  masonry from remaining close and tight. In dry tropical climates,
+  where the days are intensely hot and the nights extremely cold, the
+  rapid nocturnal contraction produces a strain so great as to rival
+  frost in its influence upon the surface of exposed rocks,
+  disintegrating them into sand, or causing them to crack or peel off in
+  skins or irregular pieces. Dr Livingstone found in Africa (12° S.
+  lat., 34° E. long.) that surfaces of rock which during the day were
+  heated up to 137° Fahr., cooled so rapidly by radiation at night that,
+  unable to sustain the strain of contraction, they split and threw off
+  sharp angular fragments from a few ounces to 100 or 200 [lb] in
+  weight. In temperate regions this action, though much less pronounced,
+  still makes itself felt. In these climates, however, and still more in
+  high latitudes, somewhat similar results are brought about by frost.
+
+  By its motion in wind the air drives loose sand over rocks, and in
+  course of time abrades and smoothes them. "Desert polish" is the name
+  given to the characteristic lustrous surface thus imparted. Holes are
+  said to be drilled in window glass at Cape Cod by the same agency.
+  Cavities are now and then hollowed out of rocks by the gyration in
+  them of little fragments of stone or grains of sand kept in motion by
+  the wind. Hurricanes form important geological agents upon land in
+  uprooting trees, and thus sometimes impeding the drainage of a country
+  and giving rise to the formation of peat mosses.
+
+  The reproductive action of the air arises partly from the effect of
+  the chemical and mechanical disintegration involved in the process of
+  "weathering," and partly from the transporting power of wind and of
+  aerial currents. The layer of soil, which covers so much of the
+  surface of the land, is the result of the decay of the underlying
+  rocks, mingled with mineral matter blown over the ground by wind, or
+  washed thither by rain, and with the mouldering remains of plants and
+  animals. The extent to which fine dust may be transported over the
+  surface of the land can hardly be realized in countries clothed with a
+  covering of vegetation, though even there, in dry weather during
+  spring, clouds of dust may often be seen blown away by wind from bare
+  ploughed fields. Intercepted by the leaves of plants and washed down
+  to their roots by rain, this dust goes to increase the soil below. In
+  arid climates, where dust clouds are dense and frequent, enormous
+  quantities of fine mineral particles are thus borne along and
+  accumulated. The remarkable deposit of "Loess," which is sometimes
+  more than 1500 ft. thick and covers extensive areas in China and other
+  countries, is regarded as due to the drifting of dust by wind. Again
+  the dunes of sand so abundant along the inner side of sandy
+  sea-beaches in many different parts of the world are attributable to
+  the same action.
+
+
+  (B) _Water._
+
+  In treating of the epigene action of water in geological processes it
+  will be convenient to deal first with its operations in traversing the
+  land, and then with those which it performs in the sea. The
+  circulation of water from land to sea and again from sea to land
+  constitutes the fundamental cause of most of the daily changes by
+  which the surface of the land is affected.
+
+  1. _Rain._--Rain effects two kinds of changes upon the surface of the
+  land. It acts _chemically_ upon soils and stones, and sinking under
+  ground continues a great series of similar reactions there. It acts
+  _mechanically_, by washing away loose materials, and thus powerfully
+  affecting the contours of the land. Its chemical action depends mainly
+  upon the nature and proportion of the substances which, in descending
+  to the earth, it abstracts from the atmosphere. Rain always absorbs a
+  little air, which, in addition to its nitrogen and oxygen, contains
+  carbonic acid, and in minute proportions, sodium chloride, sulphuric
+  acid and other ingredients, especially inorganic dust, organic
+  particles and living germs. Probably the most generally efficient of
+  these constituents are oxygen, carbonic acid and organic matter. Armed
+  with these reagents, rain effects a chemical decomposition of the
+  rocks on which it falls, and through which it sinks underground. The
+  principal changes thus produced are as follows: (a) Oxidation.--Owing
+  to the prominence of oxygen in rain-water, and its readiness to unite
+  with any substance which can contain more of it, a thin oxidized
+  pellicle is formed on the surface of many rocks on which rain falls,
+  and this oxidized layer if not at once washed off, sinks deeper until
+  a crust is formed over the stone. A familiar illustration of this
+  action is afforded by the rust, or oxide, which forms on iron when
+  exposed to moisture, though this iron may be kept long bright if
+  allowed to remain screened from moist air and rain. (b)
+  Deoxidation.--Organic matter having an affinity for more oxygen
+  decomposes peroxides by depriving them of some part of their share of
+  that element and reducing them to protoxides. These changes are
+  especially noticeable among the iron oxides so abundantly diffused
+  among rocks. Hence rain-water, in sinking through soil and obtaining
+  such organic matter, becomes thereby a reducing agent. (c)
+  Solution.--This may take place either by the simple action of the
+  water, as in the solution of rock-salt, or by the influence of the
+  carbonic acid present in the rain. (d) Formation of Carbonates.--A
+  familiar example of the action of carbonic acid in rain is to be seen
+  in the corrosion of exposed marble slabs. The carbonic acid dissolves
+  some of the lime, which, as a bicarbonate, is held in solution in the
+  carbonated water, but is deposited again when the water loses its
+  carbonic acid or evaporates. It is not merely carbonates, however,
+  which are liable to this kind of destruction. Even silicates of lime,
+  potash and soda, combinations existing abundantly as constituents of
+  rocks, are attacked; their silica is liberated, and their alkalis or
+  alkaline earths, becoming carbonates, are removed in solution. (e)
+  Hydration.--Some minerals, containing little or no water, and
+  therefore called anhydrous, when exposed to the action of the
+  atmosphere, absorb water, or become hydrous, and are then usually more
+  prone to further change. Hence the rocks of which they form part
+  become disintegrated.
+
+  Besides the reactions here enumerated, a considerable amount of decay
+  may be observed as the result of the presence of sulphuric and nitric
+  acid in the air, especially in that of large towns and manufacturing
+  districts, where much coal is consumed. Metallic surfaces, as well as
+  various kinds of stone, are there corroded, while the mortar of walls
+  may often be observed to be slowly swelling out and dropping off,
+  owing to the conversion of the lime into sulphate. Great injury is
+  likewise done from a similar cause to marble monuments in exposed
+  graveyards.
+
+  The general result of the disintegrating action of the air and of
+  rain, including also that of plants and animals, to be noticed in the
+  sequel, is denoted by the term "weathering." The amount of decay
+  depends partly on conditions of climate, especially the range of
+  temperature, the abundance of moisture, height above the sea and
+  exposure to prevalent winds. Many rocks liable to be saturated with
+  rain and rapidly dried under a warm sun are apt to disintegrate at the
+  surface with comparative rapidity. The nature and progress of the
+  weathering are mainly governed by the composition and texture of the
+  rocks exposed to it. Rocks composed of particles liable to little
+  chemical change from the influence of moisture are best fitted to
+  resist weathering, provided they possess sufficient cohesion to
+  withstand the mechanical processes of disintegration. Siliceous
+  sandstones are excellent examples of this permanence. Consisting
+  wholly or mainly of the durable mineral quartz, they are sometimes
+  able so to withstand decay that buildings made of them still retain,
+  after the lapse of centuries, the chisel-marks of the builders. Some
+  rocks, which yield with comparative rapidity to the chemical attacks
+  of moisture, may show little or no mark of disintegration on their
+  surface. This is particularly the case with certain calcareous rocks.
+  Limestone when pure is wholly soluble in acidulated water. Rain
+  falling on such a rock removes some of it in solution, and will
+  continue to do so until the whole is dissolved away. But where a
+  limestone is full of impurities, a weathered crust of more or less
+  insoluble particles remains after the solution of the calcareous part
+  of the stone. Hence the relative purity of limestones may be roughly
+  determined by examining their weathered surfaces, where, if they
+  contain much sand, the grains will be seen projecting from the
+  calcareous matrix, and where, should the rock be very ferruginous, the
+  yellow hydrous peroxide, or ochre, will be found as a powdery crust.
+  In limestones containing abundant encrinites, shells, or other organic
+  remains, the weathered surface commonly presents the fossils standing
+  out in relief. The crystalline arrangement of the lime in the organic
+  structures enables them to resist disintegration better than the
+  general mechanically aggregated matrix of the rock. An experienced
+  fossil collector will always search well such weathered surfaces, for
+  he often finds there, delicately picked out by the weather, minute
+  and frail fossils which are wholly invisible on a freshly broken
+  surface of the stone. Many rocks weather with a thick crust, or even
+  decay inwards for many feet or yards. Basalt, for example, often shows
+  a yellowish-brown ferruginous layer on its surface, formed by the
+  conversion of its felspar into kaolin, and the removal of its calcium
+  silicate as carbonate, by the hydration of its olivine and augite and
+  their conversion into serpentine, or some other hydrous magnesian
+  silicate, and by the conversion of its magnetite into limonite.
+  Granite sometimes shows in a most remarkable way the distance to which
+  weathering can reach. It may occasionally be dug into for a depth of
+  20 or 30 ft., the quartz crystals and veins retaining their original
+  positions, while the felspar is completely kaolinized. It is to the
+  endlessly varied effects of weathering that the abundant fantastic
+  shapes assumed by crags and other rocky masses are due. Most varieties
+  of rock have their own characteristic modes of weathering, whereby
+  they may be recognized even from a distance. To some of these features
+  reference will be made in Part VIII.
+
+  The mechanical action of rain, which is intimately bound up with its
+  chemical action, consists in washing off the fine superficial
+  particles of rocks which have been corroded and loosened by the
+  process of weathering, and in thus laying open fresh portions to the
+  same influences of decay. The detritus so removed is partly carried
+  down into the soil which is thereby enriched, partly held in
+  suspension in the little runnels into which the rain-drops gather as
+  they begin to flow over the land, partly pushed downwards along the
+  surface of sloping ground. A good deal of it finds its way into the
+  nearest brooks and rivers, which are consequently made muddy by heavy
+  rain.
+
+  It is natural that a casual consideration of the subject should lead
+  to an impression that, though the general result of the fall of rain
+  upon a land-surface must lead to some amount of disintegration and
+  lowering of that surface, the process must be so slow and slight as
+  hardly to be considered of much importance among geological
+  operations. But further attention will show such an impression to be
+  singularly erroneous. It loses sight of the fact that a change which
+  may be hardly appreciable within a human lifetime, or even within the
+  comparatively brief span of geological time embraced in the compass of
+  human history, may nevertheless become gigantic in its results in the
+  course of immensely protracted periods. An instructive lesson in the
+  erosive action of rain may be found in the pitted and channelled
+  surface of ground lying under the drip of the eaves of a cottage. The
+  fragments of stone and pebbles of gravel that form part of the soil
+  can there be seen sticking out of the ground, because being hard they
+  resist the impetus of the falling drops, protecting for a time the
+  earth beneath them, while that which surrounded and covered them is
+  washed away. From this familiar illustration the observer may advance
+  through every stage in the disappearance of material which once
+  covered the surface, until he comes to examples where once continuous
+  and thick sheets of solid rock have been reduced to a few fragments or
+  have been entirely removed. Since the whole land surface over which
+  rain falls is exposed to this waste, the superficial covering of
+  decayed rock or soil, as Hutton insisted, is constantly, though
+  imperceptibly, travelling outward and downward to the sea. In this
+  process of transport rain is an important carrying agent, while at the
+  same time it serves to connect the work of the other disintegrating
+  forces, and to make it conducive to the general degradation of the
+  land. Though this decay is general and constant, it is obviously not
+  uniform. In some places where, from the nature of the rock, from the
+  flatness of the ground, or from other causes, rain works under great
+  difficulties, the rate of waste may be extremely slow. In other places
+  it may be rapid enough to be appreciable from year to year. A survey
+  of this department of geological activity shows how unequal wasting by
+  rain, combined with the operations of brooks and rivers, has produced
+  the details of the present relief of the land, those tracts where the
+  destruction has been greatest forming hollows and valleys, others,
+  where it has been less, rising into ridges and hills (Part VIII.).
+
+  Rain-action is not merely destructive, but is accompanied with
+  reproductive effects, chief of which is the formation of soil. In
+  favourable situations it has gathered together accumulations of loam
+  and earth from neighbouring higher ground, such as the "brick-earth,"
+  "head," and "rain-wash" of the south of England--earthy deposits,
+  sometimes full of angular stones, derived from the subaerial waste of
+  the rocks of the neighbourhood.
+
+  2. _Underground Water._--Of the rain which falls upon the land one
+  portion flows off into brooks and rivers by which the water is
+  conducted back to the ocean; the larger part, however, sinks into the
+  ground and disappears. It is this latter part which has now to be
+  considered. Over and above the proportion of the rainfall which is
+  absorbed by living vegetation and by the soil, there is a continual
+  filtering down of the water from the surface into the rocks that lie
+  below, where it partly lodges in pores and interstices, and partly
+  finds its way into subterranean joints and fissures, in which it
+  performs an underground circulation, and ultimately issues once more
+  at the surface in the form of springs (q.v.). In the course of this
+  circulation the water performs an important geological task. Not only
+  carrying down with it the substances which the rain has abstracted
+  from the air, but obtaining more acids and organic matter from the
+  soil, it is enabled to effect chemical changes in the rocks
+  underneath, and especially to dissolve limestone and other calcareous
+  formations. So considerable is the extent of this solution in some
+  places that the springs which come to the surface, and begin there to
+  evaporate and lose some of their carbonic acid, contain more dissolved
+  lime than they can hold. They consequently deposit it in the form of
+  calcareous tuff or sinter (q.v.). Other subterranean waters issue with
+  a large proportion of iron-salts in solution which form deposits of
+  ochre. The various mineral springs so largely made use of for the
+  mitigation or cure of diseases owe their properties to the various
+  salts which they have dissolved out of rocks underground. As the
+  result of prolonged subterranean solution in limestone districts,
+  passages and caves (q.v.), sometimes of great width and length, are
+  formed. When these lie near the surface their roofs sometimes fall in
+  and engulf brooks and rivers, which then flow for some way underground
+  until the tunnels conduct them back again to daylight on some lower
+  ground.
+
+  Besides its chemical activity water exerts among subterranean rocks a
+  mechanical influence which leads to important changes in the
+  topography of the surface. In removing the mineral matter, either in
+  solution or as fine sediment, it sometimes loosens the support of
+  overlying masses of rock which may ultimately give way on sloping
+  ground, and rush down the declivities in the form of landslips. These
+  destructive effects are specially frequent on the sides of valleys in
+  mountainous countries and on lines of sea-cliff.
+
+  3. _Brooks and Rivers._--As geological agents the running waters on
+  the face of the land play an important part in epigene changes. Like
+  rain and springs they have both a chemical and a mechanical action.
+  The latter receives most attention, as it undoubtedly is the more
+  important; but the former ought not to be omitted in any survey of the
+  general waste of the earth's surface. The water of rivers must possess
+  the powers of a chemical solvent like rain and springs, though its
+  actual work in this respect can be less easily measured, seeing that
+  river water is directly derived from rain and springs, and necessarily
+  contains in solution mineral substances supplied to it by them and not
+  by its own operation. Nevertheless, it is sometimes easy to prove that
+  streams dissolve chemically the rocks of their channels. Thus, in
+  limestone districts the base of the cliffs of river ravines may be
+  found eaten away into tunnels, arches, and overhanging projections,
+  presenting in their smooth surfaces a great contrast to the angular
+  jointed faces of the same rock, where now exposed to the influence
+  only of the weather on the higher parts of the cliff.
+
+  The mechanical action of rivers consists (a) in transporting mud,
+  sand, gravel and blocks of stone from higher to lower levels; (b) in
+  using these loose materials to widen and deepen their channels by
+  erosion; (c) in depositing their load of detritus wherever possible
+  and thus to make new geological formations.
+
+  (a) _Transporting Power._--River-water is distinguished from that of
+  springs by being less transparent, because it contains more or less
+  mineral matter in suspension, derived mainly from what is washed down
+  by rain, or carried in by brooks, but partly also from the abrasion of
+  the water-channels by the erosive action of the rivers themselves. The
+  progress of this burden of detritus may be instructively followed from
+  the mountain-tributaries of a river down to the mouth of the main
+  stream. In the high grounds the water-courses may be observed to be
+  choked with large fragments of rock disengaged from the cliffs and
+  crags on either side. Traced downwards the blocks are seen to become
+  gradually smaller and more rounded. They are ground against each
+  other, and upon the rocky sides and bottom of the channel, getting
+  more and more reduced as they descend, and at the same time abrading
+  the rocks over or against which they are driven. Hence a great deal of
+  débris is produced, and is swept along by the onward and downward
+  movement of the water. The finer portions, such as mud and fine sand,
+  are carried in suspension, and impart the characteristic turbidity to
+  river-water; the coarser sand and gravel are driven along the
+  river-bottom. The proportion of suspended mineral matter has been
+  ascertained with more or less precision for a number of rivers. As an
+  illustrative example of a river draining a vast area with different
+  climates, forms of surface and geological structure the Mississippi
+  may be cited. The average proportion of sediment in its water was
+  ascertained by Humphreys and Abbot to be 1/1500 by weight or 1/2900 by
+  volume. These engineers found that, in addition to this suspended
+  material, coarse detritus is constantly being pushed forward along the
+  bed of the river into the Gulf of Mexico, to an amount which they
+  estimated at about 750,000,000 cubic ft. of sand, earth and gravel;
+  they concluded that the Mississippi carries into the gulf every year
+  an amount of mechanically transported sediment sufficient to make a
+  prism one square mile in area and 268 ft. in height.
+
+  (b) _Excavating Power._--It is by means of the sand, gravel and stones
+  which they drive against the sides and bottoms of their channels that
+  streams have hollowed out the beds in which they flow. Not only is the
+  coarse detritus reduced in size by the friction of the stones against
+  each other, but, at the same time, these materials abrade the rocks
+  against which they are driven by the current. Where, owing to the
+  shape of the bottom of the channel, the stones are caught in eddies,
+  and are kept whirling round there, they become more and more worn down
+  themselves, and at the same time scour out basin-shaped cavities, or
+  "pot-holes," in the solid rock below. The uneven bed of a swiftly
+  flowing stream may in this way be honeycombed with such eroded basins
+  which coalesce and thus appreciably lower the surface of the bed. The
+  steeper the channel, other conditions being equal, the more rapid will
+  be the erosion. Geological structure also affects the character and
+  rate of the excavation. Where the rocks are so arranged as to favour
+  the formation and persistence of a waterfall, a long chasm may be
+  hollowed out like that of the Niagara below the falls, where a hard
+  thick bed of nearly flat limestone lies on softer and more easily
+  eroded shales. The latter are scooped out from underneath the
+  limestone, which from time to time breaks off in large masses and the
+  waterfall gradually retreats up stream, while the ravine is
+  proportionately lengthened. To the excavating power of rivers the
+  origin of the valley systems of the dry land must be mainly assigned
+  (see Part VIII.).
+
+  (c) _Reproductive Power._--So long as a stream flows over a steep
+  declivity its velocity suffices to keep the sediment in suspension,
+  but when from any cause, such as a diminution of slope, the velocity
+  is checked, the transporting power is lessened and the sediment begins
+  to fall to the bottom and to remain there. Hence various river-formed
+  or "alluvial" deposits are laid down. These sometimes cover
+  considerable spaces at the foot of mountains. The floors of valleys
+  are strewn with detritus, and their level may thereby be sensibly
+  raised. In floods the ground inundated on either side of a stream
+  intercepts some part of the detritus, which is then spread over the
+  flood-plain and gradually heightens it. At the same time the stream
+  continues to erode the channel, and ultimately is unable to reach the
+  old flood-plain. It consequently forms a new plain at a lower level,
+  and thus, by degrees, it comes to be flanked on either side by a
+  series of successive terraces or platforms, each of which marks one of
+  its former levels. Where a river enters a large body of water its
+  current is checked. Some of its sediment is consequently dropped, and
+  by slow accumulation forms a delta (q.v.). On land, every lake in
+  mountain districts furnishes instances of this kind of alluvium. But
+  the most important deltas are those formed in the sea at the mouths of
+  the larger rivers of the globe. Off many coast-lines the detritus
+  washed from the land gathers into bars, which enclose long strips of
+  water more or less completely separated from the sea outside and known
+  as lagoons. A chain of such lagoon-barriers stretches for hundreds of
+  miles round the Gulf of Mexico and the eastern shores of the United
+  States.
+
+  4. _Lakes._--These sheets of water, considered as a whole, do not
+  belong to the normal system of drainage on the land whereby valleys
+  are excavated. On the contrary they are exceptional to it; for the
+  constant tendency of running water is to fill them up, or to drain
+  them by wearing down the barriers that contain them at their outflow.
+  Some of them are referable to movements of the terrestrial crust
+  whereby depressions arise on the surface of the land, as has been
+  noted after earthquakes. Others have arisen from solution such as that
+  of rock-salt or of limestone, the removal of which by underground
+  water causes a subsidence of the ground above. A third type of
+  lake-basin occurs in regions that are now or have once been subject to
+  the erosive action of glaciers (see under next subdivision,
+  _Terrestrial Ice_). Many small lakes or tarns have been caused by the
+  deposit of débris across a valley as by landslips or moraines.
+  Considered from a geological point of view, lakes perform an important
+  function in regulating the drainage of the ground below their outfall
+  and diminishing the destructive effects of floods, in filtering the
+  water received from their affluent streams, and in providing
+  undisturbed areas of deposit in which thick and extensive lacustrine
+  formations may be accumulated. In the inland basins of some dry
+  climates the lakes are salt, owing to excess of evaporation, and their
+  bottoms become the sites of chemical deposits, particularly of
+  chlorides of sodium and magnesium, and calcium sulphate and carbonate.
+
+  5. _Terrestrial Ice._--Each of the forms assumed by frozen water has
+  its own characteristic action in geological processes. Frost has a
+  powerful influence in breaking up damp soils and surfaces of stone in
+  the pores or cracks of which moisture has lodged. The water in
+  freezing expands, and in so doing pushes asunder the component
+  particles of soil or stone, or widens the space between the walls of
+  joints or crevices. When the ice melts the loosened grains remain
+  apart ready to be washed away by rain or blown off by wind, while by
+  the widening of joints large blocks of rock are detached from the
+  faces of cliffs. Where rivers or lakes are frozen over the ice exerts
+  a marked pressure on their banks; and when it breaks up large sheets
+  of it are driven ashore, pushing up quantities of gravel and stones
+  above the level of the water. The piling up of the disrupted ice
+  against obstructions in rivers ponds back the water, and often leads
+  to destructive floods when the ice barriers break. Where the ice has
+  formed round boulders in shallow water, or at the bottom
+  ("anchor-ice"), it may lift these up when the frost gives way, and may
+  transport them for some distance. Ice formed in the atmosphere, and
+  descending to the ground in the form of hail, often causes great
+  destruction to vegetation and not infrequently to animal life. Where
+  the frozen moisture reaches the earth as snow, it serves to protect
+  rock, soil and vegetation from the effects of frost; but on sloping
+  ground it is apt to give rise to destructive avalanches or landslips,
+  while indirectly, by its rapid melting, it may cause serious floods in
+  rivers.
+
+  But the most striking geological work performed by terrestrial ice is
+  that achieved by glaciers (q.v.) and ice-sheets. These vast masses of
+  moving ice, when they descend from mountains where the steeper rocks
+  are clear of snow, receive on their surface the débris detached by
+  frost from the declivities above, and bear these materials to lower
+  levels or to the sea. Enormous quantities of rock-rubbish are thus
+  transported in the Alps and other high mountain ranges. When the ice
+  retreats the boulders carried by it are dropped where it melts, and
+  left there as memorials of the former extension of the glaciers.
+  Evidence of this nature proves the much wider extent of the Alpine ice
+  at a comparatively recent geological date. It can also be shown that
+  detritus from Scandinavia has been ice-borne to the south-east of
+  England and far into the heart of Europe.
+
+  The ice, by means of grains of sand and pieces of stone which it drags
+  along, scores, scratches and polishes the surfaces of rock underneath
+  it, and, in this way, produces the abundant fine sediment that gives
+  the characteristic milky appearance to the rivers that issue from the
+  lower ends of glaciers. By such long-continued attrition the rocks are
+  worn down, portions of them of softer nature, or where the ice acts
+  with especial vigour, are hollowed out into cavities which, on the
+  disappearance of the ice, may be filled with water and become tarns or
+  lakes. Rocks over which land-ice has passed are marked by a peculiar
+  smooth, flowing outline, which forms a contrast to the more rugged
+  surface produced by ordinary weathering. They are covered with
+  groovings, which range from the finest striae left by sharp grains of
+  sand to deep ruts ground out by blocks of stone. The trend of these
+  markings shows the direction in which the ice flowed. By their
+  evidence the position and movement of former glaciers in countries
+  from which the ice has entirely vanished may be clearly determined
+  (see GLACIAL PERIOD).
+
+  6. _The Sea._--The physical features of the sea are discussed in
+  separate articles (see OCEAN AND OCEANOGRAPHY). The sea must be
+  regarded as the great regulator of temperature and climate over the
+  globe, and as thus exerting a profound influence on the distribution
+  of plant and animal life. Its distinctly geological work is partly
+  erosive and partly reproductive. As an eroding agent it must to some
+  extent effect chemical decompositions in the rocks and sediments over
+  which it spreads; but these changes have not yet been satisfactorily
+  studied. Undoubtedly, its chief destructive power is of a mechanical
+  kind, and arises from the action of its waves in beating upon
+  shore-cliffs. By the alternate compression and expansion of the air in
+  crevices of the rocks on which heavy breakers fall, and by the
+  hydraulic pressure which these masses of sea-water exert on the walls
+  of the fissures into which they rush, large masses of rock are
+  loosened and detached, and caves and tunnels are drilled along the
+  base of sea-cliffs. Probably still more efficacious are the blows of
+  the loose shingle, which, caught up and hurled forward by the waves,
+  falls with great force upon the shore rocks, battering them as with a
+  kind of artillery until they are worn away. The smooth surfaces of the
+  rocks within reach of the waves contrasted with their angular forms
+  above that limit bear witness to the amount of waste, while the
+  rounded forms of the boulders and shingle show that they too are being
+  continually reduced in size. Thus the sea, by its action on the
+  coasts, produces much sediment, which is swept away by its waves and
+  currents and strewn over its floor. Besides this material, it is
+  constantly receiving the fine silt and sand carried down by rivers. As
+  the floor of the ocean is thus the final receptacle for the waste of
+  the land, it becomes the chief era on the surface of the globe for the
+  accumulation of new stratified formations. And such has been one of
+  its great functions since the beginning of geological time, as is
+  proved by the rocks that form the visible part of the earth's crust,
+  and consist in great part of marine deposits. Chemical precipitates
+  take place more especially in enclosed parts of the sea, where
+  concentration of the water by evaporation can take place, and where
+  layers of sodium chloride, calcium sulphate and carbonate, and other
+  salts are laid down. But the chief marine accumulations are of
+  detrital origin. Near the land and for a variable distance extending
+  sometimes to 200 or 300 m. from shore the deposits consist chiefly of
+  sediments derived from the waste of the land, the finer silts being
+  transported farthest from their source. At greater depths and
+  distances the ocean floor receives a slow deposit of exceedingly fine
+  clay, which is believed to be derived from the decomposition of pumice
+  and volcanic dust from insular or submarine volcanoes. Wide tracts of
+  the bottom are covered with various forms of ooze derived from the
+  accumulation of the remains of minute organisms.
+
+
+  (C) _Life._
+
+  Among the agents by which geological changes are carried on upon the
+  surface of the globe living organisms must be enumerated. Both plants
+  and animals co-operate with the inorganic agents in promoting the
+  degradation of the land. In some cases, on the other hand, they
+  protect rocks from decay, while, by the accumulation of their remains,
+  they give rise to extensive formations both upon the land and in the
+  sea. Their operations may hence be described as alike destructive,
+  conservative and reproductive. Under this heading also the influence
+  of Man as a geological agent deserves notice.
+
+  (a) _Plants._--Vegetation promotes the disintegration of rocks and
+  soil in the following ways: (1) By keeping the surfaces of stone
+  moist, and thus promoting both mechanical and chemical dissolution, as
+  is especially shown by liverworts, mosses and other moisture-loving
+  plants. (2) By producing through their decay carbonic and other
+  acids, which, together with decaying organic matter taken up by
+  passing moisture, become potent in effecting the chemical
+  decomposition of rocks and in promoting the disintegration of soils.
+  (3) By inserting their roots or branches between joints of rock, which
+  are thereby loosened, so that large slices may be eventually wedged
+  off. (4) By attracting rain, as thick woods, forests and peat-mosses
+  do, and thus accelerating the general waste of a country by running
+  water. (5) By promoting the decay of diseased and dead plants and
+  animals, as when fungi overspread a damp rotting tree or the carcase
+  of a dead animal.
+
+  That plants also exert a conservative influence on the surface of the
+  land is shown in various ways. (1) The formation of a stratum of turf
+  protects the soil and rocks underneath from being rapidly
+  disintegrated and washed away by atmospheric action. (2) Many plants,
+  even without forming a layer of turf, serve by their roots or branches
+  to protect the loose sand or soil on which they grow from being
+  removed by wind. The common sand-carex and other arenaceous plants
+  bind the loose sand-dunes of our coasts, and give them a permanence,
+  which would at once be destroyed were the sand laid bare again to
+  storms. The growth of shrubs and brushwood along the course of a
+  stream not only keeps the alluvial banks from being so easily
+  undermined and removed as would otherwise be the case, but serves to
+  arrest the sediment in floods, filtering the water and thereby adding
+  to the height of the flood plain. (3) Some marine plants, like the
+  calcareous nullipores, afford protection to shore rocks by covering
+  them with a hard incrustation. The tangles and smaller Fuci which grow
+  abundantly on the littoral zone break the force of the waves or
+  diminish the effects of ground swell. (4) Forests and brushwood
+  protect the soil, especially on slopes, from being washed away by rain
+  or ploughed up by avalanches.
+
+  Plants contribute by the aggregation of their remains to the formation
+  of stratified deposits. Some marine algae which secrete carbonate of
+  lime not only encrust rocks but give rise to sheets of submarine
+  limestone. An analogous part is played in fresh-water lakes by various
+  lime-secreting plants, such as _Chara_. Long-continued growth of
+  vegetation has, in some regions, produced thick accumulations of a
+  dark loam, as in the black cotton soil (_regur_) of India, and the
+  black earth (_tchernozom_) of Russia. Peat-mosses are formed in
+  temperate and arctic climates by the growth of marsh-loving plants,
+  sometimes to a thickness of 40 or 50 ft. In tropical regions the
+  mangrove swamps on low moist shores form a dense jungle, sometimes 20
+  m. broad, which protects these shores from the sea until, by the
+  arrest of sediment and the constant contribution of decayed
+  vegetation, the spongy ground is at last turned into firm soil. Some
+  plants (diatoms) can abstract silica and build it into their
+  framework, so that their remains form a siliceous deposit or ooze
+  which covers spaces of the deep sea-floor estimated at more than ten
+  millions of square miles in extent.
+
+  (b) _Animals._--These exert a destructive influence in the following
+  ways: (1) By seriously affecting the composition and arrangement of
+  the vegetable soil. Worms bring up the lower portions of the soil to
+  the surface, and while thus promoting its fertility increase its
+  liability to be washed away by rain. Burrowing animals, by throwing up
+  the soil and subsoil, expose these to be dried and blown away by the
+  wind. At the same time their subterranean passages serve to drain off
+  the superficial water and to injure the stability of the surface of
+  the ground above them. In Britain the mole and rabbit are familiar
+  examples. (2) By interfering with or even diverting the flow of
+  streams. Thus beaver-dams check the current of water-courses,
+  intercept floating materials, and sometimes turn streams into new
+  channels. The embankments of the Mississippi are sometimes weakened to
+  such an extent by the burrowings of the cray-fish as to give way and
+  allow the river to inundate the surrounding country. Similar results
+  have happened in Europe from subterranean operations of rats. (3) Some
+  mollusca bore into stone or wood and by the number of contiguous
+  perforations greatly weaken the material. (4) Many animals exercise a
+  ruinously destructive influence upon vegetation. Of the numerous
+  plagues of this kind the locust, phylloxera and Colorado beetle may be
+  cited.
+
+  The most important geological function performed by animals is the
+  formation of new deposits out of their remains. It is chiefly by the
+  lower grades of the animal kingdom that this work is accomplished,
+  especially by molluscs, corals and foraminifera. Shell-banks are
+  formed abundantly in such comparatively shallow and enclosed basins as
+  that of the North Sea, and on a much more extensive scale on the floor
+  of the West Indian seas. By the coral polyps thick masses of
+  limestones have been built up in the warmer seas of the globe (see
+  CORAL REEFS). The floor of the Atlantic and other oceans is covered
+  with a fine calcareous ooze derived mainly from the remains of
+  foraminifera, while in other regions the bottom shows a siliceous ooze
+  formed almost entirely of radiolaria. Vertebrate animals give rise to
+  phosphatic deposits formed sometimes of their excrement, as in guano
+  and coprolites, sometimes of an accumulation of their bones.
+
+  (c) _Man._--No survey of the geological workings of plant and animal
+  life upon the surface of the globe can be complete which does not take
+  account of the influence of man--an influence of enormous and
+  increasing consequence in physical geography, for man has introduced,
+  as it were, an element of antagonism to nature. His interference shows
+  itself in his relations to climate, where he has affected the
+  meteorological conditions of different countries: (1) By removing
+  forests, and laying bare to the sun and winds areas which were
+  previously kept cool and damp under trees, or which, lying on the lee
+  side, were protected from tempests. It is supposed that the wholesale
+  destruction of the woodlands formerly existing in countries bordering
+  the Mediterranean has been in part the cause of the present
+  desiccation of these districts. (2) By drainage, whereby the
+  discharged rainfall is rapidly removed, and the evaporation is
+  lessened, with a consequent diminution of rainfall and some increase
+  in the general temperature of a country. (3) By the other processes of
+  agriculture, such as the transformation of moor and bog into
+  cultivated land, and the clothing of bare hillsides with green crops
+  or plantations of coniferous and hardwood trees.
+
+  Still more obvious are the results of human interference with the flow
+  of water: (1) By increasing or diminishing the rainfall man directly
+  affects the volume of rivers. (2) By his drainage operations he makes
+  the rain to run off more rapidly than before, and thereby increases
+  the magnitude of floods and of the destruction caused by them. (3) By
+  wells, bores, mines, or other subterranean works he interferes with
+  the underground waters, and consequently with the discharge of
+  springs. (4) By embanking rivers he confines them to narrow channels,
+  sometimes increasing their scour, and enabling them to carry their
+  sediment further seaward, sometimes causing them to deposit it over
+  the plains and raise their level. (5) By his engineering operations
+  for water-supply he abstracts water from its natural basins and
+  depletes the streams.
+
+  In many ways man alters the aspect of a country: (1) By changing
+  forest into bare mountain, or clothing bare mountains with forest. (2)
+  By promoting the growth or causing the removal of peat-mosses. (3) By
+  heedlessly uncovering sand-dunes, and thereby setting in motion a
+  process of destruction which may convert hundreds of acres of fertile
+  land into waste sand, or by prudently planting the dunes with
+  sand-loving vegetation and thus arresting their landward progress. (4)
+  By so guiding the course of rivers as to make them aid him in
+  reclaiming waste land, and bringing it under cultivation. (5) By piers
+  and bulwarks, whereby the ravages of the sea are stayed, or by the
+  thoughtless removal from the beach of stones which the waves had
+  themselves thrown up, and which would have served for a time to
+  protect the land. (6) By forming new deposits either designedly or
+  incidentally. The roads, bridges, canals, railways, tunnels, villages
+  and towns with which man has covered the surface of the land will in
+  many cases form a permanent record of his presence. Under his hand the
+  whole surface of civilized countries is very slowly covered with a
+  stratum, either formed wholly by him or due in great measure to his
+  operations and containing many relics of his presence. The soil of
+  ancient towns has been increased to a depth of many feet by their
+  successive destructions and renovations.
+
+  Perhaps the most subtle of human influences are to be seen in the
+  distribution of plant and animal life upon the globe. Some of man's
+  doings in this domain are indeed plain enough, such as the extirpation
+  of wild animals, the diminution or destruction of some forms of
+  vegetation, the introduction of plants and animals useful to himself,
+  and especially the enormous predominance given by him to the cereals
+  and to the spread of sheep and cattle. But no such extensive
+  disturbance of the normal conditions of the distribution of life can
+  take place without carrying with it many secondary effects, and
+  setting in motion a wide cycle of change and of reaction in the animal
+  and vegetable kingdoms. For example, the incessant warfare waged by
+  man against birds and beasts of prey in districts given up to the
+  chase leads sometimes to unforeseen results. The weak game is allowed
+  to live, which would otherwise be killed off and give more room for
+  the healthy remainder. Other animals which feed perhaps on the same
+  materials as the game are by the same cause permitted to live
+  unchecked, and thereby to act as a further hindrance to the spread of
+  the protected species. But the indirect results of man's interference
+  with the régime of plants and animals still require much prolonged
+  observation.
+
+
+PART V.--GEOTECTONIC OR STRUCTURAL GEOLOGY
+
+From a study of the nature and composition of minerals and rocks, and an
+investigation of the different agencies by which they are formed and
+modified, the geologist proceeds to inquire how these materials have
+been put together so as to build up the visible part of the earth's
+crust. He soon ascertains that they have not been thrown together wholly
+at random, but that they show a recognizable order of arrangement. Some
+of them, especially those of most recent growth, remain in their
+original condition and position, but, in proportion to their antiquity,
+they generally present increasing alteration, until it may no longer be
+possible to tell what was their pristine state. As by far the largest
+accessible portion of the terrestrial crust consists of stratified
+rocks, and as these furnish clear evidence of most of the modifications
+to which they have been subjected in the long course of geological
+history, it is convenient to take them into consideration first. They
+possess a number of structures which belong to the original conditions
+in which they were accumulated. They present in addition other
+structures which have been superinduced upon them, and which they share
+with the unstratified or igneous rocks.
+
+
+1. ORIGINAL STRUCTURES
+
+(a) _Stratified Rocks._--This extensive and important series is above
+all distinguished by possessing a prevailing stratified arrangement.
+Their materials have been laid down in laminae, layers and strata, or
+beds, pointing generally to the intermittent deposition of the sediments
+of which they consist. As this stratification was, as a rule, originally
+nearly or quite horizontal, it serves as a base from which to measure
+any subsequent disturbance which the rocks have undergone. The
+occurrence of false-bedding, i.e. bands of inclined layers between the
+normal planes of stratification, does not form any real exception; but
+indicates the action of shifting currents whereby the sediment was
+transported and thrown down. Other important records of the original
+conditions of deposit are supplied by ripple-marks, sun-cracks,
+rain-prints and concretions.
+
+  From the nature of the material further light is cast on the
+  geographical conditions in which the strata were accumulated. Thus,
+  conglomerates indicate the proximity of old shore-lines, sandstones
+  mark deposits in comparatively shallow water, clays and shales point
+  to the tranquil accumulation of fine silt at a greater depth and
+  further from land, while fossiliferous limestones bear witness to
+  clearer water in which organisms flourished at some distance from
+  deposits of sand and mud. Again, the alternation of different kinds of
+  sediment suggests a variability in the conditions of deposition, such
+  as a shifting of the sediment-bearing currents and of the areas of
+  muddy and clear water. A thick group of conformable strata, that is, a
+  series of deposits which show no discordance in their stratification,
+  may usually be regarded as having been laid down on a sea-floor that
+  was gently sinking. Here and there evidence is obtainable of the
+  limits or of the progress of the subsidence by what is called
+  "overlap." Of the absolute length of time represented by any strata or
+  groups of strata no satisfactory estimates can yet be formed. Certain
+  general conclusions may indeed be drawn, and comparisons may be made
+  between different series of rocks. Sandstones full of false-bedding
+  were probably accumulated more rapidly than finely-laminated shales or
+  clays. It is not uncommon in certain Carboniferous formations to find
+  coniferous and other trunks embedded in sandstone. Some of these trees
+  seem to have been carried along and to have sunk, their heavier or
+  root end touching the bottom and their upper end slanting upward in
+  the direction of the current, exactly as in the case of the snags of
+  the Mississippi. In other cases the trees have been submerged while
+  still in their positions of growth. The continuous deposit of sand at
+  last rose above the level of the trunks and buried them. It is clear
+  then that the rate of deposit must have been sometimes sufficiently
+  rapid to allow sand to accumulate to a depth of 30 ft. or more before
+  the decay of the wood. Modern instances are known where, under certain
+  circumstances, submerged trees may last for some centuries, but even
+  the most durable must decay in what, after all, is a brief space of
+  geological time. Since continuous layers of the same kind of deposit
+  suggest a persistence of geological conditions, while numerous
+  alternations of different kinds of sedimentary matter point to
+  vicissitudes or alternations of conditions, it may be supposed that
+  the time represented by a given thickness of similar strata was less
+  than that shown by the same thickness of dissimilar strata, because
+  the changes needed to bring new varieties of sediment into the area of
+  deposit would usually require the lapse of some time for their
+  completion. But this conclusion may often be erroneous. It will be
+  best supported when, from the very nature of the rocks, wide
+  variations in the character of the water-bottom can be established.
+  Thus a group of shales followed by a fossiliferous limestone would
+  almost always mark the lapse of a much longer period than an equal
+  depth of sandy strata. A thick mass of limestone, made up of organic
+  remains which lived and died upon the spot, and whose remains are
+  crowded together generation above generation, must have demanded many
+  years or centuries for its formation.
+
+  But in all speculations of this kind we must bear in mind that the
+  length of time represented by a given depth of strata is not to be
+  estimated merely from their thickness or lithological character. The
+  interval between the deposit of two successive laminae of shale may
+  have been as long as, or even longer than, that required for the
+  formation of one of the laminae. In like manner the interval needed
+  for the transition from one stratum or kind of strata to another may
+  often have been more than equal to the time required for the formation
+  of the strata on either side. But the relative chronological
+  importance of the bars or lines in the geological record can seldom be
+  satisfactorily discussed merely on lithological grounds. This must
+  mainly be decided on the evidence of organic remains, as shown in
+  Part VI., where the grouping of the stratified rocks into formations
+  and systems is described.
+
+(b) _Igneous Rocks._--As part of the earth's crust these rocks present
+characters by which they are strongly differentiated from the stratified
+series. While the broad petrographical distinctions of their several
+varieties remain persistent, they present sufficient local variations of
+type to point to the existence of what have been called petrographic
+provinces, in each of which the eruptive masses are connected by a
+general family relationship, differing more or less from that of a
+neighbouring province. In each region presenting a long chronological
+series of eruptive rocks a petrographical sequence can be traced, which
+is observed to be not absolutely the same everywhere, though its general
+features may be persistent. The earliest manifestations of eruptive
+material in any district appear to have been most frequently of an
+intermediate type between acid and basic, passing thence into a
+thoroughly acid series and concluding with an effusion of basic
+material.
+
+Considered as part of the architecture of the crust of the earth,
+igneous rocks are conveniently divisible into two great series: (1)
+those bodies of material which have been injected into the crust and
+have solidified there, and (2) those which have reached the surface and
+have been ejected there, either in a molten state as lava or in a
+fragmental form as dust, ashes and scoriae. The first of these divisions
+represents the plutonic, intrusive or subsequent phase of eruptivity;
+the second marks the volcanic, interstratified or contemporaneous phase.
+
+  1. The plutonic or intrusive rocks, which have been forced into the
+  crust and have consolidated there, present a wide range of texture
+  from the most coarse-grained granites to the most perfect natural
+  glass. Seeing that they have usually cooled with extreme slowness
+  underground, they are as a general rule more largely crystalline than
+  the volcanic series. The form assumed by each individual body of
+  intrusive material has depended upon the shape of the space into which
+  it has been injected, and where it has cooled and become solid. This
+  shape has been determined by the local structure of the earth's crust
+  on the one hand and by the energy of the eruptive force on the other.
+  It offers a convenient basis for the classification of the intrusive
+  rocks, which, as part of the framework of the crust, may thus be
+  grouped according to the shape of the cavity which received them, as
+  bosses, sills, dikes and necks.
+
+  Bosses, or stocks, are the largest and most shapeless extravasations
+  of erupted material. They include the great bodies of granite which,
+  in most countries of the world, have risen for many miles through the
+  stratified formations and have altered the rocks around them by
+  contact-metamorphism. Sills, or intrusive sheets, are bed-like masses
+  which have been thrust between the planes of sedimentary or even of
+  igneous rocks. The term laccolite has been applied to sills which are
+  connected with bosses. Intrusive sheets are distinguishable from true
+  contemporaneously intercalated lavas by not keeping always to the same
+  platform, but breaking across and altering the contiguous strata, and
+  by the closeness of their texture where they come in contact with the
+  contiguous rocks, which, being cold, chilled the molten material and
+  caused it to consolidate on its outer margins more rapidly than in its
+  interior. Dikes or veins are vertical walls or ramifying branches of
+  intrusive material which has consolidated in fissures or irregular
+  clefts of the crust. Necks are volcanic chimneys which have been
+  filled up with erupted material, and have now been exposed at the
+  surface after prolonged denudation has removed not only the
+  superficial volcanic masses originally associated with them, but also
+  more or less of the upper part of the vents. Plutonic rocks do not
+  present evidence of their precise geological age. All that can be
+  certainly affirmed from them is that they must be younger than the
+  rocks into which they have been intruded. From their internal
+  structure, however, and from the evidence of the rocks associated with
+  them, some more or less definite conjectures may be made as to the
+  limits of time within which they were probably injected.
+
+  2. The interstratified or volcanic series is of special importance in
+  geology, inasmuch as it contains the records of volcanic action during
+  the past history of the globe. It was pointed out in Part I. that
+  while towards the end of the 18th and in the beginning of the 19th
+  century much attention was paid by Hutton and his followers to the
+  proofs of intrusion afforded by what they called the "unerupted lavas"
+  within the earth's crust, these observers lost sight of the
+  possibility that some of these rocks might have been erupted at the
+  surface, and might thus be chronicles of volcanic action in former
+  geological periods. It is not always possible to satisfactorily
+  discriminate between the two types of contemporaneously intercalated
+  and subsequently injected material. But rocks of the former type have
+  not broken into or involved the overlying strata, and they are usually
+  marked by the characteristic structures of superficial lavas and by
+  their association with volcanic tuffs. By means of the evidence which
+  they supply, it has been ascertained that volcanic action has been
+  manifested in the globe since the earliest geological periods. In the
+  British Isles, for example, the volcanic record is remarkably full for
+  the long series of ages from Cambrian to Permian time, and again for
+  the older Tertiary period.
+
+
+2. SUBSEQUENTLY INDUCED STRUCTURES
+
+After their accumulation, whether as stratified or eruptive masses, all
+kinds of rocks have been subject to various changes, and have acquired
+in consequence a variety of superinduced structures. It has been pointed
+out in the part of this article dealing with dynamical geology that one
+of the most important forms of energy in the evolution of geological
+processes is to be found in the movements that take place within the
+crust of the earth. Some of these movements are so slight as to be only
+recognizable by means of delicate instruments; but from this inferior
+limit they range up to gigantic convulsions by which mountain-chains are
+upheaved. The crust must be regarded as in a perpetual state of strain,
+and its component materials are therefore subject to all the effects
+which flow from that condition. It is the one great object of the
+geotectonic division of geology to study the structures which have been
+developed in consequence of earth-movements, and to discover from this
+investigation the nature of the processes whereby the rocks of the crust
+have been brought into the condition and the positions in which we now
+find them. The details of this subject will be found in separate
+articles descriptive of each of the technical terms applied to the
+several kinds of superinduced structures. All that need be offered here
+is a general outline connecting the several portions of the subject
+together.
+
+  One of the most universal of these later structures is to be seen in
+  the divisional planes, usually vertical or highly inclined, by which
+  rocks are split into quadrangular or irregularly shaped blocks. To
+  these planes the name of joints has been given. They are of prime
+  importance from an industrial point of view, seeing that the art of
+  quarrying consists mainly in detecting and making proper use of them.
+  Their abundance in all kinds of rocks, from those of recent date up to
+  those of the highest antiquity, affords a remarkable testimony to the
+  strains which the terrestrial crust has suffered. They have arisen
+  sometimes from tension, such as that caused by contraction from the
+  drying and consolidation of an aqueous sediment or from the cooling of
+  a molten mass; sometimes from torsion during movements of the crust.
+
+  Although the stratified rocks were originally deposited in a more or
+  less nearly horizontal position on the floor of the sea, where now
+  visible on the dry land they are seldom found to have retained their
+  flatness. On the contrary, they are seen to have been generally tilted
+  up at various angles, sometimes even placed on end (crop, dip,
+  strike). When a sufficiently large area of ground is examined, the
+  inclination into which the strata have been thrown may be observed not
+  to continue far in the same direction, but to turn over to the
+  opposite or another quarter. It can then be seen that in reality the
+  rocks have been thrown into undulations. From the lowest and flattest
+  arches where the departure from horizontality may be only trifling,
+  every step may be followed up to intense curvature, where the strata
+  have been compressed and plicated as if they had been piles of soft
+  carpets (anticline, syncline, monocline, geo-anticline, geo-syncline,
+  isoclinal, plication, curvature, quaquaversal). It has further
+  happened abundantly all over the surface of the globe that relief from
+  internal strain in the crust has been obtained by fracture, and the
+  consequent subsidence or elevation of one or both sides of the
+  fissure. The differential movement between the two sides may be
+  scarcely perceptible in the feeblest dislocation, but in the extreme
+  cases it may amount to many thousand feet (fault, fissure,
+  dislocation, hade, slickensides). The great faults in a country are
+  among its most important structural features, and as they not
+  infrequently continue to be lines of weakness in the crust along which
+  sudden slipping may from time to time take place, they become the
+  lines of origin of earthquakes. The San Francisco earthquake of 1906,
+  already cited, affords a memorable illustration of this connexion.
+
+  It is in a great mountain-chain that the extraordinary complication of
+  plicated and faulted structures in the crust of the earth can be most
+  impressively beheld. The combination of overturned folds with rupture
+  has been already referred to as a characteristic feature in the Alps
+  (Part IV.). The gigantic folds have in many places been pushed over
+  each other so as to lie almost flat, while the upper limb has not
+  infrequently been driven for many miles beyond the lower by a rupture
+  along the axis. In this way successive slices of a thick series of
+  formations have been carried northwards on the northern slope of the
+  Alps, and have been piled so abnormally above each other that some of
+  their oldest members recur several times on different thrust-planes,
+  the whole being underlain by Tertiary strata (see ALPS). Further
+  proof of the colossal compression to which the rocks have been
+  subjected is afforded by their intense crumpling and corrugation, and
+  by the abundantly faulted and crushed condition to which they have
+  been reduced. Similar evidence as to stresses in the terrestrial crust
+  and the important changes which they produce among the rocks may also
+  be obtained on a smaller scale in many non-mountainous countries.
+
+  Another marked result of the compression of the terrestrial crust has
+  been induced in some rocks by the production of the fissile structure
+  which is typically shown in roofing-slate (cleavage). Closely
+  connected with this internal rearrangement has been the development of
+  microscopic microlites or crystals (rutile, mica, &c.) in argillaceous
+  slates which were undoubtedly originally fine marine mud and silt.
+  From this incipient form of metamorphism successive stages may be
+  traced through the various kinds of argillite and phyllite into
+  mica-schist, and thence into more crystalline gneissoid varieties
+  (foliation, slate, mica-schist, gneiss). The Alps afford excellent
+  illustrations of these transformations.
+
+  The fissures produced in the crust are sometimes clean, sharply
+  defined divisional planes, like cracks across a pane of glass. Much
+  more usually, however, the rocks on either side have been broken up by
+  the friction of movement, and the fault is marked by a variable
+  breadth of this broken material. Sometimes the walls have separated
+  and molten rock has risen from below and solidified between them as a
+  dike. Occasionally the fissures have opened to the surface, and have
+  been filled in from above with detritus, as in the sandstone-dikes of
+  Colorado and California. In mineral districts the fissures have been
+  filled with various spars and ores, forming what are known as mineral
+  veins.
+
+  Where one series of rocks is covered by another without any break or
+  discordance in the stratification they are said to be conformable. But
+  where the older series has been tilted up or visibly denuded before
+  being overlain by the younger, the latter is termed unconformable.
+  This relation is one of the greatest value in structural geology, for
+  it marks a gap in the geological record, which may represent a vast
+  lapse of time not there recorded by strata.
+
+
+PART VI.--PALEONTOLOGICAL GEOLOGY
+
+This division of the science deals with fossils, or the traces of plants
+and animals preserved in the rocks of the earth's crust, and endeavours
+to gather from them information as to the history of the globe and its
+inhabitants. The term "fossil" (Lat. _fossilis_, from _fodere_, to dig
+up), meaning literally anything "dug up," was formerly applied
+indiscriminately to any mineral substance taken out of the earth's
+crust, whether organized or not. Since the time of Lamarck, however, the
+meaning of the word has been restricted, so as to include only the
+remains or traces of plants and animals preserved in any natural
+formation whether hard rock or superficial deposit. It includes not
+merely the petrified structures of organisms, but whatever was directly
+connected with or produced by these organisms. Thus the resin which was
+exuded from trees of long-perished forests is as much a fossil as any
+portion of the stem, leaves, flowers or fruit, and in some respects is
+even more valuable to the geologist than more determinable remains of
+its parent trees, because it has often preserved in admirable perfection
+the insects which flitted about in the woodlands. The burrows and trails
+of a worm preserved in sandstone and shale claim recognition as fossils,
+and indeed are commonly the only indications to be met with of the
+existence of annelid life among old geological formations. The droppings
+of fishes and reptiles, called coprolites, are excellent fossils, and
+tell their tale as to the presence and food of vertebrate life in
+ancient waters. The little agglutinated cases of the caddis-worm remain
+as fossils in formations from which, perchance, most other traces of
+life may have passed away. Nay, the very handiwork of man, when
+preserved in any natural manner, is entitled to rank among fossils; as
+where his flint-implements have been dropped into the pre-historic
+gravels of river-valleys or where his canoes have been buried in the
+silt of lake-bottoms.
+
+  A study of the land-surfaces and sea-floors of the present time shows
+  that there are so many chances against the conservation of the remains
+  of either terrestrial or marine animals and plants that if, as is
+  probable, the same conditions existed in former geological periods, we
+  should regard the occurrence of organic remains among the stratified
+  formations of the earth's crust as generally the result of various
+  fortunate accidents.
+
+  Let us consider, in the first place, the chances for the preservation
+  of remains of the present fauna and flora of a country. The surface of
+  the land may be densely clothed with forest and abundantly peopled
+  with animal life. But the trees die and moulder into soil. The
+  animals, too, disappear, generation after generation, and leave few or
+  no perceptible traces of their existence. If we were not aware from
+  authentic records that central and northern Europe were covered with
+  vast forests at the beginning of our era, how could we know this fact?
+  What has become of the herds of wild oxen, the bears, wolves and other
+  denizens of primeval Europe? How could we prove from the examination
+  of the surface soil of any country that those creatures had once
+  abounded there? The conditions for the preservation of any relics of
+  the plant and animal life of a terrestrial surface must obviously be
+  always exceptional. They are supplied only where the organic remains
+  can be protected from the air and superficial decay. Hence they may be
+  observed in (1) the deposits on the floors of lakes; (2) in
+  peat-mosses; (3) in deltas at river-mouths; and (4) under the
+  stalagmite of caverns in limestone districts. But in these and other
+  favourable places a mere infinitesimal fraction of the fauna or flora
+  of a land-surface is likely to be entombed or preserved.
+
+  In the second place, although in the sea the conditions for the
+  preservation of organic remains are in many respects more favourable
+  than on land, they are apt to be frustrated by many adverse
+  circumstances. While the level of the land remains stationary, there
+  can be but little effective entombment of marine organisms in littoral
+  deposits; for only a limited accumulation of sediment will be formed
+  until subsidence of the sea-floor takes place. In the trifling beds of
+  sand or gravel thrown up on a stationary shore, only the harder and
+  more durable forms of life, such as gastropods and lamellibranchs,
+  which can withstand the triturating effects of the beach waves, are
+  likely to remain uneffaced.
+
+  Below tide-marks, along the margin of the land where sediment is
+  gradually deposited, the conditions are more favourable for the
+  preservation of marine organisms. In the sheets of sand and mud there
+  laid down the harder parts of many forms of life may be entombed and
+  protected from decay. But only a small proportion of the total marine
+  fauna may be expected to appear in such deposits. At the best, merely
+  littoral and shallow-water forms will occur, and, even under the most
+  favourable conditions, they will represent but a fraction of the whole
+  assemblage of life in these juxta-terrestrial parts of the ocean. As
+  we recede from the land the rate of deposition of sediment on the
+  sea-floor must become feebler, until, in the remote central abysses,
+  it reaches a hardly appreciable minimum. Except, therefore, where some
+  kind of ooze or other deposit is accumulating in these more pelagic
+  regions, the conditions must be on the whole unfavourable for the
+  preservation of any adequate representation of the deep-sea fauna.
+  Hard durable objects, such as teeth and bones, may slowly accumulate,
+  and be protected by a coating of peroxide of manganese, or of some of
+  the silicates now forming here and there over the deep-sea bottom; or
+  the rate of growth of the abysmal deposit may be so tardy that most of
+  the remains of at least the larger animals will disappear, owing to
+  decay, before they can be covered up and preserved. Any such deep-sea
+  formation, if raised into land, would supply but a meagre picture of
+  the whole life of the sea.
+
+  It would thus appear that the portion of the sea-floor best suited for
+  receiving and preserving the most varied assemblage of marine organic
+  remains is the area in front of the land, to which rivers and currents
+  bring continual supplies of sediment. The most favourable conditions
+  for the accumulation of a thick mass of marine fossiliferous strata
+  will arise when the area of deposit is undergoing a gradual
+  subsidence. If the rate of depression and that of deposit were equal,
+  or nearly so, the movement might proceed for a vast period without
+  producing any great apparent change in marine geography, and even
+  without seriously affecting the distribution of life over the
+  sea-floor within the area of subsidence. Hundreds or thousands of feet
+  of sedimentary strata might in this way be heaped up round the
+  continents, containing a fragmentary series of organic remains
+  belonging to those forms of comparatively shallow-water life which had
+  hard parts capable of preservation. There can be little doubt that
+  such has, in fact, been the history of the main mass of stratified
+  formations in the earth's crust. By far the largest proportion of
+  these piles of marine strata has unquestionably been laid down in
+  water of no great depth within the area of deposit of terrestrial
+  sediment. The enormous thickness to which they attain seems only
+  explicable by prolonged and repeated movements of subsidence,
+  interrupted, however, as we know, by other movements of a contrary
+  kind.
+
+  Since the conditions for the preservation of organic remains exist
+  more favourably under the sea than on land, marine organisms must be
+  far more abundantly conserved than those of the land. This is true
+  to-day, and has, as far as known, been true in all past geological
+  time. Hence for the purposes of the geologist the fossil remains of
+  marine forms of life far surpass all others in value. Among them there
+  will necessarily be a gradation of importance, regulated chiefly by
+  their relative abundance. Now, of all the marine tribes which live
+  within the juxta-terrestrial belt of sedimentation, unquestionably the
+  Mollusca stand in the place of pre-eminence as regards their aptitude
+  for becoming fossils. They almost all possess a hard, durable shell,
+  capable of resisting considerable abrasion and readily passing into a
+  mineralized condition. They are extremely abundant both as to
+  individuals and genera. They occur on the shore within tide mark, and
+  range thence down into the abysses. Moreover, they appear to have
+  possessed these qualifications from early geological times. In the
+  marine Mollusca, therefore, we have a common ground of comparison
+  between the stratified formations of different periods. They have been
+  styled the alphabet of palaeontological inquiry.
+
+There are two main purposes to which fossils may be put in geological
+research: (1) to throw light upon former conditions of physical
+geography, such as the presence of land, rivers, lakes and seas, in
+places where they do not now exist, changes of climate, and the former
+distribution of plants and animals; and (2) to furnish a guide in
+geological chronology whereby rocks may be classified according to
+relative date, and the facts of geological history may be arranged and
+interpreted as a connected record of the earth's progress.
+
+  1. As examples of the first of these two directions of inquiry
+  reference may be made to (a) former land-surfaces revealed by the
+  occurrence of layers of soil with tree-stumps and roots still in the
+  position of growth (see PURBECKIAN); (b) ancient lakes proved by beds
+  of marl or limestone full of lacustrine shells; (c) old sea-bottoms
+  marked by the occurrence of marine organisms; (d) variations in the
+  quality of the water, such as freshness or saltness, indicated by
+  changes in the size and shape of the fossils; (e) proximity to former
+  land, suggested by the occurrence of abundant drift-wood in the
+  strata; (f) former conditions of climate, different from the present,
+  as evidenced by such organisms as tropical types of plants and animals
+  intercalated among the strata of temperate or northern countries.
+
+  2. In applying fossils to the determination of geological chronology
+  it is first necessary to ascertain the order of superposition of the
+  rocks. Obviously, in a continuous series of undisturbed sedimentary
+  deposits the lowest must necessarily be the oldest, and the plants or
+  animals which they contain must have lived and died before any of the
+  organisms that occur in the overlying strata. This order of
+  superposition having been settled in a series of formations, it is
+  found that the fossils at the bottom are not quite the same as those
+  at the top of the series. Tracing the beds upward, we discover that
+  species after species of the lowest platforms disappears, until
+  perhaps not one of them is found. With the cessation of these older
+  species others make their entrance. These, in turn, are found to die
+  out, and to be replaced by newer forms. After patient examination of
+  the rocks, it has been ascertained that every well-marked "formation,"
+  or group of strata, is characterized by its own species or genera, or
+  by a general assemblage, or _facies_, of organic forms. Such a
+  generalization can only, of course, be determined by actual practical
+  experience over an area of some size. When the typical fossils of a
+  formation are known, they serve to identify that formation in its
+  progress across a country. Thus, in tracts where the true order of
+  superposition cannot be determined, owing to the want of sections or
+  to the disturbed condition of the rocks, fossils serve as a means of
+  identification and furnish a guide to the succession of the rocks.
+  They even demonstrate that in some mountainous ground the beds have
+  been turned completely upside down, where it can be shown that the
+  fossils in what are now the uppermost strata ought properly to lie
+  underneath those in the beds below them.
+
+  It is by their characteristic fossils that the stratified rocks of the
+  earth's crust can be most satisfactorily subdivided into convenient
+  groups of strata and classed in chronological order. Each "formation"
+  is distinguished by its own peculiar assemblage of organic remains, by
+  means of which it can be followed and recognized, even amid the
+  crumplings and dislocations of a disturbed region. The same general
+  succession of organic types can be observed over a large part of the
+  world, though, of course, with important modifications in different
+  countries. This similarity of succession has been termed _homotaxis_,
+  a term which expresses the fact that the order in which the leading
+  types of organized existence have appeared upon the earth has been
+  similar even in widely separated regions. It is evident that, in this
+  way, a reliable method of comparison is furnished, whereby the
+  stratified formations of different parts of the earth's crust can be
+  brought into relation with each other. Had the geologist continued to
+  remain, as in the days of Werner, hampered by the limitations imposed
+  by a reliance on mere lithological characters, he would have made
+  little or no progress in deciphering the record of the successive
+  phases of the history of the globe chronicled in the crust. Just as,
+  at the present time, sheets of gravel in one place are contemporaneous
+  with sheets of mud at another, so in the past all kinds of
+  sedimentation have been in progress simultaneously, and those of one
+  period may not be distinguishable in themselves from those of another.
+  Little or no reliance can be placed upon lithological resemblances or
+  differences in comparing the sedimentary formations of different
+  countries.
+
+  In making use of fossil evidence for the purpose of subdividing the
+  stratified rocks of the earth's crust, it is found to be applicable to
+  the smaller details of stratigraphy as well as to the definition of
+  large groups of strata. Thus a particular stratum may be marked by the
+  occurrence in it of various fossils, one or more of which may be
+  distinctive, either from occurring in no other bed above and below or
+  from special abundance in that stratum. One or more of these species
+  is therefore used as a guide to the occurrence of the bed in
+  question, which is called by the name of the most abundant species. In
+  this way what is called a "geological horizon," or "zone," is marked
+  off, and its exact position in the series of formations is fixed.
+
+  Perhaps the most distinctive feature in the progress of
+  palaeontological geology during the last half century has been the
+  recognition and wide application of this method of zonal stratigraphy,
+  which, in itself, was only a further development of William Smith's
+  famous idea, "Strata identified by Organized Fossils." It was first
+  carried out in detail by various palaeontologists in reference to the
+  Jurassic formations, notably by F.A. von Quenstedt and C.A. Oppel in
+  Germany and A.D. d'Orbigny in France. The publication of Oppel's
+  classic work _Die Juraformation Englands, Frankreichs und des
+  südwestlichen Deutschlands_ (1856-1858) marked an epoch in the
+  development of stratigraphical geology. Combining what had been done
+  by various observers with his own laborious researches in France,
+  England, Württemberg and Bavaria, he drew up a classification of the
+  Jurassic system, grouping its several formations into zones, each
+  characterized by some distinctly predominant fossil after which it was
+  named (see LIAS). The same method of classification was afterwards
+  extended to the Cretaceous series by A.D. d'Orbigny, E. Hébert and
+  others, until the whole Mesozoic rocks from the Trias to the top of
+  the Chalk has now been partitioned into zones, each named after some
+  characteristic species or genus of fossils. More recently the
+  principle has been extended to the Palaeozoic formations, though as
+  yet less fully than to the younger parts of the geological record. It
+  has been successfully applied by Professor C. Lapworth to the
+  investigation of the Silurian series (see SILURIAN; ORDOVICIAN
+  SYSTEM). He found that the species of graptolites have each a
+  comparatively narrow vertical range, and they may consequently be used
+  for stratigraphical purposes. Applying the method, in the first
+  instance, to the highly plicated Silurian rocks of the south of
+  Scotland, he found that by means of graptolites he was able to work
+  out the structure of the ground. Each great group of strata was seen
+  to possess its own graptolitic zones, and by their means could be
+  identified not only in the original complex Scottish area, but in
+  England and Wales and in Ireland. It was eventually ascertained that
+  the succession of zones in Great Britain could be recognized on the
+  Continent, in North America and even in Australia. The brachiopods and
+  trilobites have likewise been made use of for zonal purposes among the
+  oldest sedimentary formations. The most ancient of the Palaeozoic
+  systems has as its fitting base the _Olenellus_ zone.
+
+  Within undefined and no doubt variable geographical limits
+  palaeontological zones have been found to be remarkably persistent.
+  They follow each other in the same general order, but not always with
+  equal definiteness. The type fossil may appear in some districts on a
+  higher or a lower platform than it does in others. Only to a limited
+  degree is there any coincidence between lithological variations in the
+  strata and the sequence of the zones. In the Jurassic formations,
+  indeed, where frequent alternations of different sedimentary materials
+  are to be met with, it is in some cases possible to trace a definite
+  upward or downward limit for a zone by some abrupt change in the
+  sedimentation, such as from limestone to shale. But such a precise
+  demarcation is impossible where no distinct bands of different
+  sediments are to be seen. The zones can then only be vaguely
+  determined by finding their characteristic fossils, and noting where
+  these begin to appear in the strata and where they cease. It would
+  seem, therefore, that the sequence of palaeontological zones, or
+  life-horizons, has not depended merely upon changes in the nature of
+  the conditions under which the organisms lived. We should naturally
+  expect that these changes would have had a marked influence; that, for
+  instance, a difference should be perceptible between the character of
+  the fossils in a limestone and that of those in a shale or a
+  sandstone. The environment, when a limestone was in course of
+  deposition, would generally be one of clear water, favourable for a
+  more vigorous and more varied fauna than where a shale series was
+  accumulating, when the water would be discoloured, and only such
+  animals would continue to live in it, or on the bottom, as could
+  maintain themselves in the midst of mud. But no such lithological
+  reason, betokening geographical changes that would affect living
+  creatures, can be adduced as a universally applicable explanation of
+  the occurrence and limitation of palaeontological zones. One of these
+  zones may be only a few inches, or feet or yards in vertical extent,
+  and no obvious lithological or other cause can be seen why its
+  specially characteristic fossils should not be found just as
+  frequently in the similar strata above and below. There is often
+  little or no evidence of any serious change in the conditions of
+  sedimentation, still less of any widespread physical disturbance, such
+  as the catastrophes by which the older geologists explained the
+  extinction of successive types of life.
+
+  It has been suggested that, where the life-zones are well defined,
+  sedimentation has been extremely slow, and that though these zones
+  follow each other with no break in the sedimentation, they were really
+  separated by prolonged intervals of time during which organic
+  evolution could come effectively into play. But it is not easy to
+  explain how, for example in the Lower Lias, there could have been a
+  succession of prodigious intervals, when practically no sediment was
+  laid down, and yet that the strata should show no sign of
+  contemporaneous disturbance or denudation, but succeed each other as
+  if they had been accumulated by one continuous process of deposit. It
+  must be admitted that the problem of life-zones in stratigraphical
+  geology has not yet been solved.
+
+  As Darwin first cogently showed, the history of life has been very
+  imperfectly registered in the stratified parts of the earth's crust.
+  Apart from the fact that, even under the most favourable conditions,
+  only a small proportion of the total flora and fauna of any period
+  would be preserved in the fossil state, enormous gaps occur where no
+  record has survived at all. It is as if whole chapters and books were
+  missing from a historical work. Some of these lacunae are sufficiently
+  obvious. Thus, in some cases, powerful dislocations have thrown
+  considerable portions of the rocks out of sight. Sometimes extensive
+  metamorphism has so affected them that their original characters,
+  including their organic contents, have been destroyed. Oftenest of
+  all, denudation has come into play, and vast masses of fossiliferous
+  rock have been entirely worn away, as is demonstrated by the abundant
+  unconformabilities in the structure of the earth's crust.
+
+  While the mere fact that one series of rocks lies unconformably on
+  another proves the lapse of a considerable interval between their
+  respective dates, the relative length of this interval may sometimes
+  be proved by means of fossil evidence, and by this alone. Let us
+  suppose, for example, that a certain group of formations has been
+  disturbed, upraised, denuded and covered unconformably by a second
+  group. In lithological characters the two may closely resemble each
+  other, and there may be nothing to show that the gap represented by
+  their unconformability is of an important character. In many cases,
+  indeed, it would be quite impossible to pronounce any well-grounded
+  judgment as to the amount of interval, even measured by the vague
+  relative standards of geological chronology. But if each group
+  contains a well-preserved suite of organic remains, it may not only be
+  possible, but easy, to say exactly how much of the geological record
+  has been left out between the two sets of formations. By comparing the
+  fossils with those obtained from regions where the geological record
+  is more complete, it may be ascertained, perhaps, that the lower rocks
+  belong to a certain platform or stage in geological history which for
+  our present purpose we may call D, and that the upper rocks can in
+  like manner be paralleled with stage H. It would be then apparent that
+  at this locality the chronicles of three great geological periods E,
+  F, and G were wanting, which are elsewhere found to be intercalated
+  between D and H. The lapse of time represented by this
+  unconformability would thus be equivalent to that required for the
+  accumulation of the three missing formations in those regions where
+  sedimentation was more continuous.
+
+  Fossil evidence may be made to prove the existence of gaps which are
+  not otherwise apparent. As has been already remarked, changes in
+  organic forms must, on the whole, have been extremely slow in the
+  geological past. The whole species of a sea-floor could not pass
+  entirely away, and be replaced by other forms, without the lapse of
+  long periods of time. If then among the conformable stratified
+  formations of former ages we encounter sudden and abrupt changes in
+  the _facies_ of the fossils, we may be certain that these must mark
+  omissions in the record, which we may hope to fill in from a more
+  perfect series elsewhere. The complete biological contrasts between
+  the fossil contents of unconformable strata are sufficiently
+  explicable. It is not so easy to give a satisfactory account of those
+  which occur where the beds are strictly conformable, and where no
+  evidence can be observed of any considerable change of physical
+  conditions at the time of deposit. A group of strata having the same
+  general lithological characters throughout may be marked by a great
+  discrepance between the fossils above and below a certain line. A few
+  species may pass from the one into the other, or perhaps every species
+  may be different. In cases of this kind, when proved to be not merely
+  local but persistent over wide areas, we must admit, notwithstanding
+  the apparently undisturbed and continuous character of the original
+  deposition of the strata, that the abrupt transition from the one
+  _facies_ of fossils to the other represents a long interval of time
+  which has not been recorded by the deposit of strata. A.C. Ramsay, who
+  called attention to these gaps, termed them "breaks in the succession
+  of organic remains." He showed that they occur abundantly among the
+  Palaeozoic and Secondary rocks of England. It is obvious, of course,
+  that such breaks, even though traceable over wide regions, were not
+  general over the whole globe. There have never been any universal
+  interruptions in the continuity of the chain of being, so far as
+  geological evidence can show. But the physical changes which caused
+  the breaks may have been general over a zoological district or minor
+  region. They no doubt often caused the complete extinction of genera
+  and species which had a small geographical range.
+
+  From all these facts it is clear that the geological record, as it now
+  exists, is at the best but an imperfect chronicle of geological
+  history. In no country is it complete. The lacunae of one region must
+  be supplied from another. Yet in proportion to the geographical
+  distance between the localities where the gaps occur and those whence
+  the missing intervals are supplied, the element of uncertainty in our
+  reading of the record is increased. The most desirable method of
+  research is to exhaust the evidence for each area or province, and to
+  compare the general order of its succession as a whole with that which
+  can be established for other provinces.
+
+
+PART VII.--STRATIGRAPHICAL GEOLOGY
+
+This branch of the science arranges the rocks of the earth's crust in
+the order of their appearance, and interprets the sequence of events of
+which they form the records. Its province is to cull from the other
+departments of geology the facts which may be needed to show what has
+been the progress of our planet, and of each continent and country, from
+the earliest times of which the rocks have preserved any memorial. Thus
+from mineralogy and petrography it contains information regarding the
+origin and subsequent mutations of minerals and rocks. From dynamical
+geology it learns by what agencies the materials of the earth's crust
+have been formed, altered, broken, upheaved and melted. From geotectonic
+geology it understands the various processes whereby these materials
+were put together so as to build up the complicated crust of the earth.
+From palaeontological geology it receives in well-determined fossil
+remains a clue by which to discriminate the different stratified
+formations, and to trace the grand onward march of organized existence
+upon this planet. Stratigraphical geology thus gathers up the sum of all
+that is made known by the other departments of the science, and makes it
+subservient to the interpretation of the geological history of the
+earth.
+
+The leading principles of stratigraphy may be summed up as follows:
+
+1. In every stratigraphical research the fundamental requisite is to
+establish the order of superposition of the strata. Until this is
+accomplished it is impossible to arrange the dates, and make out the
+sequence of geological history.
+
+2. The stratified portion of the earth's crust, or what has been called
+the "geological record," can be subdivided into natural groups, or
+series of strata, characterized by distinctive organic remains and
+recognizable by these remains, in spite of great changes in lithological
+character from place to place. A bed, or a number of beds, linked
+together by containing one or more distinctive species or genera of
+fossils is termed a _zone_ or _horizon_, and usually bears the name of
+one of its more characteristic fossils, as the _Planorbis_-zone of the
+Lower Lias, which is so called from the prevalence in it of the ammonite
+_Psiloceras planorbis_. Two or more such zones related to each other by
+the possession of a number of the same characteristic species or genera
+have been designated _beds_ or an _assise_. Two or more sets of beds or
+assises similarly related form a _group_ or _stage_; a number of groups
+or stages make a _series_, _formation_ or _section_, and a succession of
+formations may be united into a _system_.
+
+3. Some living species of plants and animals can be traced downwards
+through the more recent geological formations; but the number which can
+be so followed grows smaller as the examination is pursued into more
+ancient deposits. With their disappearance other species or genera
+present themselves which are no longer living. These in turn may be
+traced backward into earlier formations, till they too cease and their
+places are taken by yet older forms. It is thus shown that the
+stratified rocks contain the records of a gradual progression of organic
+forms. A species which has once died out does not seem ever to have
+reappeared.
+
+4. When the order of succession of organic remains among the stratified
+rocks has been determined, they become an invaluable guide in the
+investigation of the relative age of rocks and the structure of the
+land. Each zone and formation, being characterized by its own species or
+genera, may be recognized by their means, and the true succession of
+strata may thus be confidently established even in a country wherein the
+rocks have been shattered by dislocation, folded, inverted or
+metamorphosed.
+
+5. Though local differences exist in regard to the precise zone in which
+a given species of organism may make its first appearance, the general
+order of succession of the organic forms found in the rocks is never
+inverted. The record is nowhere complete in any region, but the portions
+represented, even though extremely imperfect, always follow each other
+in their proper chronological order, unless where disturbance of the
+crust has intervened to destroy the original sequence.
+
+6. The relative chronological value of the divisions of the geological
+record is not to be measured by mere depth of strata. While it may be
+reasonably assumed that, in general, a great thickness of stratified
+rock must mark the passage of a long period of time, it cannot safely be
+affirmed that a much less thickness elsewhere must represent a
+correspondingly diminished period. The need for this caution may
+sometimes be made evident by an unconformability between two sets of
+rocks, as has already been explained. The total depth of both groups
+together may be, say 1000 ft. Elsewhere we may find a single unbroken
+formation reaching a depth of 10,000 ft.; but it would be unwarrantable
+to assume that the latter represents ten times the length of time
+indicated by the former two. So far from this being the case, it might
+not be difficult to show that the minor thickness of rock really denotes
+by far the longer geological interval. If, for instance, it could be
+proved that the upper part of both the sections lies on one and the same
+geological platform, but that the lower unconformable series in the one
+locality belongs to a far lower and older system of rocks than the base
+of the thick conformable series in the other, then it would be clear
+that the gap marked by the unconformability really indicates a longer
+period than the massive succession of deposits.
+
+7. Fossil evidence furnishes the chief means of comparing the relative
+value of formations and groups of rock. A "break in the succession of
+organic remains," as already explained, marks an interval of time often
+unrepresented by strata at the place where the break is found. The
+relative importance of these breaks, and therefore, probably, the
+comparative intervals of time which they mark, may be estimated by the
+difference of the _facies_ or general character of the fossils on each
+side. If, for example, in one case we find every species to be
+dissimilar above and below a certain horizon, while in another locality
+only half of the species on each side are peculiar, we naturally infer,
+if the total number of species seems large enough to warrant the
+inference, that the interval marked by the former break was much longer
+than that marked by the second. But we may go further and compare by
+means of fossil evidence the relation between breaks in the succession
+of organic remains and the depth of strata between them.
+
+  Three formations of fossiliferous strata, A, C, and H, may occur
+  conformably above each other. By a comparison of the fossil contents
+  of all parts of A, it may be ascertained that, while some species are
+  peculiar to its lower, others to its higher portions, yet the majority
+  extend throughout the formation. If now it is found that of the total
+  number of species in the upper portion of A only one-third passes up
+  into C, it may be inferred with some plausibility that the time
+  represented by the break between A and C was really longer than that
+  required for the accumulation of the whole of the formation A. It
+  might even be possible to discover elsewhere a thick intermediate
+  formation B filling up the gap between A and C. In like manner were it
+  to be discovered that, while the whole of the formation C is
+  characterized by a common suite of fossils, not one of the species and
+  only one half of the genera pass up into H, the inference could hardly
+  be resisted that the gap between the two formations marks the passage
+  of a far longer interval than was needed for the deposition of the
+  whole of C. And thus we reach the remarkable conclusion that, thick
+  though the stratified formations of a country may be, in some cases
+  they may not represent so long a total period of time as do the gaps
+  in their succession,--in other words, that non-deposition was more
+  frequent and prolonged than deposition, or that the intervals of time
+  which have been recorded by strata have not been so long as those
+  which have not been so recorded.
+
+In all speculations of this nature, however, it is necessary to reason
+from as wide a basis of observation as possible, seeing that so much of
+the evidence is negative. Especially needful is it to bear in mind that
+the cessation of one or more species at a certain line among the rocks
+of a particular district may mean nothing more than that, onward from
+the time marked by that line, these species, owing to some change in the
+conditions of life, were compelled to migrate or became locally extinct
+or, from some alteration in the conditions of fossilization, were no
+longer imbedded and preserved as fossils. They may have continued to
+flourish abundantly in neighbouring districts for a long period
+afterward. Many examples of this obvious truth might be cited. Thus in a
+great succession of mingled marine, brackish-water and terrestrial
+strata, like that of the Carboniferous Limestone series of Scotland,
+corals, crinoids and brachiopods abound in the limestones and
+accompanying shales, but disappear as the sandstones, ironstones, clays,
+coals and bituminous shales supervene. An observer meeting for the first
+time with an instance of this disappearance, and remembering what he had
+read about breaks in succession, might be tempted to speculate about the
+extinction of these organisms, and their replacement by other and later
+forms of life, such as the ferns, lycopods, estuarine or fresh-water
+shells, ganoid fishes and other fossils so abundant in the overlying
+strata. But further research would show him that high above the
+plant-bearing sandstones and coals other limestones and shales might be
+observed, once more charged with the same marine fossils as before, and
+still farther overlying groups of sandstones, coals and carbonaceous
+beds followed by yet higher marine limestones. He would thus learn that
+the same organisms, after being locally exterminated, returned again and
+again to the same area. After such a lesson he would probably pause
+before too confidently asserting that the highest bed in which we can
+detect certain fossils marks their final appearance in the history of
+life. Some breaks in the succession may thus be extremely local, one set
+of organisms having been driven to a different part of the same region,
+while another set occupied their place until the first was enabled to
+return.
+
+8. The geological record is at the best but an imperfect chronicle of
+the geological history of the earth. It abounds in gaps, some of which
+have been caused by the destruction of strata owing to metamorphism,
+denudation or otherwise, others by original non-deposition, as above
+explained. Nevertheless from this record alone can the progress of the
+earth be traced. It contains the registers of the appearance and
+disappearance of tribes of plants and animals which have from time to
+time flourished on the earth. Only a small proportion of the total
+number of species which have lived in past time have been thus
+chronicled, yet by collecting the broken fragments of the record an
+outline at least of the history of life upon the earth can be
+deciphered.
+
+It cannot be too frequently stated, nor too prominently kept in view,
+that, although gaps occur in the succession of organic remains as
+recorded in the rocks, they do not warrant the conclusion that any such
+blank intervals ever interrupted the progress of plant and animal life
+upon the globe. There is every reason to believe that the march of life
+has been unbroken, onward and upward. Geological history, therefore, if
+its records in the stratified formations were perfect, ought to show a
+blending and gradation of epoch with epoch. But the progress has been
+constantly interrupted, now by upheaval, now by volcanic outbursts, now
+by depression. These interruptions serve as natural divisions in the
+chronicle, and enable the geologist to arrange his history into periods.
+As the order of succession among stratified rocks was first made out in
+Europe, and as many of the gaps in that succession were found to be
+widespread over the European area, the divisions which experience
+established for that portion of the globe came to be regarded as
+typical, and the names adopted for them were applied to the rocks of
+other and far distant regions. This application has brought out the fact
+that some of the most marked breaks in the European series do not exist
+elsewhere, and, on the other hand, that some portions of that series are
+much more complete than the corresponding sections in other regions.
+Hence, while the general similarity of succession may remain, different
+subdivisions and nomenclature are required as we pass from continent to
+continent.
+
+The nomenclature adopted for the subdivisions of the geological record
+bears witness to the rapid growth of geology. It is a patch-work in
+which no system nor language has been adhered to, but where the
+influences by which the progress of the science has been moulded may be
+distinctly traced. Some of the earliest names are lithological, and
+remind us of the fact that mineralogy and petrography preceded geology
+in the order of birth--Chalk, Oolite, Greensand, Millstone Grit. Others
+are topographical, and often recall the labours of the early geologists
+of England--London Clay, Oxford Clay, Purbeck, Portland, Kimmeridge
+beds. Others are taken from local English provincial names, and remind
+us of the debt we owe to William Smith, by whom so many of them were
+first used--Lias, Gault, Crag, Cornbrash. Others of later date recognize
+an order of superposition as already established among formations--Old
+Red Sandstone, New Red Sandstone. By common consent it is admitted that
+names taken from the region where a formation or group of rocks is
+typically developed are best adapted for general use. Cambrian,
+Silurian, Devonian, Permian, Jurassic are of this class, and have been
+adopted all over the globe.
+
+But whatever be the name chosen to designate a particular group of
+strata, it soon comes to be used as a chronological or homotaxial term,
+apart altogether from the stratigraphical character of the strata to
+which it is applied. Thus we speak of the Chalk or Cretaceous system,
+and embrace under that term formations which may contain no chalk; and
+we may describe as Silurian a series of strata utterly unlike in
+lithological characters to the formations in the typical Silurian
+country. In using these terms we unconsciously allow the idea of
+relative date to arise prominently before us. Hence such a word as
+"chalk" or "cretaceous" does not suggest so much to us the group of
+strata so called as the interval of geological history which these
+strata represent. We speak of the Cretaceous, Jurassic, and Cambrian
+periods, and of the Cretaceous fauna, the Jurassic flora, the Cambrian
+trilobites, as if these adjectives denoted simply epochs of geological
+time.
+
+The stratified formations of the earth's crust, or geological record,
+are classified into five main divisions, which in their order of
+antiquity are as follows: (1) Archean or Pre-Cambrian, called also
+sometimes Azoic (lifeless) or Eozoic (dawn of life); (2) Palaeozoic
+(ancient life) or Primary; (3) Mesozoic (middle life) or Secondary; (4)
+Cainozoic (recent life) or Tertiary; (5) Quaternary or Post-Tertiary.
+These divisions are further ranged into systems, formations, groups or
+stages, assises and zones. Accounts of the various subdivisions named
+are given in separate articles under their own headings. In order,
+however, that the sequence of the formations and their parallelism in
+Europe and North America may be presented together a stratigraphical
+table is given on next page.
+
+
+PART VIII.--PHYSIOGRAPHICAL GEOLOGY
+
+This department of geological inquiry investigates the origin and
+history of the present topographical features of the land. As these
+features must obviously be related to those of earlier time which are
+recorded in the rocks of the earth's crust, they cannot be
+satisfactorily studied until at least the main outlines of the history
+of these rocks have been traced. Hence physiographical research comes
+appropriately after the other branches of the science have been
+considered.
+
+From the stratigraphy of the terrestrial crust we learn that by far the
+largest part of the area of dry land is built up of marine formations;
+and therefore that the present land is not an aboriginal portion of the
+earth's surface, but has been overspread by the sea in which its rocks
+were mainly accumulated. We further discover that this submergence of
+the land did not happen once only, but again and again in past ages and
+in all parts of the world. Yet although the terrestrial areas varied
+much from age to age in their extent and in their distribution, being at
+one time more continental, at another more insular, there is reason to
+believe that these successive diminutions and expansions have on the
+whole been effected within, or not far outside, the limits of the
+existing continents. There is no evidence that any portion of the
+present land ever lay under the deeper parts of the ocean. The abysmal
+deposits of the ocean-floor have no true representatives among the
+sedimentary formations anywhere visible on the land. Nor, on the other
+hand, can it be shown that any part of the existing ocean abysses ever
+rose above sea-level into dry land. Hence geologists have drawn the
+inference that the ocean basins have probably been always where they now
+are; and that although the continental areas have often been narrowed by
+submergence and by denudation, there has probably seldom or never been a
+complete disappearance of land. The fact that the sedimentary formations
+of each successive geological period consist to so large an extent of
+mechanically formed terrigenous detritus, affords good evidence of the
+coexistence of tracts of land as well as of extensive denudation.
+
+
+  _The Geological Record or Order of Succession of the Stratified
+  Formations of the Earth's Crust._
+
+  +---+---+-------------------------------------------+----------------------------------+
+  |   |   |                  Europe.                  |          North America.          |
+  +---+---+-------------------------------------------+----------------------------------+
+  | Q |    \      Historic, up to the present time.   | Similar to the European          |
+  | u |     \     Prehistoric, comprising deposits of |   development, but with scantier |
+  | a |      \     the Iron, Bronze, and later        |   traces of the presence of man. |
+  | t |       \     Stone Ages.                       |                                  |
+  | e |        \    Neolithic--alluvium, peat, lake-  |                                  |
+  | r | Recent, \   dwellings, loess, &c.             |                                  |
+  | n | Post-   | Palaeolithic--river-gravels, cave-  |                                  |
+  | a | glacial |   deposits, &c.                     |                                  |
+  | r | or      |                                     |                                  |
+  | y | Human.  |                                     |                                  |
+  |   |         |                                     |                                  |
+  | o |         |                                     |                                  |
+  | r +---------+-------------------------------------+----------------------------------+
+  |   | Pleist- | Older Loess and valley-gravels;     | As in Europe, it is hardly       |
+  | P | ocene   |   cave-deposits.                    |   possible to assign a definite  |
+  | o | or      | Strand-lines or raised beaches;     |   chronological place to each of |
+  | s | Glacial.|   youngest moraines.                |   the various deposits of this   |
+  | t |         | Upper Boulder-clays; eskers; marine |   period, terrestrial and marine.|
+  | | |         |   sands and clays.                  |   They generally resemble the    |
+  | T |         | Interglacial deposits.              |   European series. The           |
+  | e |         | Lower boulder-clay or Till, with    |   characteristic marine,         |
+  | r |         |   striated rock-surfaces below.     |   fluviatile and lacustrine      |
+  | t |         |                                     |   terraces, which overlie the    |
+  | i |        /                                      |   older drifts, have been        |
+  | a |       /                                       |   classed as the Champlain Group.|
+  | r |      /                                        |                                  |
+  | y |     /                                         |                                  |
+  | . |    /                                          |                                  |
+  +---+---+-------------------------------------------+----------------------------------+
+  |   | P | Newer:--English Forest-Bed Group; Red and | On the Atlantic border           |
+  |   | l |   Norwich Crag; Amstelian and Scaldesian  |   represented by the marine      |
+  |   | i |   groups of Belgium and Holland; Sicilian |   Floridian series; in the       |
+  |   | o |   and Astian of France and Italy.         |   interior by a subaerial and    |
+  |   | c | Older:--English Coralline Crag; Diestian  |   lacustrine series; and on the  |
+  |   | e |   of Belgium; Plaisancian of southern     |   Pacific border by the thick    |
+  |   | n |   France and Italy.                       |   marine series of San Francisco.|
+  |   | e |                                           |                                  |
+  |   | . |                                           |                                  |
+  |   +---+-------------------------------------------+----------------------------------+
+  |   | M | Wanting in Britain; well developed in     | Represented in the Eastern States|
+  |   | i |   France, S. E. Europe and Italy;         |   by a marine series (Yorktown or|
+  | C | o |   divisible into the following groups in  |   Chesapeake, Chipola and        |
+  | a | c |   descending order: (1) Pontian; (2)      |   Chattahoochee groups), and in  |
+  | i | e |   Sarmatian; (3) Tortonian; (4) Helvetian;|   the interior by the lacustrine |
+  | n | n |   (5) Langhian (Burdigalian).             |   Loup Fork (Nebraska), Deep     |
+  | o | e |                                           |   River, and John Day groups.    |
+  | z | . |                                           |                                  |
+  | o +---+-------------------------------------------+----------------------------------+
+  | i |   | In Britain the "fluvio-marine series" of  | On the Atlantic border no        |
+  | c | O |   the Isle of Wight; also the volcanic    |   equivalents have been          |
+  |   | l |   plateaux of Antrim and Inner Hebrides   |   satisfactorily recognised, but |
+  | o | i |   and those of the Faeroe Isles and       |   on the Pacific side there are  |
+  | r | g |   Iceland. In continental Europe the      |   marine deposits in N. W.       |
+  |   | o |   following subdivisions have been        |   Oregon, which may represent    |
+  | T | c |   established in descending order: (1)    |   this division. In the interior |
+  | e | e |   Aquitanian, (2) Stampian (Rupelian),    |   the equivalent is believed to  |
+  | r | n |   (3) Tongrain (Sannoisian).              |   be the fresh-water White River |
+  | t | e |                                           |   series, including (1)          |
+  | i | . |                                           |   _Protoceras_ beds, (2)         |
+  | a |   |                                           |   _Oreodon_ beds, and (3)        |
+  | r |   |                                           |   _Titanothervum_ beds.          |
+  | y +---+-------------------------------------------+----------------------------------+
+  | . |   | Barton sands and clays; Ludian series of  | Woodstock and Aquia Creek groups |
+  |   |   |   France.                                 |   of Potomac River; Vicksburg,   |
+  |   |   | Bracklesham Beds; Lutetian (Calcaire      |   Jackson, Claiborne, Buhrstone, |
+  |   | E |   grossier and Caillasses) of Paris       |   and Lignitic groups of         |
+  |   | o |   basin.                                  |   Mississippi.                   |
+  |   | c | London clay, Woolwich and Reading Beds;   | In the interior a thick series of|
+  |   | e |   Thanet sands; Ypresian or Londinian of  |   fresh-water formations,        |
+  |   | n |   N. France and Belgium; Sparnacian and   |   comprising, in descending      |
+  |   | e |   Thanetian groups.                       |   order, the Uinta, Bridger,     |
+  |   | . |                                           |   Wind River, Wasatch, Torrejon, |
+  |   |   |                                           |   and Puerco groups.             |
+  |   |   |                                           | On the Pacific side the marine   |
+  |   |   |                                           |   Tejon series of Oregon and     |
+  |   |   |                                           |   California.                    |
+  |---+---+-------------------------------------------+----------------------------------|
+  |   |   |     Upper                                 | On the Atlantic border both      |
+  |   |   |     =====                                 |   marine strata and others       |
+  |   |   | Danian--wanting in Britain; uppermost     |   containing a terrestrial flora |
+  |   |   |   limestone of Denmark.                   |   represent the Cretaceous series|
+  |   |   | Senonian--Upper Chalk with Flints of      |   of formations.                 |
+  |   |   |   England; Aturian and Emscherian stages  | In the interior there is also a  |
+  |   |   |   on the European continent.              |   commingling of marine with     |
+  |   |   | Turonian--Middle Chalk with few flints,   |   lacustrine deposits. At the top|
+  |   |   |   and comprising the Angoumian and stages.|   lies the Laramie or Lignitic   |
+  |   | C | Cenomanian--Lower Chalk and Chalk Marl.   |   series with an abundant        |
+  |   | r |                                           |   terrestrial flora, passing down|
+  |   | e |     Lower                                 |   into the lacustrine and        |
+  |   | t |     =====                                 |   brackish-water Montana series. |
+  |   | a | Albian--Upper Greensand and Gault.        |   Of older date, the Colorado    |
+  |   | c | Aptian--Lower Greensand; Marls and        |   series contains an abundant    |
+  |   | e |   limestones of Provence, &c.             |   marine fauna, yet includes also|
+  |   | o | Urgonian (Barremian)--Atherfield clay;    |   some Niobrara marls and        |
+  |   | u |   massive Hippurite limestones of         |   limestones are likewise of     |
+  |   | s |   southern France.                        |   marine origin, but the lower   |
+  |   | . | Neocomian--Weald clay and Hastings sand;  |   members of the series (Benton  |
+  |   |   |   Hauterivian and Valanginian sub-stages  |   and Dakota) show another great |
+  |   |   |   of Switzerland and France.              |   representation of fresh-water  |
+  | M |   |                                           |   sedimentation with lignites and|
+  | e |   |                                           |   coals.                         |
+  | s |   |                                           | In California a vast succession  |
+  | o |   |                                           |   of marine deposits (Shasta-    |
+  | z |   |                                           |   Chico) represents the          |
+  | o |   |                                           |   Cretaceous system; and in      |
+  | i |   |                                           |   western British N. America     |
+  | c |   |                                           |   coal-seams also occur.         |
+  |   +---+-------------------------------------------+----------------------------------+
+  | o |   | Purbeckian--Purbeck beds; Münder Mergel;  | Representatives of the Middle and|
+  | r |   |   largely present in Westphalia.          |   lower Jurassic formations have |
+  |   |   | Portlandian--Portland group of England,   |   been found in California and   |
+  | S |   |   represented in S. France by the thick   |   Oregon, and farther north among|
+  | e |   |   Tithonian limestones.                   |   the Arctic islands.            |
+  | c |   | Kimmeridgian--Kimmeridge Clay of England; | Strata containing Lower Jurassic |
+  | o |   |   Virgulian and Pterocerian groups of N.  |   marine fossils appear in       |
+  | n | J |   France; represented by thick limestones |   Wyoming and Dakota; and above  |
+  | d | u |   in the Mediterranean basin.             |   them come the _Atlantosaurus_  |
+  | a | r | Corallian--Coral Rag, Coralline Oolite;   |   and _Baptanodon_ beds, which   |
+  | r | a |   Sequanian stages of the Continent,      |   have yielded so large a        |
+  | y | s |   comprising the sub-stages of Astartian  |   variety of deinosaurs and other|
+  | . | s |   and Rauracian.                          |   vertebrates, and especially the|
+  |   | i | Oxfordian--Oxford Clay; Axgovian and      |   remains of a number of genera  |
+  |   | c |   Neuvizyan stages.                       |   of small mammals.              |
+  |   | . | Callovian--Kellaways Rock, Divesian       |                                  |
+  |   |   |   sub-stage of N. France.                 |                                  |
+  |   |   | Bathonian--series of English strata from  |                                  |
+  |   |   |   Cornbrash down to Fuller's Earth.       |                                  |
+  |   |   | Bajocian--Inferior Oolite of England.     |                                  |
+  |   |   | Lassic--divisible into (1) Upper Lias     |                                  |
+  |   |   |   or Toarcian, (2) Middle Lias, Marlstone |                                  |
+  |   |   |   or Charmouthian, (3) Lower Lias of      |                                  |
+  |   |   |   Sinemurian and Hettangian.              |                                  |
+  |   +---+-------------------------------------------+----------------------------------+
+  |   |   | In Germany and western Europe this        | In New York, Connecticut, New    |
+  |   | T |   division represents the deposits of     |   Brunswick, and Nova Scotia     |
+  |   | r |   inland seas or lagoons, and is divisible|   a series of red sandstone      |
+  |   | i |   into the following stages in descending |   (Newark series) contains land- |
+  |   | a |   order: (1) Rhaetic, (2) Keuper, (3)     |   plants and labyrinthodonts     |
+  |   | s |   Muschelkalk, (4) Bunter.  In the        |   like the lagoon type of central|
+  |   | s |   eastern Alps and the Mediterranean      |   and western Europe. On the     |
+  |   | i |   basin the contemporaneous sedimentary   |   Pacific slope, however, marine |
+  |   | c |   formations are those of open clear      |   equivalents occur, representing|
+  |   | . |   sea, in which a thickness of many       |   the pelagic type of south-     |
+  |   |   |   thousand feet of strata was accumulated.|   eastern Europe.                |
+  +---+---+-------------------------------------------+----------------------------------+
+  |   | P | Thuringian--Zechstein, Magnesian          | To this division of the geologi- |
+  |   | e |   Limestone; named from its development   |   cal record the Upper Barren    |
+  |   | r |   in Thuringia; well represented          |   Measures of the coal-fields of |
+  |   | m |   also in Saxony, Bavaria and Bohemia.    |   Pennsylvania, Prince Edward    |
+  |   | i | Saxonian--Rothliegendes Group; Red        |   Island, Nova Scotia and        |
+  |   | a |   Sandstones, &c.                         |   New Brunswick have been        |
+  |   | n | Autunian--where the strata present the    |   assigned.                      |
+  |   | . |   lagoon facies, well displayed at Autun  | Farther south in Kansas, Texas,  |
+  |   |   |   in France; where the marine type is     |   and Nebraska the representa-   |
+  |   |   |   predominant, as in Russia, the group    |   tives of the division have an  |
+  |   |   |   has been termed Artinskian.             |   abundant marine fauna.         |
+  |   +---+-------------------------------------------+----------------------------------+
+  |   | C | Stephanian or Uralian--represented in     | Upper productive Coal-measures.  |
+  |   | a |   Russia by marine formations, and in     | Lower Barren measures.           |
+  |   | r |   central and western Europe by numerous  | Lower productive Coal-measures.  |
+  |   | b |   small basins containing a peculiar      | Pottsville conglomerate.         |
+  |   | o |   flora and in some places a great variety| Mauch Chunk shales; limestones   |
+  |   | n |   of insects.                             |   of Chester, St Louis, &c.      |
+  |   | i | Westphalian or Moscovian--Coal-measures,  | Pocono series; Kinderhook        |
+  |   | f |   Millstone Grit.                         |   limestone.                     |
+  |   | e | Culm or Dinantian--Carboniferous Limestone|                                  |
+  |   | r |   and Calciferous Sandstone series.       |                                  |
+  |   | o |                                           |                                  |
+  |   | u |                                           |                                  |
+  |   | s |                                           |                                  |
+  |   | . |                                           |                                  |
+  |   +---+-------------------------------------------+----------------------------------+
+  |   |                            Devonian and Old Red Sandstone.                       |
+  | P +----------------------+------------------------+----------------------------------+
+  | a |     Devonian type.   |  Old Red Sandstone     |                                  |
+  | l |                      |       type.            |                                  |
+  | a +----------------------+------------------------+ / Catskill red sandstone; Old    |
+  | e |         / Famennian. | Yellow and red         | |   Red Sandstone type: the      |
+  | o | Upper  <             |   sandstone with       |<    strata below show the        |
+  | z |         \ Frasnian.  |   _Holoptychius_,      | |   Devonian type.               |
+  | o |                      |   _Bothriolepis_,&c.   | | Chemung Group.                 |
+  | i |                      |                        | \ Genesee   "                    |
+  | c |                      |                        |                                  |
+  |   |         / Givetian.  | Caithness Flagstones   |                                  |
+  | o | Middle <             |   with _Osteolepus_,   | / Hamilton Group.                |
+  | r |         \ Eifelian.  |   _Dipterus_,          | \ Marcellus  "                   |
+  |   |                      |   _Homosteus_, &c.     |                                  |
+  | P |                      |                        |                                  |
+  | r |                      | Red and purple         | / Corniferous Limestone. / Upper |
+  | i |        /Coblentizian.|   sandstones and       | |                        | Held- |
+  | m | Lower <              |   conglomerates with   |<  Onondaga Limestone.   <  erberg|
+  | a |        \Gedinnian.   |   _Cephalaspis_,       | |                        \ Group.|
+  | r |                      |   _Pteraspis_, &c.     | \ Oriskany Sandstone.            |
+  | y +---+------------------+------------------------+----------------------------------+
+  | . |   |                                           |  / Lower Helderberg Group.       |
+  |   | S |               / Ludlow  Group.            |  | Water-Lime.                   |
+  |   | i |   Upper      <  Wenlock   "               | <  Niagara Shale and Limestone.  |
+  |   | l |               \ Llandovery"               |  | Clinton Group.                |
+  |   | u |                                           |  \ Medina    "                   |
+  |   | r |                                           |                                  |
+  |   | i |                                           |  / Cincinnati Group.             |
+  |   | a |   Lower       / Caradoc or Bala Group.    |  | Utica        "                |
+  |   | n | (Ordovician) <  Llandeilo         "       | <  Trenton      "                |
+  |   | . |               \ Arenig            "       |  | Chazy        "                |
+  |   |   |                                           |  \ Calciferous  "                |
+  |   +---+-------------------------------------------+----------------------------------+
+  |   | C | Upper or _Olenus_ series--Tremadoc        | Upper or Potsdam series with     |
+  |   | a |   slates and _Lingula_ Flags.             |   _Olenus_ and _Dicelocephalus_  |
+  |   | m | Middle or _Pardoxides_ series--Menevian   |   fauna.                         |
+  |   | b |   Group.                                  | Middle or Acadian series with    |
+  |   | r | Lower or _Olenellus_ series--Llanberis    |   _Paradoxides_ fauna.           |
+  |   | i |   and Harlech Group, and _Olenellus_-     | Lower or Georgian series with    |
+  |   | a |   zone.                                   |   _Olenellus_ fauna.             |
+  |   | n |                                           |                                  |
+  |   | . |                                           |                                  |
+  +---+---+-------------------------------------------+----------------------------------+
+  |   |   |                       Archean, Pre-Cambrian, Eozoic.                         |
+  +---+---+-------------------------------------------+----------------------------------+
+  |   |   | In Scotland, underneath the Cambrian      | In Canada and the Lake Superior  |
+  |   |   |   Olenellus group, lies unconformably     |   region of the United States    |
+  |   |   |   a mass of red sandstone and con-        |   a vast succession of rocks of  |
+  |   |   |   glomerate (Torridonian) 8000 or 10,000  |   Pre-Cambrian age has been      |
+  |   |   |   ft. thick, which rests with a strong    |   grouped into the following     |
+  |   |   |   gneisses and schists (Lewisian).  A     |   subdivisions in descending     |
+  |   |   |   thick series of slates and phyllites    |   order: (1) Keweenwan, lying    |
+  |   |   |   lies below the oldest Palaeozoic rocks  |   unconformably on (2) Animikie, |
+  |   |   |   in central Europe, with coarse          |   separated by a strong          |
+  |   |   |   gneisses below.                         |   unconformability from (3) Upper|
+  |   |   |                                           |   Huronian, (4) Lower Huronian   |
+  |   |   |                                           |   with an unconformable base, (5)|
+  |   |   |                                           |   Goutchiching, (6) Laurentian.  |
+  |   |   |                                           |   In the eastern part of Canada, |
+  |   |   |                                           |   Newfoundland, &c., and also in |
+  |   |   |                                           |   Montana, sedimentary formations|
+  |   |   |                                           |   of great thickness below the   |
+  |   |   |                                           |   lowest Cambrian zone have been |
+  |   |   |                                           |   found to contain some obscure  |
+  |   |   |                                           |   organisms.                     |
+  +---+---+-------------------------------------------+----------------------------------+
+
+From these general considerations we proceed to inquire how the existing
+topographical features of the land arose. Obviously the co-operation of
+the two great geological agencies of hypogene and epigene energy, which
+have been at work from the beginning of our globe's decipherable
+history, must have been the cause to which these features are to be
+assigned; and the task of the geologist is to ascertain, if possible,
+the part that has been taken by each. There is a natural tendency to see
+in a stupendous piece of scenery, such as a deep ravine, a range of
+hills, a line of precipice or a chain of mountains, evidence only of
+subterranean convulsion; and before the subject was taken up as a matter
+of strict scientific induction, an appeal to former cataclysms was
+considered a sufficient solution of the problems presented by such
+features of landscape. The rise of the modern Huttonian school, however,
+led to a more careful examination of these problems. The important share
+taken by erosion in the determination of the present features of
+landscape was then recognized, while a fuller appreciation of the
+relative parts played by the hypogene and epigene causes has gradually
+been reached.
+
+1. The study of the progress of denudation at the present time has led
+to the conclusion that even if the rate of waste were not more rapid
+than it is to-day, it would yet suffice in a comparatively brief
+geological period to reduce the dry land to below the sea-level. But not
+only would the area of the land be diminished by denudation, it could
+hardly fail to be more or less involved in those widespread movements of
+subsidence, during which the thick sedimentary formations of the crust
+appear to have been accumulated. It is thus manifest that there must
+have been from time to time during the history of our globe upward
+movements of the crust, whereby the balance between land and sea was
+redressed. Proofs of such movements have been abundantly preserved among
+the stratified formations. We there learn that the uplifts have usually
+followed each other at long intervals between which subsidence
+prevailed, and thus that there has been a prolonged oscillation of the
+crust over the great continental areas of the earth's surface.
+
+An examination of that surface leads to the recognition of two great
+types of upheaval. In the one, the sea-floor, with all its thick
+accumulations of sediment, has been carried upwards, sometimes for
+several thousand feet, so equably that the strata retain their original
+flatness with hardly any sensible disturbance for hundreds of square
+miles. In the other type the solid crust has been plicated, corrugated
+and dislocated, especially along particular lines, and has attained its
+most stupendous disruption in lofty chains of mountains. Between these
+two phases of uplift many intermediate stages have been developed,
+according to the direction and intensity of the subterranean force and
+the varying nature and disposition of the rocks Of the crust.
+
+(a) Where the uplift has extended over wide spaces, without appreciable
+deformation of the crust, the flat strata have given rise to low plains,
+or if the amount of uprise has been great enough, to high plains,
+plateaux or tablelands. The plains of Russia, for example, lie for the
+most part on such tracts of equably uplifted strata. The great plains of
+the western interior of the United States form a great plateau or
+tableland, 5000 or 6000 ft. above the sea, and many thousands of square
+miles in extent, on which the Rocky Mountains have been ridged up.
+
+(b) It is in a great mountain-chain that the complicated structures
+developed during disturbances of the earth's crust can best be studied
+(see Parts IV. and V. of this article), and where the influence of these
+structures on the topography of the surface is most effectively
+displayed. Such a chain may be the result of one colossal disturbance;
+but those of high geological antiquity usually furnish proofs of
+successive uplifts with more or less intervening denudation. Formed
+along lines of continental displacement in the crust, they have again
+and again given relief from the strain of compression by fresh
+crumpling, fracture and uprise. The chief guide in tracing these
+successive stages of growth is supplied by unconformability. If, for
+example, a mountain-range consists of upraised Silurian rocks, upon the
+upturned and denuded edges of which the Carboniferous Limestone lies
+transgressively, it is clear that its original upheaval must have taken
+place in the period of geological time represented by the interval
+between the Silurian and the Carboniferous Limestone formations. If, as
+the range is followed along its course, the Carboniferous Limestone is
+found to be also highly inclined and covered unconformably by the Upper
+Coal-measures, a second uplift of that portion of the ground can be
+proved to have taken place between the time of the Limestone and that of
+the Upper Coal-measures. By this simple and obvious kind of evidence the
+relative ages of different mountain-chains may be compared. In most
+great chains, however, the rocks have been so intensely crumpled, and
+even inverted, that much labour may be required before their true
+relations can be determined.
+
+The Alps furnish an instructive example of the long series of
+revolutions through which a great mountain-system may have passed before
+reaching its present development. The first beginnings of the chain may
+have been upraised before the oldest Palaeozoic formations were laid
+down. There are at least traces of land and shore-lines in the
+Carboniferous period. Subsequent submergences and uplifts appear to have
+occurred during the Mesozoic periods. There is evidence that thereafter
+the whole region sank deep under the sea, in which the older Tertiary
+sediments were accumulated, and which seems to have spread right across
+the heart of the Old World. But after the deposition of the Eocene
+formations came the gigantic disruptions whereby all the rocks of the
+Alpine region were folded over each other, crushed, corrugated,
+fractured and displaced, some of their older portions, including the
+fundamental gneisses and schists, being squeezed up, torn off, and
+pushed horizontally for many miles over the younger rocks. But this
+upheaval, though the most momentous, was not the last which the chain
+has undergone, for at a later epoch in Tertiary time renewed disturbance
+gave rise to a further series of ruptures and plications. The chain thus
+successively upheaved has been continuously exposed to denudation and
+has consequently lost much of its original height. That it has been left
+in a state of instability is indicated by the frequent earthquakes of
+the Alpine region, which doubtless arise from the sudden snapping of
+rocks under intense strain.
+
+A distinct type of mountain due to direct hypogene action is to be seen
+in a volcano. It has been already pointed out (Part IV. sect. 1) that at
+the vents which maintain a communication between the molten magma of the
+earth's interior and the surface, eruptions take place whereby
+quantities of lava and fragmentary materials are heaped round each
+orifice of discharge. A typical volcanic mountain takes the form of a
+perfect cone, but as it grows in size and its main vent is choked, while
+the sides of the cone are unable to withstand the force of the
+explosions or the pressure of the ascending column of lava, eruptions
+take place laterally, and numerous parasitic cones arise on the flanks
+of the parent mountain. Where lava flows out from long fissures, it may
+pile up vast sheets of rock, and bury the surrounding country under
+several thousand feet of solid stone, covering many hundreds of square
+miles. In this way volcanic tablelands have been formed which, attacked
+by the denuding forces, are gradually trenched by valleys and ravines,
+until the original level surface of the lava-field may be almost or
+wholly lost. As striking examples of this physiographical type reference
+may be made to the plateau of Abyssinia, the Ghats of India, the
+plateaux of Antrim, the Inner Hebrides and Iceland, and the great
+lava-plains of the western territories of the United States.
+
+2. But while the subterranean movements have upraised portions of the
+surface of the lithosphere above the level of the ocean, and have thus
+been instrumental in producing the existing tracts of land, the detailed
+topographical features of a landscape are not solely, nor in general
+even chiefly, attributable to these movements. From the time that any
+portion of the sea-floor appears above sea-level, it undergoes erosion
+by the various epigene agents. Each climate and geological region has
+its own development of these agents, which include air, aridity, rapid
+and frequent alternations of wetness and dryness or of heat and cold,
+rain, springs, frosts, rivers, glaciers, the sea, plant and animal life.
+In a dry climate subject to great extremes of temperature the character
+and rate of decay will differ from those of a moist or an arctic
+climate. But it must be remembered that, however much they may vary in
+activity and in the results which they effect, the epigene forces work
+without intermission, while the hypogene forces bring about the upheaval
+of land only after long intervals. Hence, trifling as the results during
+a human life may appear, if we realize the multiplying influence of time
+we are led to perceive that the apparently feeble superficial agents
+can, in the course of ages, achieve stupendous transformations in the
+aspect of the land. If this efficacy may be deduced from what can be
+seen to be in progress now, it may not less convincingly be shown, from
+the nature of the sedimentary rocks of the earth's crust, to have been
+in progress from the early beginnings of geological history. Side by
+side with the various upheavals and subsidences, there has been a
+continuous removal of materials from the land, and an equally persistent
+deposit of these materials under water, with the consequent growth of
+new rocks. Denudation has been aptly compared to a process of
+sculpturing wherein, while each of the implements employed by nature,
+like a special kind of graving tool, produces its own characteristic
+impress on the land, they all combine harmoniously towards the
+achievement of their one common task. Hence the present contours of the
+land depend partly on the original configuration of the ground, and the
+influence it may have had in guiding the operations of the erosive
+agents, partly on the vigour with which these agents perform their work,
+and partly on the varying structure and powers of resistance possessed
+by the rocks on which the erosion is carried on.
+
+Where a new tract of land has been raised out of the sea by such an
+energetic movement as broke up the crust and produced the complicated
+structure and tumultuous external forms of a great mountain chain, the
+influence of the hypogene forces on the topography attains its highest
+development. But even the youngest existing chain has suffered so
+greatly from denudation that the aspect which it presented at the time
+of its uplift can only be dimly perceived. No more striking illustration
+of this feature can be found than that supplied by the Alps, nor one
+where the geotectonic structures have been so fully studied in detail.
+On the outer flanks of these mountains the longitudinal ridges and
+valleys of the Jura correspond with lines of anticline and syncline. Yet
+though the dominant topographical elements of the region have obviously
+been produced by the plication of the stratified formations, each ridge
+has suffered so large an amount of erosion that the younger rocks have
+been removed from its crest where the older members of the series are
+now exposed to view, while on every slope proofs may be seen of
+extensive denudation. If from these long wave-like undulations of the
+ground, where the relations between the disposition of the rocks below
+and the forms of the surface are so clearly traceable, the observer
+proceeds inwards to the main chain, he finds that the plications and
+displacements of the various formations assume an increasingly
+complicated character; and that although proofs of great denudation
+continue to abound, it becomes increasingly difficult to form any
+satisfactory conjecture as to the shape of the ground when the upheaval
+ended or any reliable estimate of the amount of material which has since
+then been removed. Along the central heights the mountains lift
+themselves towards the sky like the storm-swept crests of vast
+earth-billows. The whole aspect of the ground suggests intense
+commotion, and the impression thus given is often much intensified by
+the twisted and crumpled strata, visible from a long distance, on the
+crags and crests. On this broken-up surface the various agents of
+denudation have been ceaselessly engaged since it emerged from the sea.
+They have excavated valleys, sometimes along depressions provided for
+them by the subterranean disturbances, sometimes down the slopes of the
+disrupted blocks of ground. So powerful has been this erosion that
+valleys cut out along lines of anticline, which were natural ridges,
+have sometimes become more important than those in lines of syncline,
+which were structurally depressions. The same subaerial forces have
+eroded lake-basins, dug out corries or cirques, notched the ridges,
+splintered the crests and furrowed the slopes, leaving no part of the
+original surface of the uplifted chain unmodified.
+
+It has often been noted with surprise that features of underground
+structure which, it might have been confidently anticipated, should have
+exercised a marked influence on the topography of the surface have not
+been able to resist the levelling action of the denuding agents, and do
+not now affect the surface at all. This result is conspicuously seen in
+coal-fields where the strata are abundantly traversed by faults. These
+dislocations, having sometimes a displacement of several hundred feet,
+might have been expected to break up the surface into a network of
+cliffs and plains; yet in general they do not modify the level character
+of the ground above. One of the most remarkable faults in Europe is the
+great thrust which bounds the southern edge of the Belgian coal-field
+and brings the Devonian rocks above the Coal-measures. It can be traced
+across Belgium into the Boulonnais, and may not improbably run beneath
+the Secondary and Tertiary rocks of the south of England. It is crossed
+by the valleys of the Meuse and other northerly-flowing streams. Yet so
+indistinctly is it marked in the Meuse valley that no one would suspect
+its existence from any peculiarity in the general form of the ground,
+and even an experienced geologist, until he had learned the structure of
+the district, would scarcely detect any fault at all.
+
+Where faults have influenced the superficial topography, it is usually
+by giving rise to a hollow along which the subaerial agents and
+especially running water can act effectively. Such a hollow may be
+eventually widened and deepened into a valley. On bare crags and crests,
+lines of fault are apt to be marked by notches or clefts, and they thus
+help to produce the pinnacles and serrated outlines of these exposed
+uplands.
+
+It was cogently enforced by Hutton and Playfair, and independently by
+Lamarck, that no co-operation of underground agency is needed to produce
+such topography as may be seen in a great part of the world, but that if
+a tract of sea-floor were upraised into a wide plain, the fall of rain
+and the circulation of water over its surface would in the end carve out
+such a system of hills and valleys as may be seen on the dry land now.
+No such plain would be a dead-level. It would have inequalities on its
+surface which would serve as channels to guide the drainage from the
+first showers of rain. And these channels would be slowly widened and
+deepened until they would become ravines and valleys, while the ground
+between them would be left projecting as ridges and hills. Nor would the
+erosion of such a system of water-courses require a long series of
+geological periods for its accomplishment. From measurements and
+estimates of the amount of erosion now taking place in the basin of the
+Mississippi river it has been computed that valleys 800 ft. deep might
+be carved out in less than a million years. In the vast tablelands of
+Colorado and other western regions of the United States an impressive
+picture is presented of the results of mere subaerial erosion on
+undisturbed and nearly level strata. Systems of stream-courses and
+valleys, river gorges unexampled elsewhere in the world for depth and
+length, vast winding lines of escarpment, like ranges of sea-cliffs,
+terraced slopes rising from plateau to plateau, huge buttresses and
+solitary stacks standing like islands out of the plains, great
+mountain-masses towering into picturesque peaks and pinnacles cleft by
+innumerable gullies, yet everywhere marked by the parallel bars of the
+horizontal strata out of which they have been carved--these are the
+orderly symmetrical characteristics of a country where the scenery is
+due entirely to the action of subaerial agents on the one hand and the
+varying resistance of perfectly regular stratified rocks on the other.
+
+The details of the sculpture of the land have mainly depended on the
+nature of the materials on which nature's erosive tools have been
+employed. The joints by which all rocks are traversed have been
+especially serviceable as dominant lines down which the rain has
+filtered, up which the springs have risen and into which the frost
+wedges have been driven. On the high bare scarps of a lofty mountain the
+inner structure of the mass is laid open, and there the system of joints
+even more than faults is seen to have determined the lines of crest, the
+vertical walls of cliff and precipice, the forms of buttress and recess,
+the position of cleft and chasm, the outline of spire and pinnacle. On
+the lower slopes, even under the tapestry of verdure which nature
+delights to hang where she can over her naked rocks, we may detect the
+same pervading influence of the joints upon the forms assumed by ravines
+and crags. Each kind of stone, too, gives rise to its own characteristic
+form of scenery. Massive crystalline rocks, such as granite, break up
+along their joints and often decay into sand or earth along their
+exposed surfaces, giving rise to rugged crags with long talus slopes at
+their base. The stratified rocks besides splitting at their joints are
+especially distinguished by parallel ledges, cornices and recesses,
+produced by the irregular decay of their component strata, so that they
+often assume curiously architectural types of scenery. But besides this
+family feature they display many minor varieties of aspect according to
+their lithological composition. A range of sandstone hills, for example,
+presents a marked contrast to one of limestone, and a line of chalk
+downs to the escarpments formed by alternating bands of harder and
+softer clays and shales.
+
+It may suffice here merely to allude to a few of the more important
+parts of the topography of the land in their relation to physiographical
+geology. A true mountain-chain, viewed from the geological side, is a
+mass of high ground which owes its prominence to a ridging-up of the
+earth's crust, and the intense plication and rupture of the rocks of
+which it is composed. But ranges of hills almost mountainous in their
+bulk may be formed by the gradual erosion of valleys out of a mass of
+original high ground, such as a high plateau or tableland. Eminences
+which have been isolated by denudation from the main mass of the
+formations of which they originally formed part are known as "outliers"
+or "hills of circumdenudation."
+
+Tablelands, as already pointed out, may be produced either by the
+upheaval of tracts of horizontal strata from the sea-floor into land; or
+by the uprise of plains of denudation, where rocks of various
+composition, structure and age have been levelled down to near or below
+the level of the sea by the co-operation of the various erosive agents.
+Most of the great tablelands of the globe are platforms of
+little-disturbed strata which have been upraised bodily to a
+considerable elevation. No sooner, however, are they placed in that
+position than they are attacked by running water, and begin to be
+hollowed out into systems of valleys. As the valleys sink, the platforms
+between them grow into narrower and more definite ridges, until
+eventually the level tableland is converted into a complicated network
+of hills and valleys, wherein, nevertheless, the key to the whole
+arrangement is furnished by a knowledge of the disposition and effects
+of the flow of water. The examples of this process brought to light in
+Colorado, Wyoming, Nevada and the other western regions by Newberry,
+King, Hayden, Powell and other explorers, are among the most striking
+monuments of geological operations in the world.
+
+Examples of ancient and much decayed tablelands formed by the denudation
+of much disturbed rocks are furnished by the Highlands of Scotland and
+of Norway. Each of these tracts of high ground consists of some of the
+oldest and most dislocated formations of Europe, which at a remote
+period were worn down into a plain, and in that condition may have lain
+long submerged under the sea and may possibly have been overspread there
+with younger formations. Having at a much later time been raised several
+thousand feet above sea-level the ancient platforms of Britain and
+Scandinavia have been since exposed to denudation, whereby each of them
+has been so deeply channeled into glens and fjords that it presents
+to-day a surface of rugged hills, either isolated or connected along the
+flanks, while only fragments of the general surface of the tableland can
+here and there be recognized amidst the general destruction.
+
+Valleys have in general been hollowed out by the greater erosive action
+of running water along the channels of drainage. Their direction has
+been probably determined in the great majority of cases by
+irregularities of the surface along which the drainage flowed on the
+first emergence of the land. Sometimes these irregularities have been
+produced by folds of the terrestrial crust, sometimes by faults,
+sometimes by the irregularities on the surface of an uplifted platform
+of deposition or of denudation. Two dominant trends may be observed
+among them. Some are longitudinal and run along the line of flexures in
+the upraised tract of land, others are transverse where the drainage has
+flowed down the slopes of the ridges into the longitudinal valleys or
+into the sea. The forms of valleys have been governed partly by the
+structure and composition of the rocks, and partly by the relative
+potency of the different denuding agents. Where the influence of rain
+and frost has been slight, and the streams, supplied from distant
+sources, have had sufficient declivity, deep, narrow, precipitous
+ravines or gorges have been excavated. The canyons of the arid region of
+the Colorado are a magnificent example of this result. Where, on the
+other hand, ordinary atmospheric action has been more rapid, the sides
+of the river channels have been attacked, and open sloping glens and
+valleys have been hollowed out. A gorge or defile is usually due to the
+action of a waterfall, which, beginning with some abrupt declivity or
+precipice in the course of the river when it first commenced to flow, or
+caused by some hard rock crossing the channel, has eaten its way
+backward.
+
+Lakes have been already referred to, and their modes of origin have been
+mentioned. As they are continually being filled up with the detritus
+washed into them from the surrounding regions they cannot be of any
+great geological antiquity, unless where by some unknown process their
+basins are from time to time widened and deepened.
+
+In the general subaerial denudation of a country, innumerable minor
+features are worked out as the structure of the rocks controls the
+operations of the eroding agents. Thus, among comparatively undisturbed
+strata, a hard bed resting upon others of a softer kind is apt to form
+along its outcrop a line of cliff or escarpment. Though a long range of
+such cliffs resembles a coast that has been worn by the sea, it may be
+entirely due to mere atmospheric waste. Again, the more resisting
+portions of a rock may be seen projecting as crags or knolls. An igneous
+mass will stand out as a bold hill from amidst the more decomposable
+strata through which it has risen. These features, often so marked on
+the lower grounds, attain their most conspicuous development among the
+higher and barer parts of the mountains, where subaerial disintegration
+is most rapid. The torrents tear out deep gullies from the sides of the
+declivities. Corries or cirques are scooped out on the one hand and
+naked precipices are left on the other. The harder bands of rock project
+as massive ribs down the slopes, shoot up into prominent _aiguilles_, or
+help to give to the summits the notched saw-like outlines they so often
+present.
+
+The materials worn from the surface of the higher are spread out over
+the lower grounds. The streams as they descend begin to drop their
+freight of sediment when, by the lessening of their declivity, their
+carrying power is diminished. The great plains of the earth's surface
+are due to this deposit of gravel, sand and loam. They are thus
+monuments at once of the destructive and reproductive processes which
+have been in progress unceasingly since the first land rose above the
+sea and the first shower of rain fell. Every pebble and particle of
+their soil, once part of the distant mountains, has travelled slowly and
+fitfully to lower levels. Again and again have these materials been
+shifted, ever moving downward and sea-ward. For centuries, perhaps, they
+have taken their share in the fertility of the plains and have
+ministered to the nurture of flower and tree, of the bird of the air,
+the beast of the field and of man himself. But their destiny is still
+the great ocean. In that bourne alone can they find undisturbed repose,
+and there, slowly accumulating in massive beds, they will remain until,
+in the course of ages, renewed upheaval shall raise them into future
+land, there once more to pass through the same cycle of change. (A. Ge.)
+
+  LITERATURE.--_Historical_: The standard work is Karl A. von Zittel's
+  _Geschichte der Geologie und Paläontologie_ (1899), of which there is
+  an abbreviated, but still valuable, English translation; D'Archiac,
+  _Histoire des progrčs de la géologie_, deals especially with the
+  period 1834-1850; Keferstein, _Geschichte und Literatur der
+  Geognosie_, gives a summary up to 1840; while Sir A. Geikie's
+  _Founders of Geology_ (1897; 2nd ed., 1906) deals more particularly
+  with the period 1750-1820. General treatises: Sir Charles Lyell's
+  _Principles of Geology_ is a classic. Of modern English works, Sir A.
+  Geikie's _Text Book of Geology_ (4th ed., 1903) occupies the first
+  place; the work of T.C. Chamberlin and R.D. Salisbury, _Geology; Earth
+  History_ (3 vols., 1905-1906), is especially valuable for American
+  geology. A. de Lapparent's _Traité de géologie_ (5th ed., 1906), is
+  the standard French work. H. Credner's _Elemente der Geologie_ has
+  gone through several editions in Germany. Dynamical and
+  physiographical geology are elaborately treated by E. Suess, _Das
+  Antlitz der Erde_, translated into English, with the title _The Face
+  of the Earth_. The practical study of the science is treated of by F.
+  von Richthofen, _Führer für Forschungsreisende_ (1886); G.A. Cole,
+  _Aids in Practical Geology_ (5th ed., 1906); A. Geikie, _Outlines of
+  Field Geology_ (5th ed., 1900). The practical applications of Geology
+  are discussed by J.V. Elsden, _Applied Geology_ (1898-1899). The
+  relations of Geology to scenery are dealt with by Sir A. Geikie,
+  _Scenery of Scotland_ (3rd ed., 1901); J.E. Marr, _The Scientific
+  Study of Scenery_ (1900); Lord Avebury, _The Scenery of Switzerland_
+  (1896); _The Scenery of England_ (1902); and J. Geikie, _Earth
+  Sculpture_ (1898). A detailed bibliography is given in Sir A. Geikie's
+  _Text Book of Geology_. See also the separate articles on geological
+  subjects for special references to authorities.
+
+
+FOOTNOTES:
+
+  [1] In De Luc's _Lettres physiques et morales sur les montagnes_
+    (1778), the word "cosmology" is used for our science, the author
+    stating that "geology" is more appropriate, but it "was not a word in
+    use." In a completed edition, published in 1779, the same statement
+    is made, but "geology" occurs in the text; in the same year De
+    Saussure used the word without any explanation, as if it were well
+    known.
+
+  [2] The subject of the age of the earth has also been discussed by
+    Professor J. Joly and Professor W.J. Sollas. The former geologist,
+    approaching the question from a novel point of view, has estimated
+    the total quantity of sodium in the water of the ocean and the
+    quantity of that element received annually by the ocean from the
+    denudation of the land. Dividing the one sum by the other, he arrives
+    at the result that the probable age of the earth is between 90 and
+    100 millions of years (_Trans. Roy. Dublin Soc._ ser. ii. vol. vii.,
+    1899, p. 23: _Geol. Mag._, 1900, p. 220). Professor Sollas believes
+    that this limit exceeds what is required for the evolution of
+    geological history, that the lower limit assigned by Lord Kelvin
+    falls short of what the facts demand, and that geological time will
+    probably be found to have been comprised within some indeterminate
+    period between these limits. (Address to Section C, _Brit. Assoc.
+    Report_, 1900; _Age of the Earth_, London, 1905.)
+
+
+
+
+GEOMETRICAL CONTINUITY. In a report of the Institute prefixed to Jean
+Victor Poncelet's _Traité des propriétés projectives des figures_
+(Paris, 1822), it is said that he employed "ce qu'il appelle le principe
+de continuité." The law or principle thus named by him had, he tells us,
+been tacitly assumed as axiomatic by "les plus savans géomčtres." It had
+in fact been enunciated as "lex continuationis," and "la loi de la
+continuité," by Gottfried Wilhelm Leibnitz (Oxf. N.E.D.), and previously
+under another name by Johann Kepler in cap. iv. 4 of his _Ad Vitellionem
+paralipomena quibus astronomiae pars optica traditur_ (Francofurti,
+1604). Of sections of the cone, he says, there are five species from the
+"recta linea" or line-pair to the circle. From the line-pair we pass
+through an infinity of hyperbolas to the parabola, and thence through an
+infinity of ellipses to the circle. Related to the sections are certain
+remarkable points which have no name. Kepler calls them foci. The circle
+has one focus at the centre, an ellipse or hyperbola two foci
+equidistant from the centre. The parabola has one focus within it, and
+another, the "caecus focus," which may be imagined to be _at infinity_
+on the axis _within or without the curve_. The line from it to any point
+of the section is parallel to the axis. To carry out the analogy we must
+speak paradoxically, and say that the line-pair likewise has foci, which
+in this case coalesce as in the circle and fall upon the lines
+themselves; for our geometrical terms should be subject to analogy.
+Kepler dearly loves analogies, his most trusty teachers, acquainted with
+all the secrets of nature, "_omnium naturae arcanorum conscios_." And
+they are to be especially regarded in geometry as, by the use of
+"however absurd expressions," classing extreme limiting forms with an
+infinity of intermediate cases, and placing the whole essence of a thing
+clearly before the eyes.
+
+Here, then, we find formulated by Kepler the doctrine of the concurrence
+of parallels at a single point at infinity and the principle of
+continuity (under the name analogy) in relation to the infinitely great.
+Such conceptions so strikingly propounded in a famous work could not
+escape the notice of contemporary mathematicians. Henry Briggs, in a
+letter to Kepler from Merton College, Oxford, dated "10 Cal. Martiis
+1625," suggests improvements in the _Ad Vitellionem paralipomena_, and
+gives the following construction: Draw a line CBADC, and let an ellipse,
+a parabola, and a hyperbola have B and A for focus and vertex. Let CC
+be the other foci of the ellipse and the hyperbola. Make AD equal to AB,
+and with centres CC and radius in each case equal to CD describe
+circles. Then any point of the ellipse is equidistant from the focus B
+and one circle, and any point of the hyperbola from the focus B and the
+other circle. Any point P of the parabola, in which the second focus is
+missing or infinitely distant, is equidistant from the focus B and the
+line through D which we call the directrix, this taking the place of
+either circle when its centre C is at infinity, and every line CP being
+then parallel to the axis. Thus Briggs, and we know not how many "savans
+géomčtres" who have left no record, had already taken up the new
+doctrine in geometry in its author's lifetime. Six years after Kepler's
+death in 1630 Girard Desargues, "the Monge of his age," brought out the
+first of his remarkable works founded on the same principles, a short
+tract entitled _Méthode universelle de mettre en perspective les objets
+donnés réellement ou en devis_ (Paris, 1636); but "Le privilége étoit de
+1630." (Poudra, _[OE]uvres de Des._, i. 55). Kepler as a modern geometer
+is best known by his _New Stereometry of Wine Casks_ (Lincii, 1615), in
+which he replaces the circuitous Archimedean method of exhaustion by a
+direct "royal road" of infinitesimals, treating a vanishing arc as a
+straight line and regarding a curve as made up of a succession of short
+chords. Some 2000 years previously one Antipho, probably the well-known
+opponent of Socrates, has regarded a circle in like manner as the
+limiting form of a many-sided inscribed rectilinear figure. Antipho's
+notion was rejected by the men of his day as unsound, and when
+reproduced by Kepler it was again stoutly opposed as incapable of any
+sort of geometrical demonstration--not altogether without reason, for it
+rested on an assumed law of continuity rather than on palpable proof.
+
+To complete the theory of continuity, the one thing needful was the idea
+of imaginary points implied in the algebraical geometry of René
+Descartes, in which equations between variables representing
+co-ordinates were found often to have imaginary roots. Newton, in his
+two sections on "Inventio orbium" (_Principia_ i. 4, 5), shows in his
+brief way that he is familiar with the principles of modern geometry. In
+two propositions he uses an auxiliary line which is supposed to cut the
+conic in X and Y, but, as he remarks at the end of the second (prop.
+24), it may not cut it at all. For the sake of brevity he passes on at
+once with the observation that the required constructions are evident
+from the case in which the line cuts the trajectory. In the scholium
+appended to prop. 27, after saying that an asymptote is a tangent at
+infinity, he gives an unexplained general construction for the axes of a
+conic, which seems to imply that it has asymptotes. In all such cases,
+having equations to his loci in the background, he may have thought of
+elements of the figure as passing into the imaginary state in such
+manner as not to vitiate conclusions arrived at on the hypothesis of
+their reality.
+
+Roger Joseph Boscovich, a careful student of Newton's works, has a full
+and thorough discussion of geometrical continuity in the third and last
+volume of his _Elementa universae matheseos_ (ed. prim. Venet, 1757),
+which contains _Sectionum conicarum elementa nova quadam methodo
+concinnata et dissertationem de transformatione locorum geometricorum,
+ubi de continuitatis lege, et de quibusdam infiniti mysteriis_. His
+first principle is that all varieties of a defined locus have the same
+properties, so that what is demonstrable of one should be demonstrable
+in like manner of all, although some artifice may be required to bring
+out the underlying analogy between them. The opposite extremities of an
+infinite straight line, he says, are to be regarded as joined, as if the
+line were a circle having its centre at the infinity on either side of
+it. This leads up to the idea of a _veluti plus quam infinita extensio_,
+a line-circle containing, as we say, the line infinity. Change from the
+real to the imaginary state is contingent upon the passage of some
+element of a figure through zero or infinity and never takes place _per
+saltum_. Lines being some positive and some negative, there must be
+negative rectangles and negative squares, such as those of the exterior
+diameters of a hyperbola. Boscovich's first principle was that of
+Kepler, by whose _quantumvis absurdis locutionibus_ the boldest
+applications of it are covered, as when we say with Poncelet that all
+concentric circles in a plane touch one another in two imaginary fixed
+points at infinity. In G.K. Ch. von Staudt's _Geometrie der Lage and
+Beiträge zur G. der L._ (Nürnberg, 1847, 1856-1860) the geometry of
+position, including the extension of the field of pure geometry to the
+infinite and the imaginary, is presented as an independent science,
+"welche des Messens nicht bedarf." (See GEOMETRY: _Projective_.)
+
+Ocular illusions due to distance, such as Roger Bacon notices in the
+_Opus majus_ (i. 126, ii. 108, 497; Oxford, 1897), lead up to or
+illustrate the mathematical uses of the infinite and its reciprocal the
+infinitesimal. Specious objections can, of course, be made to the
+anomalies of the law of continuity, but they are inherent in the higher
+geometry, which has taught us so much of the "secrets of nature."
+Kepler's excursus on the "analogy" between the conic sections
+hereinbefore referred to is given at length in an article on "The
+Geometry of Kepler and Newton" in vol. xviii. of the _Transactions of
+the Cambridge Philosophical Society_ (1900). It had been generally
+overlooked, until attention was called to it by the present writer in a
+note read in 1880 (_Proc. C.P.S._ iv. 14-17), and shortly afterwards in
+_The Ancient and Modern Geometry of Conics, with Historical Notes and
+Prolegomena_ (Cambridge 1881).     (C. T.*)
+
+
+
+
+GEOMETRY, the general term for the branch of mathematics which has for
+its province the study of the properties of space. From experience, or
+possibly intuitively, we characterize existent space by certain
+fundamental qualities, termed axioms, which are insusceptible of proof;
+and these axioms, in conjunction with the mathematical entities of the
+point, straight line, curve, surface and solid, appropriately defined,
+are the premises from which the geometer draws conclusions. The
+geometrical axioms are merely conventions; on the one hand, the system
+may be based upon inductions from experience, in which case the deduced
+geometry may be regarded as a branch of physical science; or, on the
+other hand, the system may be formed by purely logical methods, in which
+case the geometry is a phase of pure mathematics. Obviously the geometry
+with which we are most familiar is that of existent space--the
+three-dimensional space of experience; this geometry may be termed
+Euclidean, after its most famous expositor. But other geometries exist,
+for it is possible to frame systems of axioms which definitely
+characterize some other kind of space, and from these axioms to deduce a
+series of non-contradictory propositions; such geometries are called
+non-Euclidean.
+
+It is convenient to discuss the subject-matter of geometry under the
+following headings:
+
+I. _Euclidean Geometry_: a discussion of the axioms of existent space
+and of the geometrical entities, followed by a synoptical account of
+Euclid's Elements.
+
+II. _Projective Geometry_: primarily Euclidean, but differing from I. in
+employing the notion of geometrical continuity (q.v.)--points and lines
+at infinity.
+
+III. _Descriptive Geometry_: the methods for representing upon planes
+figures placed in space of three dimensions.
+
+IV. _Analytical Geometry_: the representation of geometrical figures and
+their relations by algebraic equations.
+
+V. _Line Geometry_: an analytical treatment of the line regarded as the
+space element.
+
+VI. _Non-Euclidean Geometry_: a discussion of geometries other than that
+of the space of experience.
+
+VII. _Axioms of Geometry_: a critical analysis of the foundations of
+geometry.
+
+  Special subjects are treated under their own headings: e.g.
+  PROJECTION, PERSPECTIVE; CURVE, SURFACE; CIRCLE, CONIC SECTION;
+  TRIANGLE, POLYGON, POLYHEDRON; there are also articles on special
+  curves and figures, e.g. ELLIPSE, PARABOLA, HYPERBOLA; TETRAHEDRON,
+  CUBE, OCTAHEDRON, DODECAHEDRON, ICOSAHEDRON; CARDIOID, CATENARY,
+  CISSOID, CONCHOID, CYCLOID, EPICYCLOID, LIMAĒON, OVAL, QUADRATRIX,
+  SPIRAL, &c.
+
+_History._--The origin of geometry (Gr. [Greek: gź], earth, [Greek:
+metron], a measure) is, according to Herodotus, to be found in the
+etymology of the word. Its birthplace was Egypt, and it arose from the
+need of surveying the lands inundated by the Nile floods. In its
+infancy it therefore consisted of a few rules, very rough and
+approximate, for computing the areas of triangles and quadrilaterals;
+and, with the Egyptians, it proceeded no further, the geometrical
+entities--the point, line, surface and solid--being only discussed in so
+far as they were involved in practical affairs. The point was realized
+as a mark or position, a straight line as a stretched string or the
+tracing of a pole, a surface as an area; but these units were not
+abstracted; and for the Egyptians geometry was only an art--an auxiliary
+to surveying.[1] The first step towards its elevation to the rank of a
+science was made by Thales (q.v.) of Miletus, who transplanted the
+elementary Egyptian mensuration to Greece. Thales clearly abstracted the
+notions of points and lines, founding the geometry of the latter unit,
+and discovering _per saltum_ many propositions concerning areas, the
+circle, &c. The empirical rules of the Egyptians were corrected and
+developed by the Ionic School which he founded, especially by
+Anaximander and Anaxagoras, and in the 6th century B.C. passed into the
+care of the Pythagoreans. From this time geometry exercised a powerful
+influence on Greek thought. Pythagoras (q.v.), seeking the key of the
+universe in arithmetic and geometry, investigated logically the
+principles underlying the known propositions; and this resulted in the
+formulation of definitions, axioms and postulates which, in addition to
+founding a _science_ of geometry, permitted a crystallization,
+fractional, it is true, of the amorphous collection of material at hand.
+Pythagorean geometry was essentially a geometry of areas and solids; its
+goal was the regular solids--the tetrahedron, cube, octahedron,
+dodecahedron and icosahedron--which symbolized the five elements of
+Greek cosmology. The geometry of the circle, previously studied in Egypt
+and much more seriously by Thales, was somewhat neglected, although this
+curve was regarded as the most perfect of all plane figures and the
+sphere the most perfect of all solids. The circle, however, was taken up
+by the Sophists, who made most of their discoveries in attempts to solve
+the classical problems of squaring the circle, doubling the cube and
+trisecting an angle. These problems, besides stimulating pure geometry,
+i.e. the geometry of constructions made by the ruler and compasses,
+exercised considerable influence in other directions. The first problem
+led to the discovery of the method of _exhaustion_ for determining
+areas. Antiphon inscribed a square in a circle, and on each side an
+isosceles triangle having its vertex on the circle; on the sides of the
+octagon so obtained, isosceles triangles were again constructed, the
+process leading to inscribed polygons of 8, 16 and 32 sides; and the
+areas of these polygons, which are easily determined, are successive
+approximations to the area of the circle. Bryson of Heraclea took an
+important step when he circumscribed, in addition to inscribing,
+polygons to a circle, but he committed an error in treating the circle
+as the mean of the two polygons. The method of Antiphon, in assuming
+that by continued division a polygon can be constructed coincident with
+the circle, demanded that magnitudes are not infinitely divisible. Much
+controversy ranged about this point; Aristotle supported the doctrine of
+infinite divisibility; Zeno attempted to show its absurdity. The
+mechanical tracing of loci, a principle initiated by Archytas of
+Tarentum to solve the last two problems, was a frequent subject for
+study, and several mechanical curves were thus discovered at subsequent
+dates (cissoid, conchoid, quadratrix). Mention may be made of
+Hippocrates, who, besides developing the known methods, made a study of
+similar figures, and, as a consequence, of proportion. This step is
+important as bringing into line discontinuous number and continuous
+magnitude.
+
+A fresh stimulus was given by the succeeding Platonists, who, accepting
+in part the Pythagorean cosmology, made the study of geometry
+preliminary to that of philosophy. The many discoveries made by this
+school were facilitated in no small measure by the clarification of the
+axioms and definitions, the logical sequence of propositions which was
+adopted, and, more especially, by the formulation of the analytic
+method, i.e. of assuming the truth of a proposition and then reasoning
+to a known truth. The main strength of the Platonist geometers lies in
+stereometry or the geometry of solids. The Pythagoreans had dealt with
+the sphere and regular solids, but the pyramid, prism, cone and cylinder
+were but little known until the Platonists took them in hand. Eudoxus
+established their mensuration, proving the pyramid and cone to have
+one-third the content of a prism and cylinder on the same base and of
+the same height, and was probably the discoverer of a proof that the
+volumes of spheres are as the cubes of their radii. The discussion of
+sections of the cone and cylinder led to the discovery of the three
+curves named the parabola, ellipse and hyperbola (see CONIC SECTION); it
+is difficult to over-estimate the importance of this discovery; its
+investigation marks the crowning achievement of Greek geometry, and led
+in later years to the fundamental theorems and methods of modern
+geometry.
+
+The presentation of the subject-matter of geometry as a connected and
+logical series of propositions, prefaced by [Greek: Horoi] or
+foundations, had been attempted by many; but it is to Euclid that we owe
+a complete exposition. Little indeed in the _Elements_ is probably
+original except the arrangement; but in this Euclid surpassed such
+predecessors as Hippocrates, Leon, pupil of Neocleides, and Theudius of
+Magnesia, devising an apt logical model, although when scrutinized in
+the light of modern mathematical conceptions the proofs are riddled with
+fallacies. According to the commentator Proclus, the _Elements_ were
+written with a twofold object, first, to introduce the novice to
+geometry, and secondly, to lead him to the regular solids; conic
+sections found no place therein. What Euclid did for the line and
+circle, Apollonius did for the conic sections, but there we have a
+discoverer as well as editor. These two works, which contain the
+greatest contributions to ancient geometry, are treated in detail in
+Section I. _Euclidean Geometry_ and the articles EUCLID; CONIC SECTION;
+APPOLONIUS. Between Euclid and Apollonius there flourished the
+illustrious Archimedes, whose geometrical discoveries are mainly
+concerned with the mensuration of the circle and conic sections, and of
+the sphere, cone and cylinder, and whose greatest contribution to
+geometrical method is the elevation of the method of exhaustion to the
+dignity of an instrument of research. Apollonius was followed by
+Nicomedes, the inventor of the conchoid; Diocles, the inventor of the
+cissoid; Zenodorus, the founder of the study of isoperimetrical figures;
+Hipparchus, the founder of trigonometry; and Heron the elder, who wrote
+after the manner of the Egyptians, and primarily directed attention to
+problems of practical surveying.
+
+Of the many isolated discoveries made by the later Alexandrian
+mathematicians, those of Menelaus are of importance. He showed how to
+treat spherical triangles, establishing their properties and determining
+their congruence; his theorem on the products of the segments in which
+the sides of a triangle are cut by a line was the foundation on which
+Carnot erected his theory of transversals. These propositions, and also
+those of Hipparchus, were utilized and developed by Ptolemy (q.v.), the
+expositor of trigonometry and discoverer of many isolated propositions.
+Mention may be made of the commentator Pappus, whose _Mathematical
+Collections_ is valuable for its wealth of historical matter; of Theon,
+an editor of Euclid's _Elements_ and commentator of Ptolemy's
+_Almagest_; of Proclus, a commentator of Euclid; and of Eutocius, a
+commentator of Apollonius and Archimedes.
+
+The Romans, essentially practical and having no inclination to study
+science _qua_ science, only had a geometry which sufficed for surveying;
+and even here there were abundant inaccuracies, the empirical rules
+employed being akin to those of the Egyptians and Heron. The Hindus,
+likewise, gave more attention to computation, and their geometry was
+either of Greek origin or in the form presented in trigonometry, more
+particularly connected with arithmetic. It had no logical foundations;
+each proposition stood alone; and the results were empirical. The Arabs
+more closely followed the Greeks, a plan adopted as a sequel to the
+translation of the works of Euclid, Apollonius, Archimedes and many
+others into Arabic. Their chief contribution to geometry is exhibited in
+their solution of algebraic equations by intersecting conics, a step
+already taken by the Greeks in isolated cases, but only elevated into a
+_method_ by Omar al Hayyami, who flourished in the 11th century. During
+the middle ages little was added to Greek and Arabic geometry. Leonardo
+of Pisa wrote a _Practica geometriae_ (1220), wherein Euclidean methods
+are employed; but it was not until the 14th century that geometry,
+generally Euclid's _Elements_, became an essential item in university
+curricula. There was, however, no sign of original development, other
+branches of mathematics, mainly algebra and trigonometry, exercising a
+greater fascination until the 16th century, when the subject again came
+into favour.
+
+The extraordinary mathematical talent which came into being in the 16th
+and 17th centuries reacted on geometry and gave rise to all those
+characters which distinguish modern from ancient geometry. The first
+innovation of moment was the formulation of the principle of geometrical
+continuity by Kepler. The notion of infinity which it involved permitted
+generalizations and systematizations hitherto unthought of (see
+GEOMETRICAL CONTINUITY); and the method of indefinite division applied
+to rectification, and quadrature and cubature problems avoided the
+cumbrous method of exhaustion and provided more accurate results.
+Further progress was made by Bonaventura Cavalieri, who, in his
+_Geometria indivisibilibus continuorum_ (1620), devised a method
+intermediate between that of exhaustion and the infinitesimal calculus
+of Leibnitz and Newton. The logical basis of his system was corrected by
+Roberval and Pascal; and their discoveries, taken in conjunction with
+those of Leibnitz, Newton, and many others in the fluxional calculus,
+culminated in the branch of our subject known as differential geometry
+(see INFINITESIMAL CALCULUS; CURVE; SURFACE).
+
+A second important advance followed the recognition that conics could be
+regarded as projections of a circle, a conception which led at the hands
+of Desargues and Pascal to modern _projective geometry_ and
+_perspective_. A third, and perhaps the most important, advance attended
+the application of algebra to geometry by Descartes, who thereby founded
+_analytical geometry_. The new fields thus opened up were diligently
+explored, but the calculus exercised the greatest attraction and
+relatively little progress was made in geometry until the beginning of
+the 19th century, when a new era opened.
+
+Gaspard Monge was the first important contributor, stimulating
+analytical and differential geometry and founding _descriptive geometry_
+in a series of papers and especially in his lectures at the École
+polytechnique. Projective geometry, founded by Desargues, Pascal, Monge
+and L.N.M. Carnot, was crystallized by J.V. Poncelet, the creator of the
+modern methods. In his _Traité des propriétés des figures_ (1822) the
+line and circular points at infinity, imaginaries, polar reciprocation,
+homology, cross-ratio and projection are systematically employed. In
+Germany, A.F. Möbius, J. Plücker and J. Steiner were making far-reaching
+contributions. Möbius, in his _Barycentrische Calcul_ (1827), introduced
+homogeneous co-ordinates, and also the powerful notion of geometrical
+transformation, including the special cases of collineation and duality;
+Plücker, in his _Analytisch-geometrische Entwickelungen_ (1828-1831),
+and his _System der analytischen Geometrie_ (1835), introduced the
+abridged notation, line and plane co-ordinates, and the conception of
+generalized space elements; while Steiner, besides enriching geometry in
+numerous directions, was the first to systematically generate figures by
+projective pencils. We may also notice M. Chasles, whose _Aperēu
+historique_ (1837) is a classic. Synthetic geometry, characterized by
+its fruitfulness and beauty, attracted most attention, and it so
+happened that its originally weak logical foundations became replaced by
+a more substantial set of axioms. These were found in the anharmonic
+ratio, a device leading to the liberation of synthetic geometry from
+metrical relations, and in involution, which yielded rigorous
+definitions of imaginaries. These innovations were made by K.J.C. von
+Staudt. Analytical geometry was stimulated by the algebra of invariants,
+a subject much developed by A. Cayley, G. Salmon, S.H. Aronhold, L.O.
+Hesse, and more particularly by R.F.A. Clebsch.
+
+The introduction of the line as a space element, initiated by H.
+Grassmann (1844) and Cayley (1859), yielded at the hands of Plücker a
+new geometry, termed _line geometry_, a subject developed more notably
+by F. Klein, Clebsch, C.T. Reye and F.O.R. Sturm (see Section V., _Line
+Geometry_).
+
+_Non-euclidean geometries_, having primarily their origin in the
+discussion of Euclidean parallels, and treated by Wallis, Saccheri and
+Lambert, have been especially developed during the 19th century. Four
+lines of investigation may be distinguished:--the naļve-synthetic,
+associated with Lobatschewski, Bolyai, Gauss; the metric differential,
+studied by Riemann, Helmholtz, Beltrami; the projective, developed by
+Cayley, Klein, Clifford; and the critical-synthetic, promoted chiefly by
+the Italian mathematicians Peano, Veronese, Burali-Forte, Levi Civittą,
+and the Germans Pasch and Hilbert.     (C. E.*)
+
+
+I. EUCLIDEAN GEOMETRY
+
+This branch of the science of geometry is so named since its methods and
+arrangement are those laid down in Euclid's _Elements_.
+
+§ 1. _Axioms._--The object of geometry is to investigate the properties
+of space. The first step must consist in establishing those fundamental
+properties from which all others follow by processes of deductive
+reasoning. They are laid down in the Axioms, and these ought to form
+such a system that nothing need be added to them in order fully to
+characterize space, and that nothing may be omitted without making the
+system incomplete. They must, in fact, completely "define" space.
+
+§ 2. _Definitions._--The axioms of Euclidean Geometry are obtained from
+inspection of existent space and of solids in existent space,--hence
+from experience. The same source gives us the notions of the geometrical
+entities to which the axioms relate, viz. solids, surfaces, lines or
+curves, and points. A solid is directly given by experience; we have
+only to abstract all material from it in order to gain the notion of a
+geometrical solid. This has shape, size, position, and may be moved. Its
+boundary or boundaries are called surfaces. They separate one part of
+space from another, and are said to have no thickness. Their boundaries
+are curves or lines, and these have length only. Their boundaries,
+again, are points, which have no magnitude but only position. We thus
+come in three steps from solids to points which have no magnitude; in
+each step we lose one extension. Hence we say a solid has three
+dimensions, a surface two, a line one, and a point none. Space itself,
+of which a solid forms only a part, is also said to be of three
+dimensions. The same thing is intended to be expressed by saying that a
+solid has length, breadth and thickness, a surface length and breadth, a
+line length only, and a point no extension whatsoever.
+
+Euclid gives the essence of these statements as definitions:--
+
+  Def. 1, I. _A point is that which has no parts, or which has no
+  magnitude._
+
+  Def. 2, I. _A line is length without breadth._
+
+  Def. 5, I. _A superficies is that which has only length and breadth._
+
+  Def. 1, XI. _A solid is that which has length, breadth and thickness._
+
+It is to be noted that the synthetic method is adopted by Euclid; the
+analytical derivation of the successive ideas of "surface," "line," and
+"point" from the experimental realization of a "solid" does not find a
+place in his system, although possessing more advantages.
+
+If we allow motion in geometry, we may generate these entities by moving
+a point, a line, or a surface, thus:--
+
+  The path of a moving point is a line.
+
+  The path of a moving line is, in general, a surface.
+
+  The path of a moving surface is, in general, a solid.
+
+And we may then assume that the lines, surfaces and solids, as defined
+before, can all be generated in this manner. From this generation of the
+entities it follows again that the boundaries--the first and last
+position of the moving element--of a line are points, and so on; and
+thus we come back to the considerations with which we started.
+
+Euclid points this out in his definitions,--Def. 3, I., Def. 6, I., and
+Def. 2, XI. He does not, however, show the connexion which these
+definitions have with those mentioned before. When points and lines have
+been defined, a statement like Def. 3, I., "The extremities of a line
+are points," is a proposition which either has to be proved, and then it
+is a theorem, or which has to be taken for granted, in which case it is
+an axiom. And so with Def. 6, I., and Def. 2, XI.
+
+§ 3. Euclid's definitions mentioned above are attempts to describe, in a
+few words, notions which we have obtained by inspection of and
+abstraction from solids. A few more notions have to be added to these,
+principally those of the simplest line--the straight line, and of the
+simplest surface--the flat surface or plane. These notions we possess,
+but to define them accurately is difficult. Euclid's Definition 4, I.,
+"A straight line is that which lies evenly between its extreme points,"
+must be meaningless to any one who has not the notion of straightness in
+his mind. Neither does it state a property of the straight line which
+can be used in any further investigation. Such a property is given in
+Axiom 10, I. It is really this axiom, together with Postulates 2 and 3,
+which characterizes the straight line.
+
+Whilst for the straight line the verbal definition and axiom are kept
+apart, Euclid mixes them up in the case of the plane. Here the
+Definition 7, I., includes an axiom. It defines a plane as a surface
+which has the property that every straight line which joins any two
+points in it lies altogether in the surface. But if we take a straight
+line and a point in such a surface, and draw all straight lines which
+join the latter to all points in the first line, the surface will be
+fully determined. This construction is therefore sufficient as a
+definition. That every other straight line which joins any two points in
+this surface lies altogether in it is a further property, and to assume
+it gives another axiom.
+
+Thus a number of Euclid's axioms are hidden among his first definitions.
+A still greater confusion exists in the present editions of Euclid
+between the postulates and axioms so called, but this is due to later
+editors and not to Euclid himself. The latter had the last three axioms
+put together with the postulates [Greek: (aitźmata)], so that these were
+meant to include all assumptions relating to space. The remaining
+assumptions, which relate to magnitudes in general, viz. the first eight
+"axioms" in modern editions, were called "common notions" [Greek:
+(koivai ennoiai)]. Of the latter a few may be said to be definitions.
+Thus the eighth might be taken as a definition of "equal," and the
+seventh of "halves." If we wish to collect the axioms used in Euclid's
+_Elements_, we have therefore to take the three postulates, the last
+three axioms as generally given, a few axioms hidden in the definitions,
+and an axiom used by Euclid in the proof of Prop. 4, I, and on a few
+other occasions, viz. that figures may be moved in space without change
+of shape or size.
+
+  § 4. _Postulates._--The assumptions actually made by Euclid may be
+  stated as follows:--
+
+  (1) Straight lines exist which have the property that any one of them
+  may be produced both ways without limit, that through any two points
+  in space such a line may be drawn, and that any two of them coincide
+  throughout their indefinite extensions as soon as two points in the
+  one coincide with two points in the other. (This gives the contents of
+  Def. 4, part of Def. 35, the first two Postulates, and Axiom 10.)
+
+  (2) Plane surfaces or planes exist having the property laid down in
+  Def. 7, that every straight line joining any two points in such a
+  surface lies altogether in it.
+
+  (3) Right angles, as defined in Def. 10, are possible, and all right
+  angles are equal; that is to say, wherever in space we take a plane,
+  and wherever in that plane we construct a right angle, all angles thus
+  constructed will be equal, so that any one of them may be made to
+  coincide with any other. (Axiom 11.)
+
+  (4) The 12th Axiom of Euclid. This we shall not state now, but only
+  introduce it when we cannot proceed any further without it.
+
+  (5) Figures maybe freely moved in space without change of shape or
+  size. This is assumed by Euclid, but not stated as an axiom.
+
+  (6) In any plane a circle may be described, having any point in that
+  plane as centre, and its distance from any other point in that plane
+  as radius. (Postulate 3.)
+
+The definitions which have not been mentioned are all "nominal
+definitions," that is to say, they fix a name for a thing described.
+Many of them overdetermine a figure.
+
+§ 5. Euclid's _Elements_ (see EUCLID) are contained in thirteen books.
+Of these the first four and the sixth are devoted to "plane geometry,"
+as the investigation of figures in a plane is generally called. The 5th
+book contains the theory of proportion which is used in Book VI. The
+7th, 8th and 9th books are purely arithmetical, whilst the 10th contains
+a most ingenious treatment of geometrical irrational quantities. These
+four books will be excluded from our survey. The remaining three books
+relate to figures in space, or, as it is generally called, to "solid
+geometry." The 7th, 8th, 9th, 10th, 13th and part of the 11th and 12th
+books are now generally omitted from the school editions of the
+_Elements_. In the first four and in the 6th book it is to be understood
+that all figures are drawn in a plane.
+
+
+  BOOK I. OF EUCLID'S "ELEMENTS."
+
+  § 6. According to the third postulate it is possible to draw in any
+  plane a circle which has its centre at any given point, and its radius
+  equal to the distance of this point from any other point given in the
+  plane. This makes it possible (Prop. 1) to construct on a given line
+  AB an equilateral triangle, by drawing first a circle with A as centre
+  and AB as radius, and then a circle with B as centre and BA as radius.
+  The point where these circles intersect--that they intersect Euclid
+  quietly assumes--is the vertex of the required triangle. Euclid does
+  not suppose, however, that a circle may be drawn which has its radius
+  equal to the distance between any two points unless one of the points
+  be the centre. This implies also that we are not supposed to be able
+  to make any straight line equal to any other straight line, or to
+  carry a distance about in space. Euclid therefore next solves the
+  problem: It is required along a given straight line from a point in it
+  to set off a distance equal to the length of another straight line
+  given anywhere in the plane. This is done in two steps. It is shown in
+  Prop. 2 how a straight line may be drawn from a given point equal in
+  length to another given straight line not drawn from that point. And
+  then the problem itself is solved in Prop. 3, by drawing first through
+  the given point some straight line of the required length, and then
+  about the same point as centre a circle having this length as radius.
+  This circle will cut off from the given straight line a length equal
+  to the required one. Nowadays, instead of going through this long
+  process, we take a pair of compasses and set off the given length by
+  its aid. This assumes that we may move a length about without changing
+  it. But Euclid has not assumed it, and this proceeding would be fully
+  justified by his desire not to take for granted more than was
+  necessary, if he were not obliged at his very next step actually to
+  make this assumption, though without stating it.
+
+  § 7. We now come (in Prop. 4) to the first theorem. It is the
+  fundamental theorem of Euclid's whole system, there being only a very
+  few propositions (like Props. 13, 14, 15, I.), except those in the 5th
+  book and the first half of the 11th, which do not depend upon it. It
+  is stated very accurately, though somewhat clumsily, as follows:--
+
+  _If two triangles have two sides of the one equal to two sides of the
+  other, each to each, and have also the angles contained by those sides
+  equal to one another, they shall also have their bases or third sides
+  equal; and the two triangles shall be equal; and their other angles
+  shall be equal, each to each, namely, those to which the equal sides
+  are opposite._
+
+  That is to say, the triangles are "identically" equal, and one may be
+  considered as a copy of the other. The proof is very simple. The first
+  triangle is taken up and placed on the second, so that the parts of
+  the triangles which are known to be equal fall upon each other. It is
+  then easily seen that also the remaining parts of one coincide with
+  those of the other, and that they are therefore equal. This process of
+  applying one figure to another Euclid scarcely uses again, though many
+  proofs would be simplified by doing so. The process introduces motion
+  into geometry, and includes, as already stated, the axiom that figures
+  may be moved without change of shape or size.
+
+  If the last proposition be applied to an isosceles triangle, which has
+  two sides equal, we obtain the theorem (Prop. 5), _if two sides of a
+  triangle are equal, then the angles opposite these sides are equal_.
+
+  Euclid's proof is somewhat complicated, and a stumbling-block to many
+  schoolboys. The proof becomes much simpler if we consider the
+  isosceles triangle ABC (AB = AC) twice over, once as a triangle BAC,
+  and once as a triangle CAB; and now remember that AB, AC in the first
+  are equal respectively to AC, AB in the second, and the angles
+  included by these sides are equal. Hence the triangles are equal, and
+  the angles in the one are equal to those in the other, viz. those
+  which are opposite equal sides, i.e. angle ABC in the first equals
+  angle ACB in the second, as they are opposite the equal sides AC and
+  AB in the two triangles.
+
+  There follows the converse theorem (Prop. 6). _If two angles in a
+  triangle are equal, then the sides opposite them are equal_,--i.e. the
+  triangle is isosceles. The proof given consists in what is called a
+  _reductio ad absurdum_, a kind of proof often used by Euclid, and
+  principally in proving the converse of a previous theorem. It assumes
+  that the theorem to be proved is wrong, and then shows that this
+  assumption leads to an absurdity, i.e. to a conclusion which is in
+  contradiction to a proposition proved before--that therefore the
+  assumption made cannot be true, and hence that the theorem is true. It
+  is often stated that Euclid invented this kind of proof, but the
+  method is most likely much older.
+
+  § 8. It is next proved that _two triangles which have the three sides
+  of the one equal respectively to those of the other are identically
+  equal, hence that the angles of the one are equal respectively to
+  those of the other, those being equal which are opposite equal sides_.
+  This is Prop. 8, Prop. 7 containing only a first step towards its
+  proof.
+
+  These theorems allow now of the solution of a number of problems,
+  viz.:--
+
+  _To bisect a given angle_ (Prop. 9).
+
+  _To bisect a given finite straight line_ (Prop. 10).
+
+  _To draw a straight line perpendicularly to a given straight line
+  through a given point in it_ (Prop. 11), _and also through a given
+  point not in it_ (Prop. 12).
+
+  The solutions all depend upon properties of isosceles triangles.
+
+  § 9. The next three theorems relate to angles only, and might have
+  been proved before Prop. 4, or even at the very beginning. The first
+  (Prop. 13) says, _The angles which one straight line makes with
+  another straight line on one side of it either are two right angles or
+  are together equal to two right angles_. This theorem would have been
+  unnecessary if Euclid had admitted the notion of an angle such that
+  its two limits are in the same straight line, and had besides defined
+  the sum of two angles.
+
+  Its converse (Prop. 14) is of great use, inasmuch as it enables us in
+  many cases to prove that two straight lines drawn from the same point
+  are one the continuation of the other. So also is
+
+  Prop. 15. _If two straight lines cut one another, the vertical or
+  opposite angles shall be equal._
+
+  § 10. Euclid returns now to properties of triangles. Of great
+  importance for the next steps (though afterwards superseded by a more
+  complete theorem) is
+
+  Prop. 16. _If one side of a triangle be produced, the exterior angle
+  shall be greater than either of the interior opposite angles._
+
+  Prop. 17. _Any two angles of a triangle are together less than two
+  right angles, is an immediate consequence of it._ By the aid of these
+  two, the following fundamental properties of triangles are easily
+  proved:--
+
+  Prop. 18. _The greater side of every triangle has the greater angle
+  opposite to it_;
+
+  Its converse, Prop. 19. _The greater angle of every triangle is
+  subtended by the greater side, or has the greater side opposite to
+  it_;
+
+  Prop. 20. _Any two sides of a triangle are together greater than the
+  third side_;
+
+  And also Prop. 21. _If from the ends of the side of a triangle there
+  be drawn two straight lines to a point within the triangle, these
+  shall be less than the other two sides of the triangle, but shall
+  contain a greater angle._
+
+  § 11. Having solved two problems (Props. 22, 23), he returns to two
+  triangles which have two sides of the one equal respectively to two
+  sides of the other. It is known (Prop. 4) that if the included angles
+  are equal then the third sides are equal; and conversely (Prop. 8), if
+  the third sides are equal, then the angles included by the first sides
+  are equal. From this it follows that if the included angles are not
+  equal, the third sides are not equal; and conversely, that if the
+  third sides are not equal, the included angles are not equal. Euclid
+  now completes this knowledge by proving, that "_if the included angles
+  are not equal, then the third side in that triangle is the greater
+  which contains the greater angle_"; and conversely, that "_if the
+  third sides are unequal, that triangle contains the greater angle
+  which contains the greater side_." These are Prop. 24 and Prop. 25.
+
+  § 12. The next theorem (Prop. 26) says that _if two triangles have one
+  side and two angles of the one equal respectively to one side and two
+  angles of the other, viz. in both triangles either the angles adjacent
+  to the equal side, or one angle adjacent and one angle opposite it,
+  then the two triangles are identically equal_.
+
+  This theorem belongs to a group with Prop. 4 and Prop. 8. Its first
+  case might have been given immediately after Prop. 4, but the second
+  case requires Prop. 16 for its proof.
+
+  § 13. We come now to the investigation of parallel straight lines,
+  i.e. of straight lines which lie in the same plane, and cannot be made
+  to meet however far they be produced either way. The investigation
+  which starts from Prop. 16, will become clearer if a few names be
+  explained which are not all used by Euclid. If two straight lines be
+  cut by a third, the latter is now generally called a "transversal" of
+  the figure. It forms at the two points where it cuts the given lines
+  four angles with each. Those of the angles which lie between the given
+  lines are called interior angles, and of these, again, any two which
+  lie on opposite sides of the transversal but one at each of the two
+  points are called "alternate angles."
+
+  We may now state Prop. 16 thus:--_If two straight lines which meet are
+  cut by a transversal, their alternate angles are unequal_. For the
+  lines will form a triangle, and one of the alternate angles will be an
+  exterior angle to the triangle, the other interior and opposite to it.
+
+  From this follows at once the theorem contained in Prop. 27. _If two
+  straight lines which are cut by a transversal make alternate angles
+  equal, the lines cannot meet, however far they be produced, hence they
+  are parallel._ This proves the existence of parallel lines.
+
+  Prop. 28 states the same fact in different forms. _If a straight line,
+  falling on two other straight lines, make the exterior angle equal to
+  the interior and opposite angle on the same side of the line, or make_
+  _the interior angles on the same side together equal to two right
+  angles, the two straight lines shall be parallel to one another_.
+
+  Hence we know that, "if two straight lines which are cut by a
+  transversal meet, their alternate angles are not equal"; and hence
+  that, "if alternate angles are equal, then the lines are parallel."
+
+  The question now arises, Are the propositions converse to these true
+  or not? That is to say, "If alternate angles are unequal, do the lines
+  meet?" And "if the lines are parallel, are alternate angles
+  necessarily equal?"
+
+  The answer to either of these two questions implies the answer to the
+  other. But it has been found impossible to prove that the negation or
+  the affirmation of either is true.
+
+  The difficulty which thus arises is overcome by Euclid assuming that
+  the first question has to be answered in the affirmative. This gives
+  his last axiom (12), which we quote in his own words.
+
+  Axiom 12.--_If a straight line meet two straight lines, so as to make
+  the two interior angles on the same side of it taken together less
+  than two right angles, these straight lines, being continually
+  produced, shall at length meet on that side on which are the angles
+  which are less than two right angles._
+
+  The answer to the second of the above questions follows from this, and
+  gives the theorem Prop. 29:--_If a straight line fall on two parallel
+  straight lines, it makes the alternate angles equal to one another,
+  and the exterior angle equal to the interior and opposite angle on the
+  same side, and also the two interior angles on the same side together
+  equal to two right angles_.
+
+  § 14. With this a new part of elementary geometry begins. The earlier
+  propositions are independent of this axiom, and would be true even if
+  a wrong assumption had been made in it. They all relate to figures in
+  a plane. But a plane is only one among an infinite number of
+  conceivable surfaces. We may draw figures on any one of them and study
+  their properties. We may, for instance, take a sphere instead of the
+  plane, and obtain "spherical" in the place of "plane" geometry. If on
+  one of these surfaces lines and figures could be drawn, answering to
+  all the definitions of our plane figures, and if the axioms with the
+  exception of the last all hold, then all propositions up to the 28th
+  will be true for these figures. This is the case in spherical geometry
+  if we substitute "shortest line" or "great circle" for "straight
+  line," "small circle" for "circle," and if, besides, we limit all
+  figures to a part of the sphere which is less than a hemisphere, so
+  that two points on it cannot be opposite ends of a diameter, and
+  therefore determine always one and only one great circle.
+
+  For spherical triangles, therefore, all the important propositions 4,
+  8, 26; 5 and 6; and 18, 19 and 20 will hold good.
+
+  This remark will be sufficient to show the impossibility of proving
+  Euclid's last axiom, which would mean proving that this axiom is a
+  consequence of the others, and hence that the theory of parallels
+  would hold on a spherical surface, where the other axioms do hold,
+  whilst parallels do not even exist.
+
+  It follows that the axiom in question states an inherent difference
+  between the plane and other surfaces, and that the plane is only fully
+  characterized when this axiom is added to the other assumptions.
+
+  § 15. The introduction of the new axiom and of parallel lines leads to
+  a new class of propositions.
+
+  After proving (Prop. 30) that "_two lines which are each parallel to a
+  third are parallel to each other_," we obtain the new properties of
+  triangles contained in Prop. 32. Of these the second part is the most
+  important, viz. the theorem, _The three interior angles of every
+  triangle are together equal to two right angles_.
+
+  As easy deductions not given by Euclid but added by Simson follow the
+  propositions about the angles in polygons, they are given in English
+  editions as corollaries to Prop. 32.
+
+  These theorems do not hold for spherical figures. The sum of the
+  interior angles of a spherical triangle is always greater than two
+  right angles, and increases with the area.
+
+  § 16. The theory of parallels as such may be said to be finished with
+  Props. 33 and 34, which state properties of the parallelogram, i.e. of
+  a quadrilateral formed by two pairs of parallels. They are--
+
+  Prop. 33. _The straight lines which join the extremities of two equal
+  and parallel straight lines towards the same parts are themselves
+  equal and parallel_; and
+
+  Prop. 34. _The opposite sides and angles of a parallelogram are equal
+  to one another, and the diameter (diagonal) bisects the parallelogram,
+  that is, divides it into two equal parts._
+
+  § 17. The rest of the first book relates to areas of figures.
+
+  The theory is made to depend upon the theorems--
+
+  Prop. 35. _Parallelograms on the same base and between the same
+  parallels are equal to one another_; and
+
+  Prop. 36. _Parallelograms on equal bases and between the same
+  parallels are equal to one another_.
+
+  As each parallelogram is bisected by a diagonal, the last theorems
+  hold also if the word parallelogram be replaced by "triangle," as is
+  done in Props. 37 and 38.
+
+  It is to be remarked that Euclid proves these propositions only in the
+  case when the parallelograms or triangles have their bases in the same
+  straight line.
+
+  The theorems converse to the last form the contents of the next three
+  propositions, viz.: Props, 40 and 41.--_Equal triangles, on the same
+  or on equal bases, in the same straight line, and on the same side of
+  it, are between the same parallels_.
+
+  That the two cases here stated are given by Euclid in two separate
+  propositions proved separately is characteristic of his method.
+
+  § 18. To compare areas of other figures, Euclid shows first, in Prop.
+  42, how _to draw a parallelogram which is equal in area to a given
+  triangle, and has one of its angles equal to a given angle_. If the
+  given angle is right, then the problem is solved _to draw a
+  "rectangle" equal in area to a given triangle_.
+
+  Next this parallelogram is transformed into another parallelogram,
+  _which has one of its sides equal to a given straight line_, whilst
+  its angles remain unaltered. This may be done by aid of the theorem in
+
+  Prop. 43. _The complements of the parallelograms which are about the
+  diameter of any parallelogram are equal to one another._
+
+  Thus the problem (Prop. 44) is solved to _construct a parallelogram on
+  a given line, which is equal in area to a given triangle, and which
+  has one angle equal to a given angle_ (generally a right angle).
+
+  As every polygon can be divided into a number of triangles, we can now
+  construct a parallelogram having a given angle, say a right angle, and
+  being equal in area to a given polygon. For each of the triangles into
+  which the polygon has been divided, a parallelogram may be
+  constructed, having one side equal to a given straight line and one
+  angle equal to a given angle. If these parallelograms be placed side
+  by side, they may be added together to form a single parallelogram,
+  having still one side of the given length. This is done in Prop. 45.
+
+  Herewith a means is found to compare areas of different polygons. We
+  need only construct two rectangles equal in area to the given
+  polygons, and having each one side of given length. By comparing the
+  unequal sides we are enabled to judge whether the areas are equal, or
+  which is the greater. Euclid does not state this consequence, but the
+  problem is taken up again at the end of the second book, where it is
+  shown how to construct a square equal in area to a given polygon.
+
+  Prop. 46 is: _To describe a square on a given straight line_.
+
+  § 19. The first book concludes with one of the most important theorems
+  in the whole of geometry, and one which has been celebrated since the
+  earliest times. It is stated, but on doubtful authority, that
+  Pythagoras discovered it, and it has been called by his name. If we
+  call that side in a right-angled triangle which is opposite the right
+  angle the hypotenuse, we may state it as follows:--
+
+  Theorem of Pythagoras (Prop. 47).--_In every right-angled triangle the
+  square on the hypotenuse is equal to the sum of the squares of the
+  other sides._
+
+  And conversely--
+
+  Prop. 48. _If the square described on one of the sides of a triangle
+  be equal to the squares described on the other sides, then the angle
+  contained by these two sides is a right angle._
+
+  On this theorem (Prop. 47) almost all geometrical measurement depends,
+  which cannot be directly obtained.
+
+
+  BOOK II.
+
+  § 20. The propositions in the second book are very different in
+  character from those in the first; they all relate to areas of
+  rectangles and squares. Their true significance is best seen by
+  stating them in an algebraic form. This is often done by expressing
+  the lengths of lines by aid of numbers, which tell how many times a
+  chosen unit is contained in the lines. If there is a unit to be found
+  which is contained an exact number of times in each side of a
+  rectangle, it is easily seen, and generally shown in the teaching of
+  arithmetic, that the rectangle contains a number of unit squares equal
+  to the product of the numbers which measure the sides, a unit square
+  being the square on the unit line. If, however, no such unit can be
+  found, this process requires that connexion between lines and numbers
+  which is only established by aid of ratios of lines, and which is
+  therefore at this stage altogether inadmissible. But there exists
+  another way of connecting these propositions with algebra, based on
+  modern notions which seem destined greatly to change and to simplify
+  mathematics. We shall introduce here as much of it as is required for
+  our present purpose.
+
+  At the beginning of the second book we find a definition according to
+  which "a rectangle is said to be 'contained' by the two sides which
+  contain one of its right angles"; in the text this phraseology is
+  extended by speaking of rectangles contained by any two straight
+  lines, meaning the rectangle which has two adjacent sides equal to the
+  two straight lines.
+
+  We shall denote a finite straight line by a single small letter, a, b,
+  c, ... x, and the area of the rectangle contained by two lines a and b
+  by ab, and this we shall call the product of the two lines a and b. It
+  will be understood that this definition has nothing to do with the
+  definition of a product of numbers.
+
+  We define as follows:--
+
+  The _sum_ of two straight lines a and b means a straight line c which
+  may be divided in two parts equal respectively to a and b. This sum is
+  denoted by a + b.
+
+  The _difference_ of two lines a and b (in symbols, a-b) means a line c
+  which when added to b gives a; that is,
+
+    a - b = c if b + c = a.
+
+  The _product_ of two lines a and b (in symbols, ab) means the area of
+  the rectangle contained by the lines a and b. For aa, which means the
+  square on the line a, we write a².
+
+  § 21. The first ten of the fourteen propositions of the second book
+  may then be written in the form of formulae as follows:--
+
+    Prop. 1. a(b + c + d + ... ) = ab + ac + ad + ...
+
+      "   2. ab + ac = a² if b + c = a.
+
+      "   3. a(a + b) = a² + ab.
+
+      "   4. (a + b)² = a² + 2ab + b².
+
+      "   5. (a + b)(a - b) + b² = a².
+
+      "   6. (a + b)(a - b) + b² = a².
+
+      "   7. a² + (a - b)² = 2a(a - b) + b².
+
+      "   8. 4(a + b)a + b² = (2a + b)².
+
+      "   9. (a + b)² + (a - b)² = 2a² + 2b².
+
+      "  10. (a + b)² + (a - b)² = 2a² + 2b².
+
+  It will be seen that 5 and 6, and also 9 and 10, are identical. In
+  Euclid's statement they do not look the same, the figures being
+  arranged differently.
+
+  If the letters a, b, c, ... denoted numbers, it follows from algebra
+  that each of these formulae is true. But this does not prove them in
+  our case, where the letters denote lines, and their products areas
+  without any reference to numbers. To prove them we have to discover
+  the laws which rule the operations introduced, viz. addition and
+  multiplication of segments. This we shall do now; and we shall find
+  that these laws are the same with those which hold in algebraical
+  addition and multiplication.
+
+  § 22. In a sum of numbers we may change the order in which the numbers
+  are added, and we may also add the numbers together in groups and then
+  add these groups. But this also holds for the sum of segments and for
+  the sum of rectangles, as a little consideration shows. That the sum
+  of rectangles has always a meaning follows from the Props. 43-45 in
+  the first book. These laws about addition are reducible to the two--
+
+    a + b = b + a            (1),
+
+    a + (b + c) = a + b + c  (2);
+
+  or, when expressed for rectangles,
+
+    ab + ed = ed + ab              (3),
+
+    ab + (cd + ef) = ab + cd + ef  (4).
+
+  The brackets mean that the terms in the bracket have been added
+  together before they are added to another term. The more general cases
+  for more terms may be deduced from the above.
+
+  For the product of two numbers we have the law that it remains
+  unaltered if the factors be interchanged. This also holds for our
+  geometrical product. For if ab denotes the area of the rectangle which
+  has a as base and b as altitude, then ba will denote the area of the
+  rectangle which has b as base and a as altitude. But in a rectangle we
+  may take either of the two lines which contain it as base, and then
+  the other will be the altitude. This gives
+
+    ab = ba       (5).
+
+  In order further to multiply a sum by a number, we have in algebra the
+  rule:--Multiply each term of the sum, and add the products thus
+  obtained. That this holds for our geometrical products is shown by
+  Euclid in his first proposition of the second book, where he proves
+  that the area of a rectangle whose base is the sum of a number of
+  segments is equal to the sum of rectangles which have these segments
+  separately as bases. In symbols this gives, in the simplest case,
+
+     a(b + c) = ab + ac   \
+                           >  (6).
+  and (b + c)a = ba + ca  /
+
+  To these laws, which have been investigated by Sir William Hamilton
+  and by Hermann Grassmann, the former has given special names. He calls
+  the laws expressed in
+
+    (1) and (3) the commutative law for addition;
+
+            (5)          "       "   "  multiplication;
+
+    (2) and (4) the associative laws for addition;
+
+            (6) the distributive law.
+
+  § 23. Having proved that these six laws hold, we can at once prove
+  every one of the above propositions in their algebraical form.
+
+  The first is proved geometrically, it being one of the fundamental
+  laws. The next two propositions are only special cases of the first.
+  Of the others we shall prove one, viz. the fourth:--
+
+         (a + b)² = (a + b)(a + b) = (a + b)a + (a + b)b   by (6).
+
+  But                     (a + b)a = aa + ba       by (6),
+                                   = aa + ab       by (5);
+
+  and                     (a + b)b = ab + bb       by (6).
+
+  Therefore     (a + b)² = aa + ab + (ab + bb) \
+                         = aa + (ab + ab) + bb  >  by (4).
+                         = aa + 2ab + bb       /
+
+  This gives the theorem in question.
+
+  In the same manner every one of the first ten propositions is proved.
+
+  It will be seen that the operations performed are exactly the same as
+  if the letters denoted numbers.
+
+  Props. 5 and 6 may also be written thus--
+
+    (a + b)(a - b) = a² - b².
+
+  Prop. 7, which is an easy consequence of Prop. 4, may be transformed.
+  If we denote by c the line a + b, so that
+
+    c = a + b, a = c - b,
+
+  we get
+
+    c² + (c - b)² = 2c(c - b) + b²
+                  = 2c² - 2bc + b².
+
+  Subtracting c² from both sides, and writing a for c, we get
+
+    (a - b)² = a² - 2ab + b².
+
+  In Euclid's _Elements_ this form of the theorem does not appear, all
+  propositions being so stated that the notion of subtraction does not
+  enter into them.
+
+  § 24. The remaining two theorems (Props. 12 and 13) connect the square
+  on one side of a triangle with the sum of the squares on the other
+  sides, in case that the angle between the latter is acute or obtuse.
+  They are important theorems in trigonometry, where it is possible to
+  include them in a single theorem.
+
+  § 25. There are in the second book two problems, Props. 11 and 14.
+
+  If written in the above symbolic language, the former requires to find
+  a line x such that a(a - x) = x². Prop. 11 contains, therefore, the
+  solution of a quadratic equation, which we may write x² + ax = a². The
+  solution is required later on in the construction of a regular
+  decagon.
+
+  More important is the problem in the last proposition (Prop. 14). It
+  requires the construction of a square equal in area to a given
+  rectangle, hence a solution of the equation
+
+    x² = ab.
+
+  In Book I., 42-45, it has been shown how a rectangle may be
+  constructed equal in area to a given figure bounded by straight lines.
+  By aid of the new proposition we may therefore now determine a line
+  such that the square on that line is equal in area to any given
+  rectilinear figure, or we can _square_ any such figure.
+
+  As of two squares that is the greater which has the greater side, it
+  follows that now the comparison of two areas has been reduced to the
+  comparison of two lines.
+
+  The problem of reducing other areas to squares is frequently met with
+  among Greek mathematicians. We need only mention the problem of
+  squaring the circle (see CIRCLE).
+
+  In the present day the comparison of areas is performed in a simpler
+  way by reducing all areas to rectangles having a common base. Their
+  altitudes give then a measure of their areas.
+
+  The construction of a rectangle having the base u, and being equal in
+  area to a given rectangle, depends upon Prop. 43, I. This therefore
+  gives a solution of the equation
+
+    ab = ux,
+
+  where x denotes the unknown altitude.
+
+
+  BOOK III.
+
+  § 26. The third book of the _Elements_ relates exclusively to
+  properties of the circle. A circle and its circumference have been
+  defined in Book I., Def. 15. We restate it here in slightly different
+  words:--
+
+  _Definition_.--The circumference of a circle is a plane curve such
+  that all points in it have the same distance from a fixed point in the
+  plane. This point is called the "centre" of the circle.
+
+  Of the new definitions, of which eleven are given at the beginning of
+  the third book, a few only require special mention. The first, which
+  says that circles with equal radii are equal, is in part a theorem,
+  but easily proved by applying the one circle to the other. Or it may
+  be considered proved by aid of Prop. 24, equal circles not being used
+  till after this theorem.
+
+  In the second definition is explained what is meant by a line which
+  "touches" a circle. Such a line is now generally called a tangent to
+  the circle. The introduction of this name allows us to state many of
+  Euclid's propositions in a much shorter form.
+
+  For the same reason we shall call a straight line joining two points
+  on the circumference of a circle a "chord."
+
+  Definitions 4 and 5 may be replaced with a slight generalization by
+  the following:--
+
+  _Definition_.--By the distance of a point from a line is meant the
+  length of the perpendicular drawn from the point to the line.
+
+  § 27. From the definition of a circle it follows that every circle has
+  a centre. Prop. 1 requires to find it when the circle is given, i.e.
+  when its circumference is drawn.
+
+  To solve this problem a chord is drawn (that is, any two points in the
+  circumference are joined), and through the point where this is
+  bisected a perpendicular to it is erected. Euclid then proves, first,
+  that no point off this perpendicular can be the centre, hence that the
+  centre must lie in this line; and, secondly, that of the points on the
+  perpendicular one only can be the centre, viz. the one which bisects
+  the parts of the perpendicular bounded by the circle. In the second
+  part Euclid silently assumes that the perpendicular there used does
+  cut the circumference in two, and only in two points. The proof
+  therefore is incomplete. The proof of the first part, however, is
+  exact. By drawing two non-parallel chords, and the perpendiculars
+  which bisect them, the centre will be found as the point where these
+  perpendiculars intersect.
+
+  § 28. In Prop. 2 it is proved that a chord of a circle lies altogether
+  within the circle.
+
+  What we have called the first part of Euclid's solution of Prop. 1 may
+  be stated as a theorem:--
+
+  _Every straight line which bisects a chord, and is at right angles to
+  it, passes through the centre of the circle._
+
+  The converse to this gives Prop. 3, which may be stated thus:--
+
+  _If a straight line through the centre of a circle bisect a chord,
+  then it is perpendicular to the chord, and if it be perpendicular to
+  the chord it bisects it._
+
+  An easy consequence of this is the following theorem, which is
+  essentially the same as Prop. 4:--
+
+  _Two chords of a circle, of which neither passes through the centre,
+  cannot bisect each other._
+
+  These last three theorems are fundamental for the theory of the
+  circle. It is to be remarked that Euclid never proves that a straight
+  line cannot have more than two points in common with a circumference.
+
+  § 29. The next two propositions (5 and 6) might be replaced by a
+  single and a simpler theorem, viz:--
+
+  _Two circles which have a common centre, and whose circumferences have
+  one point in common, coincide._
+
+  Or, more in agreement with Euclid's form:--
+
+  _Two different circles, whose circumferences have a point in common,
+  cannot have the same centre._
+
+  That Euclid treats of two cases is characteristic of Greek
+  mathematics.
+
+  The next two propositions (7 and 8) again belong together. They may be
+  combined thus:--
+
+  _If from a point in a plane of a circle, which is not the centre,
+  straight lines be drawn to the different points of the circumference,
+  then of all these lines one is the shortest, and one the longest, and
+  these lie both in that straight line which joins the given point to
+  the centre. Of all the remaining lines each is equal to one and only
+  one other, and these equal lines lie on opposite sides of the shortest
+  or longest, and make equal angles with them._
+
+  Euclid distinguishes the two cases where the given point lies within
+  or without the circle, omitting the case where it lies in the
+  circumference.
+
+  From the last proposition it follows that if from a point more than
+  two equal straight lines can be drawn to the circumference, this point
+  must be the centre. This is Prop. 9.
+
+  As a consequence of this we get
+
+  _If the circumferences of the two circles have three points in common
+  they coincide._
+
+  For in this case the two circles have a common centre, because from
+  the centre of the one three equal lines can be drawn to points on the
+  circumference of the other. But two circles which have a common
+  centre, and whose circumferences have a point in common, coincide.
+  (Compare above statement of Props. 5 and 6.)
+
+  This theorem may also be stated thus:--
+
+  _Through three points only one circumference may be drawn; or, Three
+  points determine a circle._
+
+  Euclid does not give the theorem in this form. He proves, however,
+  _that the two circles cannot cut another in more than two points_
+  (Prop. 10), and _that two circles cannot touch one another in more
+  points than one_ (Prop. 13).
+
+  § 30. Propositions 11 and 12 assert that _if two circles touch, then
+  the point of contact lies on the line joining their centres_. This
+  gives two propositions, because the circles may touch either
+  internally or externally.
+
+  § 31. Propositions 14 and 15 relate to the length of chords. The first
+  says _that equal chords are equidistant from the centre, and that
+  chords which are equidistant from the centre are equal_;
+
+  Whilst Prop. 15 compares unequal chords, viz. _Of all chords the
+  diameter is the greatest, and of other chords that is the greater
+  which is nearer to the centre_; and conversely, _the greater chord is
+  nearer to the centre_.
+
+  § 32. In Prop. 16 the tangent to a circle is for the first time
+  introduced. The proposition is meant to show that the straight line at
+  the end point of the diameter and at right angles to it is a tangent.
+  The proposition itself does not state this. It runs thus:--
+
+  Prop. 16. _The straight line drawn at right angles to the diameter of
+  a circle, from the extremity of it, falls without the circle; and no
+  straight line can be drawn from the extremity, between that straight
+  line and the circumference, so as not to cut the circle._
+
+  _Corollary_.--The straight line at right angles to a diameter drawn
+  through the end point of it touches the circle.
+
+  The statement of the proposition and its whole treatment show the
+  difficulties which the tangents presented to Euclid.
+
+  Prop. 17 solves the problem _through a given point, either in the
+  circumference or without it, to draw a tangent to a given circle_.
+
+  Closely connected with Prop. 16 are Props. 18 and 19, which state
+  (Prop. 18), _that the line joining the centre of a circle to the point
+  of contact of a tangent is perpendicular to the tangent_; and
+  conversely (Prop. 19), _that the straight line through the point of
+  contact of, and perpendicular to, a tangent to a circle passes through
+  the centre of the circle_.
+
+  § 33. The rest of the book relates to angles connected with a circle,
+  viz. angles which have the vertex either at the centre or on the
+  circumference, and which are called respectively angles at the centre
+  and angles at the circumference. Between these two kinds of angles
+  exists the important relation expressed as follows:--
+
+  Prop. 20. _The angle at the centre of a circle is double of the angle
+  at the circumference on the same base, that is, on the same arc._
+
+  This is of great importance for its consequences, of which the two
+  following are the principal:--
+
+  Prop. 21. _The angles in the same segment of a circle are equal to one
+  another_;
+
+  Prop. 22. _The opposite angles of any quadrilateral figure inscribed
+  in a circle are together equal to two right angles._
+
+  Further consequences are:--
+
+  Prop. 23. _On the same straight line, and on the same side of it,
+  there cannot be two similar segments of circles, not coinciding with
+  one another_;
+
+  Prop. 24. _Similar segments of circles on equal straight lines are
+  equal to one another._
+
+  The problem Prop. 25. _A segment of a circle being given to describe
+  the circle of which it is a segment_, may be solved much more easily
+  by aid of the construction described in relation to Prop. 1, III., in
+  § 27.
+
+  § 34. There follow four theorems connecting the angles at the centre,
+  the arcs into which they divide the circumference, and the chords
+  subtending these arcs. They are expressed for angles, arcs and chords
+  in equal circles, but they hold also for angles, arcs and chords in
+  the same circle.
+
+  The theorems are:--
+
+  Prop. 26. _In equal circles equal angles stand on equal arcs, whether
+  they be at the centres or circumferences_;
+
+  Prop. 27. (converse to Prop. 26). _In equal circles the angles which
+  stand on equal arcs are equal to one another, whether they be at the
+  centres or the circumferences_;
+
+  Prop. 28. _In equal circles equal straight lines_ (equal chords) _cut
+  off equal arcs, the greater equal to the greater, and the less equal
+  to the less_;
+
+  Prop. 29 (converse to Prop. 28). _In equal circles equal arcs are
+  subtended by equal straight lines._
+
+  § 35. Other important consequences of Props. 20-22 are:--
+
+  Prop. 31. _In a circle the angle in a semicircle is a right angle; but
+  the angle in a segment greater than a semicircle is less than a right
+  angle; and the angle in a segment less than a semicircle is greater
+  than a right angle_;
+
+  Prop. 32. _If a straight line touch a circle, and from the point of
+  contact a straight line be drawn cutting the circle, the angles which
+  this line makes with the line touching the circle shall be equal to
+  the angles which are in the alternate segments of the circle._
+
+  § 36. Propositions 30, 33, 34, contain problems which are solved by
+  aid of the propositions preceding them:--
+
+  Prop. 30. _To bisect a given arc, that is, to divide it into two equal
+  parts_;
+
+  Prop. 33. _On a given straight line to describe a segment of a circle
+  containing an angle equal to a given rectilineal angle_;
+
+  Prop. 34. _From a given circle to cut off a segment containing an
+  angle equal to a given rectilineal angle_.
+
+  § 37. If we draw chords through a point A within a circle, they will
+  each be divided by A into two segments. Between these segments the law
+  holds that the rectangle contained by them has the same area on
+  whatever chord through A the segments are taken. The value of this
+  rectangle changes, of course, with the position of A.
+
+  A similar theorem holds if the point A be taken without the circle. On
+  every straight line through A, which cuts the circle in two points B
+  and C, we have two segments AB and AC, and the rectangles contained by
+  them are again equal to one another, and equal to the square on a
+  tangent drawn from A to the circle.
+
+  The first of these theorems gives Prop. 35, and the second Prop. 36,
+  with its corollary, whilst Prop. 37, the last of Book III., gives the
+  converse to Prop. 36. The first two theorems may be combined in one:--
+
+  _If through a point A in the plane of a circle a straight line be
+  drawn cutting the circle in B and C, then the rectangle AB·AC has a
+  constant value so long as the point A be fixed; and if from A a
+  tangent AD can be drawn to the circle, touching at D, then the above
+  rectangle equals the square on AD._
+
+  Prop. 37 may be stated thus:--
+
+  _If from a point A without a circle a line be drawn cutting the circle
+  in B and C, and another line to a point D on the circle, and AB·AC =
+  AD², then the line AD touches the circle at D._
+
+  It is not difficult to prove also the converse to the general
+  proposition as above stated. This proposition and its converse may be
+  expressed as follows:--
+
+  _If four points ABCD be taken on the circumference of a circle, and if
+  the lines AB, CD, produced if necessary, meet at E, then_
+
+    EA·EB = EC·ED;
+
+  _and conversely, if this relation holds then the four points lie on a
+  circle, that is, the circle drawn through three of them passes through
+  the fourth._
+
+  That a circle may always be drawn through three points, provided that
+  they do not lie in a straight line, is proved only later on in Book
+  IV.
+
+
+  BOOK IV.
+
+  § 38. The fourth book contains only problems, all relating to the
+  construction of triangles and polygons inscribed in and circumscribed
+  about circles, and of circles inscribed in or circumscribed about
+  triangles and polygons. They are nearly all given for their own sake,
+  and not for future use in the construction of figures, as are most of
+  those in the former books. In seven definitions at the beginning of
+  the book it is explained what is understood by figures inscribed in or
+  described about other figures, with special reference to the case
+  where one figure is a circle. Instead, however, of saying that one
+  figure is described about another, it is now generally said that the
+  one figure is circumscribed about the other. We may then state the
+  definitions 3 or 4 thus:--
+
+  _Definition._--A polygon is said to be inscribed in a circle, and the
+  circle is said to be circumscribed about the polygon, if the vertices
+  of the polygon lie in the circumference of the circle.
+
+  And definitions 5 and 6 thus:--
+
+  _Definition._--A polygon is said to be circumscribed about a circle,
+  and a circle is said to be inscribed in a polygon, if the sides of the
+  polygon are tangents to the circle.
+
+  § 39. The first problem is merely constructive. It requires to draw in
+  a given circle a chord equal to a given straight line, which is not
+  greater than the diameter of the circle. The problem is not a
+  determinate one, inasmuch as the chord may be drawn from any point in
+  the circumference. This may be said of almost all problems in this
+  book, especially of the next two. They are:--
+
+  Prop. 2. _In a given circle to inscribe a triangle equiangular to a
+  given triangle;_
+
+  Prop. 3. _About a given circle to circumscribe a triangle equiangular
+  to a given triangle._
+
+  § 40. Of somewhat greater interest are the next problems, where the
+  triangles are given and the circles to be found.
+
+  Prop. 4. _To inscribe a circle in a given triangle._
+
+  The result is that the problem has always a solution, viz. the centre
+  of the circle is the point where the bisectors of two of the interior
+  angles of the triangle meet. The solution shows, though Euclid does
+  not state this, that the problem has but one solution; and also,
+
+  _The three bisectors of the interior angles of any triangle meet in a
+  point, and this is the centre of the circle inscribed in the
+  triangle._
+
+  The solutions of most of the other problems contain also theorems. Of
+  these we shall state those which are of special interest; Euclid does
+  not state any one of them.
+
+  § 41. Prop. 5. _To circumscribe a circle about a given triangle._
+
+  The one solution which always exists contains the following:--
+
+  _The three straight lines which bisect the sides of a triangle at
+  right angles meet in a point, and this point is the centre of the
+  circle circumscribed about the triangle._
+
+  Euclid adds in a corollary the following property:--
+
+  The centre of the circle circumscribed about a triangle lies within,
+  on a side of, or without the triangle, according as the triangle is
+  acute-angled, right-angled or obtuse-angled.
+
+  § 42. Whilst it is always possible to draw a circle which is inscribed
+  in or circumscribed about a given triangle, this is not the case with
+  quadrilaterals or polygons of more sides. Of those for which this is
+  possible the regular polygons, i.e. polygons which have all their
+  sides and angles equal, are the most interesting. In each of them a
+  circle may be inscribed, and another may be circumscribed about it.
+
+  Euclid does not use the word regular, but he describes the polygons in
+  question as _equiangular_ and _equilateral_. We shall use the name
+  regular polygon. The regular triangle is equilateral, the regular
+  quadrilateral is the square.
+
+  Euclid considers the regular polygons of 4, 5, 6 and 15 sides. For
+  each of the first three he solves the problems--(1) to inscribe such a
+  polygon in a given circle; (2) to circumscribe it about a given
+  circle; (3) to inscribe a circle in, and (4) to circumscribe a circle
+  about, such a polygon.
+
+  For the regular triangle the problems are not repeated, because more
+  general problems have been solved.
+
+  Props. 6, 7, 8 and 9 solve these problems for the square.
+
+  The general problem of inscribing in a given circle a regular polygon
+  of n sides depends upon the problem of dividing the circumference of a
+  circle into n equal parts, or what comes to the same thing, of drawing
+  from the centre of the circle n radii such that the angles between
+  consecutive radii are equal, that is, to divide the space about the
+  centre into n equal angles. Thus, if it is required to inscribe a
+  square in a circle, we have to draw four lines from the centre, making
+  the four angles equal. This is done by drawing two diameters at right
+  angles to one another. The ends of these diameters are the vertices of
+  the required square. If, on the other hand, tangents be drawn at these
+  ends, we obtain a square circumscribed about the circle.
+
+  § 43. To construct a _regular pentagon_, we find it convenient first
+  to construct a _regular decagon_. This requires to divide the space
+  about the centre into ten equal angles. Each will be 1/10th of a right
+  angle, or 1/5th of two right angles. If we suppose the decagon
+  constructed, and if we join the centre to the end of one side, we get
+  an isosceles triangle, where the angle at the centre equals 1/5th of
+  two right angles; hence each of the angles at the base will be 2/5ths
+  of two right angles, as all three angles together equal two right
+  angles. Thus we have to construct an isosceles triangle, having the
+  angle at the vertex equal to half an angle at the base. This is solved
+  in Prop. 10, by aid of the problem in Prop. 11 of the second book. If
+  we make the sides of this triangle equal to the radius of the given
+  circle, then the base will be the side of the regular decagon
+  inscribed in the circle. This side being known the decagon can be
+  constructed, and if the vertices are joined alternately, leaving out
+  half their number, we obtain the regular pentagon. (Prop. 11.)
+
+  Euclid does not proceed thus. He wants the pentagon before the
+  decagon. This, however, does not change the real nature of his
+  solution, nor does his solution become simpler by not mentioning the
+  decagon.
+
+  Once the regular pentagon is inscribed, it is easy to circumscribe
+  another by drawing tangents at the vertices of the inscribed pentagon.
+  This is shown in Prop. 12.
+
+  Props. 13 and 14 teach how a circle may be inscribed in or
+  circumscribed about any given regular pentagon.
+
+  § 44. The _regular hexagon_ is more easily constructed, as shown in
+  Prop. 15. The result is that the side of the regular hexagon inscribed
+  in a circle is equal to the radius of the circle.
+
+  For this polygon the other three problems mentioned are not solved.
+
+  § 45. The book closes with Prop. 16. To inscribe a regular quindecagon
+  in a given circle. If we inscribe a regular pentagon and a regular
+  hexagon in the circle, having one vertex in common, then the arc from
+  the common vertex to the next vertex of the pentagon is 1/5th of the
+  circumference, and to the next vertex of the hexagon is 1/6th of the
+  circumference. The difference between these arcs is, therefore, 1/5 -
+  1/6 = 1/30th of the circumference. The latter may, therefore, be
+  divided into thirty, and hence also in fifteen equal parts, and the
+  regular quindecagon be described.
+
+  § 46. We conclude with a few theorems about regular polygons which are
+  not given by Euclid.
+
+  _The straight lines perpendicular to and bisecting the sides of any
+  regular polygon meet in a point. The straight lines bisecting the
+  angles in the regular polygon meet in the same point. This point is
+  the centre of the circles circumscribed about and inscribed in the
+  regular polygon._
+
+  We can bisect any given arc (Prop. 30, III.). Hence we can divide a
+  circumference into 2n equal parts as soon as it has been divided into
+  n equal parts, or as soon as a regular polygon of n sides has been
+  constructed. Hence--
+
+  _If a regular polygon of n sides has been constructed, then a regular
+  polygon of 2n sides, of 4n, of 8n sides, &c., may also be
+  constructed._ Euclid shows how to construct regular polygons of 3, 4,
+  5 and 15 sides. It follows that we can construct regular polygons of
+
+     3,  6, 12,  24  sides
+     4,  8, 16,  32    "
+     5, 10, 20,  40    "
+    15, 30, 60, 120    "
+
+  The construction of any new regular polygon not included in one of
+  these series will give rise to a new series. Till the beginning of the
+  19th century nothing was added to the knowledge of regular polygons as
+  given by Euclid. Then Gauss, in his celebrated _Arithmetic_, proved
+  that every regular polygon of 2^n + 1 sides may be constructed if this
+  number 2^n + 1 be prime, and that no others except those with 2^m(2^n
+  + 1) sides can be constructed by elementary methods. This shows that
+  regular polygons of 7, 9, 13 sides cannot thus be constructed, but
+  that a regular polygon of 17 sides is possible; for 17 = 2^4 + 1. The
+  next polygon is one of 257 sides. The construction becomes already
+  rather complicated for 17 sides.
+
+
+  BOOK V.
+
+  § 47. The fifth book of the _Elements_ is not exclusively geometrical.
+  It contains the theory of ratios and proportion of quantities in
+  general. The treatment, as here given, is admirable, and in every
+  respect superior to the algebraical method by which Euclid's theory is
+  now generally replaced. We shall treat the subject in order to show
+  why the usual algebraical treatment of proportion is not really sound.
+  We begin by quoting those definitions at the beginning of Book V.
+  which are most important. These definitions have given rise to much
+  discussion.
+
+  The only definitions which are essential for the fifth book are Defs.
+  1, 2, 4, 5, 6 and 7. Of the remainder 3, 8 and 9 are more than
+  useless, and probably not Euclid's, but additions of later editors, of
+  whom Theon of Alexandria was the most prominent. Defs. 10 and 11
+  belong rather to the sixth book, whilst all the others are merely
+  nominal. The really important ones are 4, 5, 6 and 7.
+
+  § 48. To define a magnitude is not attempted by Euclid. The first two
+  definitions state what is meant by a "part," that is, a submultiple or
+  measure, and by a "multiple" of a given magnitude. The meaning of Def.
+  4 is that two given quantities can have a ratio to one another only in
+  case that they are comparable as to their magnitude, that is, if they
+  are of the same kind.
+
+  Def. 3, which is probably due to Theon, professes to define a ratio,
+  but is as meaningless as it is uncalled for, for all that is wanted is
+  given in Defs. 5 and 7.
+
+  In Def. 5 it is explained what is meant by saying that two magnitudes
+  have the same ratio to one another as two other magnitudes, and in
+  Def. 7 what we have to understand by a greater or a less ratio. The
+  6th definition is only nominal, explaining the meaning of the word
+  _proportional_.
+
+  Euclid represents magnitudes by lines, and often denotes them either
+  by single letters or, like lines, by two letters. We shall use only
+  single letters for the purpose. If a and b denote two magnitudes of
+  the same kind, their ratio will be denoted by a : b; if c and d are
+  two other magnitudes of the same kind, but possibly of a different
+  kind from a and b, then if c and d have the same ratio to one another
+  as a and b, this will be expressed by writing--
+
+    a : b :: c : d.
+
+  Further, if m is a (whole) number, ma shall denote the multiple of a
+  which is obtained by taking it m times.
+
+  § 49. The whole theory of ratios is based on Def. 5.
+
+  Def. 5. _The first of four magnitudes is said to have the same ratio
+  to the second that the third has to the fourth when, any equimultiples
+  whatever of the first and the third being taken, and any equimultiples
+  whatever of the second and the fourth, if the multiple of the first be
+  less than that of the second, the multiple of the third is also less
+  than that of the fourth; and if the multiple of the first is equal to
+  that of the second, the multiple of the third is also equal to that of
+  the fourth; and if the multiple of the first is greater than that of
+  the second, the multiple of the third is also greater than that of the
+  fourth._
+
+  It will be well to show at once in an example how this definition can
+  be used, by proving the first part of the first proposition in the
+  sixth book. _Triangles of the same altitude are to one another as
+  their bases_, or if a and b are the bases, and [alpha] and ß the
+  areas, of two triangles which have the same altitude, then a : b ::
+  [alpha] : ß.
+
+  To prove this, we have, according to Definition 5, to show--
+
+    if ma > nb, then m[alpha] > nß,
+    if ma = nb, then m[alpha] = nß,
+    if ma < nb, then m[alpha] < nß.
+
+  That this is true is in our case easily seen. We may suppose that the
+  triangles have a common vertex, and their bases in the same line. We
+  set off the base a along the line containing the bases m times; we
+  then join the different parts of division to the vertex, and get m
+  triangles all equal to [alpha]. The triangle on ma as base equals,
+  therefore, m[alpha]. If we proceed in the same manner with the base b,
+  setting it off n times, we find that the area of the triangle on the
+  base nb equals nß, the vertex of all triangles being the same. But if
+  two triangles have the same altitude, then their areas are equal if
+  the bases are equal; hence m[alpha] = nß if ma = nb, and if their
+  bases are unequal, then that has the greater area which is on the
+  greater base; in other words, m[alpha] is greater than, equal to, or
+  less than nß, according as ma is greater than, equal to, or less than
+  nb, which was to be proved.
+
+  § 50. It will be seen that even in this example it does not become
+  evident what a ratio really is. It is still an open question whether
+  ratios are magnitudes which we can compare. We do not know whether the
+  ratio of two lines is a magnitude of the same kind as the ratio of two
+  areas. Though we might say that Def. 5 defines _equal _ratios, still
+  we do not know whether they are equal in the sense of the axiom, that
+  two things which are equal to a third are equal to one another. That
+  this is the case requires a proof, and until this proof is given we
+  shall use the :: instead of the sign = , which, however, we shall
+  afterwards introduce.
+
+  As soon as it has been established that all ratios are like
+  magnitudes, it becomes easy to show that, in some cases at least, they
+  are numbers. This step was never made by Greek mathematicians. They
+  distinguished always most carefully between continuous magnitudes and
+  the discrete series of numbers. In modern times it has become the
+  custom to ignore this difference.
+
+  If, in determining the ratio of two lines, a common measure can be
+  found, which is contained m times in the first, and n times in the
+  second, then the ratio of the two lines equals the ratio of the two
+  numbers m : n. This is shown by Euclid in Prop. 5, X. But the ratio of
+  two numbers is, as a rule, a fraction, and the Greeks did not, as we
+  do, consider fractions as numbers. Far less had they any notion of
+  introducing irrational numbers, which are neither whole nor
+  fractional, as we are obliged to do if we wish to say that all ratios
+  are numbers. The incommensurable numbers which are thus introduced as
+  ratios of incommensurable quantities are nowadays as familiar to us as
+  fractions; but a proof is generally omitted that we may apply to them
+  the rules which have been established for rational numbers only.
+  Euclid's treatment of ratios avoids this difficulty. His definitions
+  hold for commensurable as well as for incommensurable quantities. Even
+  the notion of incommensurable quantities is avoided in Book V. But he
+  proves that the more elementary rules of algebra hold for ratios. We
+  shall state all his propositions in that algebraical form to which we
+  are now accustomed. This may, of course, be done without changing the
+  character of Euclid's method.
+
+  §. 51. Using the notation explained above we express the first
+  propositions as follows:--
+
+  Prop. 1. If   a = ma', b = mb', c = mc',
+  then          a + b + c = m(a' + b' + c').
+
+  Prop. 2. If   a = mb, and c = md,
+                e = nb, and f = nd,
+
+  then a + e is the same multiple of b as c + f is of d, viz.:--
+
+    a + e = (m + n)b, and c + f = (m + n)d.
+
+  Prop. 3. If a = mb, c = md, then is na the same multiple of b that nc
+  is of d, viz. na = nmb, nc = nmd.
+
+  Prop. 4. If   a : b :: c : d,
+  then          ma : nb :: mc : nd.
+
+  Prop. 5. If   a = mb, and c = md,
+  then          a - c = m(b - d).
+
+  Prop. 6. If   a = mb, c = md,
+
+  then are a - nb and c - nd either equal to, or equimultiples of, b and
+  d, viz. a - nb = (m - n)b and c - nd = (m - n)d, where m - n may be
+  unity.
+
+  All these propositions relate to _equimultiples_. Now follow
+  propositions about ratios which are compared as to their magnitude.
+
+  § 52. Prop. 7. If a = b, then a : c :: b : c and c : a :: c : b.
+
+  The proof is simply this. As a = b we know that ma = mb; therefore
+
+    if   ma > nc, then mb > nc,
+    if   ma = nc, then mb = nc,
+    if   ma < nc, then mb < nc,
+
+  therefore the first proportion holds by Definition 5.
+
+  Prop. 8. If a > b, then a : c > b : c,
+  and                     c : a < c : b.
+
+  The proof depends on Definition 7.
+
+  Prop. 9 (converse to Prop. 7). If
+             a : c :: b : c,
+  or if      c : a :: c : b, then a = b.
+
+  Prop. 10 (converse to Prop. 8). If
+             a : c > b : c, then a > b,
+  and if     c : a < c : b, then a < b.
+
+  Prop. 11. If  a : b :: c : d,
+  and           a : b :: e : f,
+  then          c : d :: e : f.
+
+  In words, _if too ratios are equal to a third, they are equal to one
+  another_. After these propositions have been proved, we have a right
+  to consider a ratio as a _magnitude_, for only now can we consider a
+  ratio as something for which the axiom about magnitudes holds: things
+  which are equal to a third are equal to one another.
+
+  We shall indicate this by writing in future the sign = instead of ::.
+  The remaining propositions, which explain themselves, may then be
+  stated as follows:
+
+  § 53. Prop. 12. If  a : b = c : d = e : f,
+  then                a + c + e : b + d + f = a : b.
+
+  Prop. 13. If    a : b = c : d and c : d > e : f,
+  then            a : b > e : f.
+
+  Prop. 14. If    a : b = c : d, and a > c, then b > d.
+
+  Prop. 15. Magnitudes have the same ratio to one another that their
+  equimultiples have--
+
+  ma : mb = a : b.
+
+  Prop. 16. If a, b, c, d are magnitudes of the same kind, and if
+                  a : b = c : d,
+  then            a : c = b : d.
+
+  Prop. 17. If    a + b : b = c + d : d,
+  then            a : b = c : d.
+
+  Prop. 18 (converse to 17). If
+                  a : b = c : d
+  then            a + b : b = c + d : d.
+
+  Prop. 19. If a, b, c, d are quantities of the same kind, and if
+                  a : b = c : d,
+  then            a - c : b - d = a : b.
+
+  § 54. Prop. 20. _If there be three magnitudes, and another three,
+  which have the same ratio, taken two and two, then if the first be
+  greater than the third, the fourth shall be greater than the sixth:
+  and if equal, equal; and if less, less._
+
+  If we understand by
+
+    a : b : c : d : e : ... = a' : b' : c' : d' : e' : ...
+
+  that the ratio of any two consecutive magnitudes on the first side
+  equals that of the corresponding magnitudes on the second side, we may
+  write this theorem in symbols, thus:--
+
+  If a, b, c be quantities of one, and d, e, f magnitudes of the same or
+  any other kind, such that
+
+           a : b : c = d : e : f,
+  and if   a > c, then d > f,
+  but if   a = c, then d = f,
+  and if   a < c, then d < f.
+
+  Prop. 21. If  a : b = e : f and b : c = d : e,
+  or if         a : b : c = 1/f : 1/e : 1/d,
+  and if        a > c, then d > f,
+  but if        a = c, then d = f,
+  and if        a < c, then d < f.
+
+  By aid of these two propositions the following two are proved.
+
+  § 55. Prop. 22. _If there be any number of magnitudes, and as many
+  others, which have the same ratio, taken two and two in order, the
+  first shall have to the last of the first magnitudes the same ratio
+  which the first of the others has to the last._
+
+  We may state it more generally, thus:
+
+  If   a : b : c : d : e: ... = a' : b' : c' : d' : e' : ... ,
+
+  then not only have two consecutive, but any two magnitudes on the
+  first side, the same ratio as the corresponding magnitudes on the
+  other. For instance--
+
+    a : c = a' : c'; b : e = b' : e', &c.
+
+  Prop. 23 we state only in symbols, viz.:--
+
+  If  a : b : c : d : e : ... = 1/a' : 1/b' : 1/c' : 1/d' : 1/e' ...,
+
+  then  a : c = c' : a',
+        b : e = e' : b',
+
+  and so on.
+
+  Prop. 24 comes to this: If a : b = c : d and e : b = f : d, then
+
+    a + e : b = c + f : d.
+
+  Some of the proportions which are considered in the above propositions
+  have special names. These we have omitted, as being of no use, since
+  algebra has enabled us to bring the different operations contained in
+  the propositions under a common point of view.
+
+  § 56. The last proposition in the fifth book is of a different
+  character.
+
+  Prop. 25. _If four magnitudes of the same kind be proportional, the
+  greatest and least of them together shall be greater than the other
+  two together._ In symbols--
+
+  If a, b, c, d be magnitudes of the same kind, and if a : b = c : d,
+  and if a is the greatest, hence d the least, then a + d > b + c.
+
+  § 57. We return once again to the question. What is a ratio? We have
+  seen that we may treat ratios as magnitudes, and that all ratios are
+  magnitudes of the same kind, for we may compare any two as to their
+  magnitude. It will presently be shown that ratios of lines may be
+  considered as _quotients_ of lines, so that a ratio appears as answer
+  to the question, How often is one line contained in another? But the
+  answer to this question is given by a number, at least in some cases,
+  and in all cases if we admit incommensurable numbers. Considered from
+  this point of view, we may say the fifth book of the _Elements_ shows
+  that some of the simpler algebraical operations hold for
+  incommensurable numbers. In the ordinary algebraical treatment of
+  numbers this proof is altogether omitted, or given by a process of
+  limits which does not seem to be natural to the subject.
+
+
+  BOOK VI.
+
+  § 58. The sixth book contains the theory of similar figures. After a
+  few definitions explaining terms, the first proposition gives the
+  first application of the theory of proportion.
+
+  Prop. 1. _Triangles and parallelograms of the same altitude are to one
+  another as their bases._
+
+  The proof has already been considered in § 49.
+
+  From this follows easily the important theorem
+
+  Prop. 2. _If a straight line be drawn parallel to one of the sides of
+  a triangle it shall cut the other sides, or those sides produced,
+  proportionally; and if the sides or the sides produced be cut
+  proportionally, the straight line which joins the points of section
+  shall be parallel to the remaining side of the triangle._
+
+  § 59. The next proposition, together with one added by Simson as Prop.
+  A, may be expressed more conveniently if we introduce a modern
+  phraseology, viz. if in a line AB we assume a point C between A and B,
+  we shall say that C divides AB internally in the ratio AC : CB; but if
+  C be taken in the line AB produced, we shall say that AB is divided
+  externally in the ratio AC : CB.
+
+  The two propositions then come to this:
+
+  Prop. 3. _The bisector of an angle in a triangle divides the opposite
+  side internally in a ratio equal to the ratio of the two sides
+  including that angle;_ and conversely, _if a line through the vertex
+  of a triangle divide the base internally in the ratio of the two other
+  sides, then that line bisects the angle at the vertex_.
+
+  Simson's Prop. A. _The line which bisects an exterior angle of a
+  triangle divides the opposite side externally in the ratio of the
+  other sides;_ and conversely, _if a line through the vertex of a
+  triangle divide the base externally in the ratio of the sides, then it
+  bisects an exterior angle at the vertex of the triangle_.
+
+  If we combine both we have--
+
+  _The two lines which bisect the interior and exterior angles at one
+  vertex of a triangle divide the opposite side internally and
+  externally in the same ratio, viz. in the ratio of the other two
+  sides._
+
+  § 60. The next four propositions contain the theory of similar
+  triangles, of which four cases are considered. They may be stated
+  together.
+
+  _Two triangles are similar_,--
+
+  1. (Prop. 4). _If the triangles are equiangular:_
+
+  2. (Prop. 5). _If the sides of the one are proportional to those of
+  the other_;
+
+  3. (Prop. 6). _If two sides in one are proportional to two sides in
+  the other, and if the angles contained by these sides are equal_;
+
+  4. (Prop. 7). _If two sides in one are proportional to two sides in
+  the other, if the angles opposite homologous sides are equal, and if
+  the angles opposite the other homologous sides are both acute, both
+  right or both obtuse; homologous sides being in each case those which
+  are opposite equal angles_.
+
+  An important application of these theorems is at once made to a
+  right-angled triangle, viz.:--
+
+  Prop. 8. _In a right-angled triangle, if a perpendicular be drawn from
+  the right angle to the base, the triangles on each side of it are
+  similar to the whole triangle, and to one another_.
+
+  _Corollary._--From this it is manifest that the perpendicular drawn
+  from the right angle of a right-angled triangle to the base is a mean
+  proportional between the segments of the base, and also that each of
+  the sides is a mean proportional between the base and the segment of
+  the base adjacent to that side.
+
+  § 61. There follow four propositions containing problems, in language
+  slightly different from Euclid's, viz.:--
+
+  Prop. 9. _To divide a straight line into a given number of equal
+  parts_.
+
+  Prop. 10. _To divide a straight line in a given ratio_.
+
+  Prop. 11. _To find a third proportional to two given straight lines_.
+
+  Prop. 12. _To find a fourth proportional to three given straight
+  lines_.
+
+  Prop. 13. _To find a mean proportional between two given straight
+  lines_.
+
+  The last three may be written as equations with one unknown
+  quantity--viz. if we call the given straight lines a, b, c, and the
+  required line x, we have to find a line x so that
+
+  Prop. 11.  a : b = b : x;
+
+  Prop. 12.  a : b = c : x;
+
+  Prop. 13.  a : x = x : b.
+
+  We shall see presently how these may be written without the signs of
+  ratios.
+
+  § 62. Euclid considers next proportions connected with parallelograms
+  and triangles which are equal in area.
+
+  Prop. 14. _Equal parallelograms which have one angle of the one equal
+  to one angle of the other have their sides about the equal angles
+  reciprocally proportional; and parallelograms which have one angle of
+  the one equal to one angle of the other, and their sides about the
+  equal angles reciprocally proportional, are equal to one another_.
+
+  Prop. 15. _Equal triangles which have one angle of the one equal to
+  one angle of the other, have their sides about the equal angles
+  reciprocally proportional; and triangles which have one angle of the
+  one equal to one angle of the other, and their sides about the equal
+  angles reciprocally proportional, are equal to one another_.
+
+  [Illustration]
+
+  The latter proposition is really the same as the former, for if, as in
+  the accompanying diagram, in the figure belonging to the former the
+  two equal parallelograms AB and BC be bisected by the lines DF and EG,
+  and if EF be drawn, we get the figure belonging to the latter.
+
+  It is worth noticing that the lines FE and DG are parallel. We may
+  state therefore the theorem--
+
+  _If two triangles are equal in area, and have one angle in the one
+  vertically opposite to one angle in the other, then the two straight
+  lines which join the remaining two vertices of the one to those of the
+  other triangle are parallel_.
+
+  § 63. A most important theorem is
+
+  _Prop. 16. If four straight lines be proportionals, the rectangle
+  contained by the extremes is equal to the rectangle contained by the
+  means; and if the rectangle contained by the extremes be equal to the
+  rectangle contained by the means, the four straight lines are
+  proportionals_.
+
+  In symbols, if a, b, c, d are the four lines, and
+  if                  a : b = c : d,
+  then                ad = bc;
+  and conversely, if  ad = bc,
+  then                a : b = c : d,
+
+  where ad and bc denote (as in § 20), the areas of the rectangles
+  contained by a and d and by b and c respectively.
+
+  This allows us to transform every proportion between four lines into
+  an equation between two products.
+
+  It shows further that the operation of forming a product of two lines,
+  and the operation of forming their ratio are each the inverse of the
+  other.
+
+  If we now define a quotient a/b of two lines as the _number_ which
+  multiplied into b gives a, so that
+
+    a
+    -- b = a,
+    b
+
+  we see that from the equality of two quotients
+
+    a    c
+    -- = --
+    b    d
+
+  follows, if we multiply both sides by bd,
+
+    a        c
+    -- b·d = -- d·b,
+    b        d
+
+    ad = cb.
+
+  But from this it follows, according to the last theorem, that
+
+    a : b = c : d.
+
+  Hence we conclude that the quotient a/b and the ratio a : b are
+  different forms of the same magnitude, only with this important
+  difference that the quotient a/b would have a meaning only if a and b
+  have a common measure, until we introduce incommensurable numbers,
+  while the ratio a : b has always a meaning, and thus gives rise to the
+  introduction of incommensurable numbers.
+
+  Thus it is really the theory of ratios in the fifth book which enables
+  us to extend the geometrical calculus given before in connexion with
+  Book II. It will also be seen that if we write the ratios in Book V.
+  as quotients, or rather as fractions, then most of the theorems state
+  properties of quotients or of fractions.
+
+  § 64. Prop. 17. _If three straight lines are proportional the
+  rectangle contained by the extremes is equal to the square on the
+  mean;_ and conversely, is only a special case of 16. After the
+  problem, Prop. 18, _On a given straight line to describe a rectilineal
+  figure similar and similarly situated to a given rectilineal figure_,
+  there follows another fundamental theorem:
+
+  Prop. 19. _Similar triangles are to one another in the duplicate ratio
+  of their homologous sides._ In other words, the areas of similar
+  triangles are to one another as the squares on homologous sides. This
+  is generalized in:
+
+  Prop. 20. _Similar polygons may be divided into the same number of
+  similar triangles, having the same ratio to one another that the
+  polygons have; and the polygons are to one another in the duplicate
+  ratio of their homologous sides._
+
+  § 65. Prop. 21. _Rectilineal figures which are similar to the same
+  rectilineal figure are also similar to each other_, is an immediate
+  consequence of the definition of similar figures. As similar figures
+  may be said to be equal in "shape" but not in "size," we may state it
+  also thus:
+
+  "Figures which are equal in shape to a third are equal in shape to
+  each other."
+
+  Prop. 22. _If four straight lines be proportionals, the similar
+  rectilineal figures similarly described on them shall also be
+  proportionals; and if the similar rectilineal figures similarly
+  described on four straight lines be proportionals, those straight
+  lines shall be proportionals._
+
+  This is essentially the same as the following:--
+
+  _If_    a : b = c : d,
+  _then_  a² : b² = c² : d².
+
+  § 66. Now follows a proposition which has been much discussed with
+  regard to Euclid's exact meaning in saying that a ratio is
+  _compounded_ of two other ratios, viz.:
+
+  Prop. 23. _Parallelograms which are equiangular to one another, have
+  to one another the ratio which is compounded of the ratios of their
+  sides._
+
+  The proof of the proposition makes its meaning clear. In symbols the
+  ratio a : c is compounded of the two ratios a : b and b : c, and if a
+  : b = a' : b', b : c = b" : c", then a : c is compounded of a' : b'
+  and b" : c".
+
+  If we consider the ratios as numbers, we may say that the one ratio is
+  the product of those of which it is compounded, or in symbols,
+
+    a    a    b    a'   b"     a    a'     b    b"
+    -- = -- · -- = -- · --, if -- = -- and -- = --.
+    c    b    c    b'   c"     b    b'     c    c"
+
+  The theorem in Prop. 23 is the foundation of all mensuration of areas.
+  From it we see at once that two rectangles have the ratio of their
+  areas compounded of the ratios of their sides.
+
+  If A is the area of a rectangle contained by a and b, and B that of a
+  rectangle contained by c and d, so that A = ab, B = cd, then A : B =
+  ab : cd, and this is, the theorem says, compounded of the ratios a : c
+  and b : d. In forms of quotients,
+
+    a    b    ab
+    -- · -- = --.
+    c    d    cd
+
+  This shows how to multiply quotients in our geometrical calculus.
+
+  Further, _Two triangles have the ratios of their areas compounded of
+  the ratios of their bases and their altitude._ For a triangle is equal
+  in area to half a parallelogram which has the same base and the same
+  altitude.
+
+  § 67. To bring these theorems to the form in which they are usually
+  given, we assume a straight line u as our unit of length (generally an
+  inch, a foot, a mile, &c.), and determine the number [alpha] which
+  expresses how often u is contained in a line a, so that [alpha]
+  denotes the ratio a : u whether commensurable or not, and that a =
+  [alpha]u. We call this number [alpha] the numerical value of a. If in
+  the same manner ß be the numerical value of a line b we have
+
+    a : b = [alpha] : ß;
+
+  in words: _The ratio of two lines (and of two like quantities in
+  general) is equal to that of their numerical values._
+
+  This is easily proved by observing that a = [alpha]u, b = ßu,
+  therefore a : b = [alpha]u : ßu, and this may without difficulty be
+  shown to equal [alpha] : ß.
+
+  If now a, b be base and altitude of one, a', b' those of another
+  parallelogram, [alpha], ß and [alpha]', ß' their numerical values
+  respectively, and A, A' their areas, then
+
+    A    a    b    [alpha]    ß     [alpha]ß
+    -- = -- · -- = -------- · -- = ----------.
+    A'   a'   b'   [alpha]'   ß'   [alpha]'ß'
+
+  In words: _The areas of two parallelograms are to each other as the
+  products of the numerical values of their bases and altitudes._
+
+  If especially the second parallelogram is the unit square, i.e. a
+  square on the unit of length, then [alpha]' = ß' = 1, A' = u², and we
+  have
+
+    A
+    -- = [alpha]ß or A = [alpha]ß · u².
+    A'
+
+  This gives the theorem: The number of unit squares contained in a
+  parallelogram equals the product of the numerical values of base and
+  altitude, and similarly the number of unit squares contained in a
+  triangle equals half the product of the numerical values of base and
+  altitude.
+
+  This is often stated by saying that the area of a parallelogram is
+  equal to the product of the base and the altitude, meaning by this
+  product the product of the numerical values, and not the product as
+  defined above in § 20.
+
+  § 68. Propositions 24 and 26 relate to parallelograms about diagonals,
+  such as are considered in Book I., 43. They are--
+
+  Prop. 24. _Parallelograms about the diameter of any parallelogram are
+  similar to the whole parallelogram and to one another_; and its
+  converse (Prop. 26), _If two similar parallelograms have a common
+  angle, and be similarly situated, they are about the same diameter._
+
+  Between these is inserted a problem.
+
+  Prop. 25. _To describe a rectilineal figure which shall be similar to
+  one given rectilinear figure, and equal to another given rectilineal
+  figure_.
+
+  § 69. Prop. 27 contains a theorem relating to the theory of maxima and
+  minima. We may state it thus:
+
+  Prop. 27. _If a parallelogram be divided into two by a straight line
+  cutting the base, and if on half the base another parallelogram be
+  constructed similar to one of those parts, then this third
+  parallelogram is greater than the other part._
+
+  Of far greater interest than this general theorem is a special case of
+  it, where the parallelograms are changed into rectangles, and where
+  one of the parts into which the parallelogram is divided is made a
+  square; for then the theorem changes into one which is easily
+  recognized to be identical with the following:--
+
+  _Of all rectangles which have the same perimeter the square has the
+  greatest area._
+
+  This may also be stated thus:--
+
+  _Of all rectangles which have the same area the square has the least
+  perimeter._
+
+  § 70. The next three propositions contain problems which may be said
+  to be solutions of quadratic equations. The first two are, like the
+  last, involved in somewhat obscure language. We transcribe them as
+  follows:
+
+  _Problem_.--To describe on a given base a parallelogram, and to divide
+  it either internally (Prop. 28) or externally (Prop. 29) from a point
+  on the base into two parallelograms, of which the one has a given size
+  (is equal in area to a given figure), whilst the other has a given
+  shape (is similar to a given parallelogram).
+
+  If we express this again in symbols, calling the given base a, the one
+  part x, and the altitude y, we have to determine x and y in the first
+  case from the equations
+
+    (a - x)y = k²,
+
+    x    p
+    -- = --,
+    y    q
+
+  k² being the given size of the first, and p and q the base and
+  altitude of the parallelogram which determine the shape of the second
+  of the required parallelograms.
+
+  If we substitute the value of y, we get
+
+               pk²
+    (a - x)x = ---,
+                q
+
+  or,
+
+    ax - x² = b²,
+
+  where a and b are known quantities, taking b² = pk²/q.
+
+  The second case (Prop. 29) gives rise, in the same manner, to the
+  quadratic
+
+    ax + x² = b².
+
+  The next problem--
+
+  Prop. 30. _To cut a given straight line in extreme and mean ratio_,
+  leads to the equation
+
+    ax + x² = a².
+
+  This is, therefore, only a special case of the last, and is, besides,
+  an old acquaintance, being essentially the same problem as that
+  proposed in II. 11.
+
+  Prop. 30 may therefore be solved in two ways, either by aid of Prop.
+  29 or by aid of II. 11. Euclid gives both solutions.
+
+  § 71. Prop. 31 (Theorem). _In any right-angled triangle, any
+  rectilineal figure described on the side subtending the right angle is
+  equal to the similar and similarly-described figures on the sides
+  containing the right angle_,--is a pretty generalization of the
+  theorem of Pythagoras (I. 47).
+
+  Leaving out the next proposition, which is of little interest, we come
+  to the last in this book.
+
+  Prop. 33. _In equal circles angles, whether at the centres or the
+  circumferences, have the same ratio which the arcs on which they stand
+  have to one another; so also have the sectors_.
+
+  Of this, the part relating to angles at the centre is of special
+  importance; it enables us to measure angles by arcs.
+
+  With this closes that part of the _Elements_ which is devoted to the
+  study of figures in a plane.
+
+
+  BOOK XI.
+
+  § 72. In this book figures are considered which are not confined to a
+  plane, viz. first relations between lines and planes in space, and
+  afterwards properties of solids.
+
+  Of new definitions we mention those which relate to the
+  perpendicularity and the inclination of lines and planes.
+
+  Def. 3. _A straight line is perpendicular, or at right angles, to a
+  plane when it makes right angles with every straight line meeting it
+  in that plane_.
+
+  The definition of perpendicular planes (Def. 4) offers no difficulty.
+  Euclid defines the inclination of lines to planes and of planes to
+  planes (Defs. 5 and 6) by aid of plane angles, included by straight
+  lines, with which we have been made familiar in the first books.
+
+  The other important definitions are those of parallel planes, which
+  never meet (Def. 8), and of solid angles formed by three or more
+  planes meeting in a point (Def. 9).
+
+  To these we add the definition of a line parallel to a plane as a line
+  which does not meet the plane.
+
+  § 73. Before we investigate the contents of Book XI., it will be well
+  to recapitulate shortly what we know of planes and lines from the
+  definitions and axioms of the first book. There a plane has been
+  defined as a surface which has the property that every straight line
+  which joins two points in it lies altogether in it. This is equivalent
+  to saying that a straight line which has two points in a plane has all
+  points in the plane. Hence, a straight line which does not lie in the
+  plane cannot have more than one point in common with the plane. This
+  is virtually the same as Euclid's Prop. 1, viz.:--
+
+  Prop. 1. _One part of a straight line cannot be in a plane and another
+  part without it_.
+
+  It also follows, as was pointed out in § 3, in discussing the
+  definitions of Book I., that a plane is determined already by one
+  straight line and a point without it, viz. if all lines be drawn
+  through the point, and cutting the line, they will form a plane.
+
+  This may be stated thus:--
+
+  _A plane is determined_--
+
+  1st, _By a straight line and a point which does not lie on it;_
+
+  2nd, _By three points which do not lie in a straight line_; for if two
+  of these points be joined by a straight line we have case 1;
+
+  3rd, _By two intersecting straight lines_; for the point of
+  intersection and two other points, one in each line, give case 2;
+
+  4th, _By two parallel lines_ (Def. 35, I.).
+
+  The third case of this theorem is Euclid's
+
+  Prop. 2. _Two straight lines which cut one another are in one plane,
+  and three straight lines which meet one another are in one plane_.
+
+  And the fourth is Euclid's
+
+  Prop. 7. _If two straight lines be parallel, the straight line drawn
+  from any point in one to any point in the other is in the same plane
+  with the parallels_. From the definition of a plane further follows
+
+  Prop. 3. _If two planes cut one another, their common section is a
+  straight line_.
+
+  § 74. Whilst these propositions are virtually contained in the
+  definition of a plane, the next gives us a new and fundamental
+  property of space, showing at the same time that it is possible to
+  have a straight line perpendicular to a plane, according to Def. 3. It
+  states--
+
+  Prop. 4. _If a straight line is perpendicular to two straight lines in
+  a plane which it meets, then it is perpendicular to all lines in the
+  plane which it meets, and hence it is perpendicular to the plane_.
+
+  Def. 3 may be stated thus: If a straight line is perpendicular to a
+  plane, then it is perpendicular to every line in the plane which it
+  meets. The converse to this would be
+
+  _All straight lines which meet a given straight line in the same
+  point, and are perpendicular to it, lie in a plane which is
+  perpendicular to that line_.
+
+  This Euclid states thus:
+
+  Prop. 5. _If three straight lines meet all at one point, and a
+  straight line stands at right angles to each of them at that point,
+  the three straight lines shall be in one and the same plane_.
+
+  § 75. There follow theorems relating to the theory of parallel lines
+  in space, viz.:--
+
+  Prop. 6. _Any two lines which are perpendicular to the same plane are
+  parallel to each other;_ and conversely
+
+  Prop. 8. _If of two parallel straight lines one is perpendicular to a
+  plane, the other is so also._
+
+  Prop. 7. _If two straight lines are parallel, the straight line which
+  joins any point in one to any point in the other is in the same plane
+  as the parallels._ (See above, § 73.)
+
+  Prop. 9. _Two straight lines which are each of them parallel to the
+  same straight line, and not in the same plane with it, are parallel to
+  one another;_ where the words, "and not in the same plane with it,"
+  may be omitted, for they exclude the case of three parallels in a
+  plane, which has been proved before; and
+
+  Prop. 10. _If two angles in different planes have the two limits of
+  the one parallel to those of the other, then the angles are equal._
+  That their planes are parallel is shown later on in Prop. 15.
+
+  This theorem is not necessarily true, for the angles in question may
+  be supplementary; but then the one angle will be equal to that which
+  is adjacent and supplementary to the other, and this latter angle will
+  also have its limits parallel to those of the first.
+
+  From this theorem it follows that if we take any two straight lines in
+  space which do not meet, and if we draw through any point P in space
+  two lines parallel to them, then the angle included by these lines
+  will always be the same, whatever the position of the point P may be.
+  This angle has in modern times been called the angle between the given
+  lines:--
+
+  _By the angles between two not intersecting lines we understand the
+  angles which two intersecting lines include that are parallel
+  respectively to the two given lines._
+
+  § 76. It is now possible to solve the following two problems:--
+
+  _To draw a straight line perpendicular to a given plane from a given
+  point which lies_
+
+  1. _Not in the plane_ (Prop. 11).
+
+  2. _In the plane_ (Prop. 12).
+
+  The second case is easily reduced to the first--viz. if by aid of the
+  first we have drawn any perpendicular to the plane from some point
+  without it, we need only draw through the given point in the plane a
+  line parallel to it, in order to have the required perpendicular
+  given. The solution of the first part is of interest in itself. It
+  depends upon a construction which may be expressed as a theorem.
+
+  _If from a point A without a plane a perpendicular AB be drawn to the
+  plane, and if from the foot B of this perpendicular another
+  perpendicular BC be drawn to any straight line in the plane, then the
+  straight line joining A to the foot C of this second perpendicular
+  will also be perpendicular to the line in the plane._
+
+  The theory of perpendiculars to a plane is concluded by the theorem--
+
+  Prop. 13. _Through any point in space, whether in or without a plane,
+  only one straight line can be drawn perpendicular to the plane._
+
+  § 77. The next four propositions treat of parallel planes. It is shown
+  _that planes which have a common perpendicular are parallel_ (Prop.
+  14); _that two planes are parallel if two intersecting straight lines
+  in the one are parallel respectively to two straight lines in the
+  other plane_ (Prop. 15); _that parallel planes are cut by any plane in
+  parallel straight lines_ (Prop. 16); and lastly, _that any two
+  straight lines are cut proportionally by a series of parallel planes_
+  (Prop. 17).
+
+  This theory is made more complete by adding the following theorems,
+  which are easy deductions from the last: _Two parallel planes have
+  common perpendiculars_ (converse to 14); and _Two planes which are
+  parallel to a third plane are parallel to each other._
+
+  It will be noted that Prop. 15 at once allows of the solution of the
+  problem: "Through a given point to draw a plane parallel to a given
+  plane." And it is also easily proved that this problem allows always
+  of one, and only of one, solution.
+
+  § 78. We come now to planes which are perpendicular to one another.
+  Two theorems relate to them.
+
+  Prop. 18. _If a straight line be at right angles to a plane, every
+  plane which passes through it shall be at right angles to that plane._
+
+  Prop. 19. _If two planes which cut one another be each of them
+  perpendicular to a third plane, their common section shall be
+  perpendicular to the same plane._
+
+  § 79. If three planes pass through a common point, and if they bound
+  each other, a solid angle of three faces, or a _trihedral_ angle, is
+  formed, and similarly by more planes a solid angle of more faces, or a
+  _polyhedral_ angle. These have many properties which are quite
+  analogous to those of triangles and polygons in a plane. Euclid states
+  some, viz.:--
+
+  Prop. 20. _If a solid angle be contained by three plane angles, any
+  two of them are together greater than the third._
+
+  But the next--
+
+  Prop. 21. _Every solid angle is contained by plane angles, which are
+  together less than four right angles_--has no analogous theorem in the
+  plane.
+
+  We may mention, however, that the theorems about triangles contained
+  in the propositions of Book I., which do not depend upon the theory of
+  parallels (that is all up to Prop. 27), have their corresponding
+  theorems about trihedral angles. The latter are formed, if for "side
+  of a triangle" we write "plane angle" or "face" of trihedral angle,
+  and for "angle of triangle" we substitute "angle between two faces"
+  where the planes containing the solid angle are called its _faces_. We
+  get, for instance, from I. 4, the theorem, _If two trihedral angles
+  have the angles of two faces in the one equal to the angles of two
+  faces in the other, and have likewise the angles included by these
+  faces equal, then the angles in the remaining faces are equal, and the
+  angles between the other faces are equal each to each, viz. those
+  which are opposite equal faces._ The solid angles themselves are not
+  necessarily equal, for they may be only symmetrical like the right
+  hand and the left.
+
+  The connexion indicated between triangles and trihedral angles will
+  also be recognized in
+
+  Prop. 22. _If every two of three plane angles be greater than the
+  third, and if the straight lines which contain them be all equal, a
+  triangle may be made of the straight lines that join the extremities
+  of those equal straight lines._
+
+  And Prop. 23 solves the problem, _To construct a trihedral angle
+  having the angles of its faces equal to three given plane angles, any
+  two of them being greater than the third._ It is, of course, analogous
+  to the problem of constructing a triangle having its sides of given
+  length.
+
+  Two other theorems of this kind are added by Simson in his edition of
+  Euclid's _Elements_.
+
+  § 80. These are the principal properties of lines and planes in space,
+  but before we go on to their applications it will be well to define
+  the word _distance_. In geometry distance means always "shortest
+  distance"; viz. the distance of a point from a straight line, or from
+  a plane, is the length of the perpendicular from the point to the line
+  or plane. The distance between two non-intersecting lines is the
+  length of their common perpendicular, there being but one. The
+  distance between two parallel lines or between two parallel planes is
+  the length of the common perpendicular between the lines or the
+  planes.
+
+  § 81. _Parallelepipeds_.--The rest of the book is devoted to the study
+  of the parallelepiped. In Prop. 24 the possibility of such a solid is
+  proved, viz.:--
+
+  Prop. 24. _If a solid be contained by six planes two and two of which
+  are parallel, the opposite planes are similar and equal
+  parallelograms._
+
+  Euclid calls this solid henceforth a parallelepiped, though he never
+  defines the word. Either face of it may be taken as _base_, and its
+  distance from the opposite face as _altitude_.
+
+  Prop. 25. _If a solid parallelepiped be cut by a plane parallel to two
+  of its opposite planes, it divides the whole into two solids, the base
+  of one of which shall be to the base of the other as the one solid is
+  to the other_.
+
+  This theorem corresponds to the theorem (VI. 1) that parallelograms
+  between the same parallels are to one another as their bases. A
+  similar analogy is to be observed among a number of the remaining
+  propositions.
+
+  § 82. After solving a few problems we come to
+
+  Prop. 28. _If a solid parallelepiped be cut by a plane passing through
+  the diagonals of two of the opposite planes, it shall be cut in two
+  equal parts._
+
+  In the proof of this, as of several other propositions, Euclid
+  neglects the difference between solids which are symmetrical like the
+  right hand and the left.
+
+  Prop. 31. _Solid parallelepipeds, which are upon equal bases, and of
+  the same altitude, are equal to one another._
+
+  Props. 29 and 30 contain special cases of this theorem leading up to
+  the proof of the general theorem.
+
+  As consequences of this fundamental theorem we get
+
+  Prop. 32. _Solid parallelepipeds, which have the same altitude, are to
+  one another as their bases;_ and
+
+  Prop. 33. _Similar solid parallelepipeds are to one another in the
+  triplicate ratio of their homologous sides._
+
+  If we consider, as in § 67, the ratios of lines as numbers, we may
+  also say--
+
+  _The ratio of the volumes of similar parallelepipeds is equal to the
+  ratio of the third powers of homologous sides._
+
+  Parallelepipeds which are not similar but equal are compared by aid of
+  the theorem
+
+  Prop. 34. _The bases and altitudes of equal solid parallelepipeds are
+  reciprocally proportional; and if the bases and altitudes be
+  reciprocally proportional, the solid parallelepipeds are equal._
+
+  § 83. Of the following propositions the 37th and 40th are of special
+  interest.
+
+  Prop. 37. _If four straight lines be proportionals, the similar solid
+  parallelepipeds, similarly described from them, shall also be
+  proportionals; and if the similar parallelepipeds similarly described
+  from four straight lines be proportionals, the straight lines shall be
+  proportionals._
+
+  In symbols it says--
+
+    If a : b = c : d, then a³ : b³ = c³ : d³.
+
+  Prop. 40 teaches how to compare the volumes of triangular prisms with
+  those of parallelepipeds, by proving _that a triangular prism is equal
+  in volume to a parallelepiped, which has its altitude and its base
+  equal to the altitude and the base of the triangular prism._
+
+  § 84. From these propositions follow all results relating to the
+  mensuration of volumes. We shall state these as we did in the case of
+  areas. The starting-point is the "rectangular" parallelepiped, which
+  has every edge perpendicular to the planes it meets, and which takes
+  the place of the rectangle in the plane. If this has all its edges
+  equal we obtain the "cube."
+
+  If we take a certain line u as unit length, then the square on u is
+  the unit of area, and the cube on u the unit of volume, that is to
+  say, if we wish to measure a volume we have to determine how many unit
+  cubes it contains.
+
+  A rectangular parallelepiped has, as a rule, the three edges unequal,
+  which meet at a point. Every other edge is equal to one of them. If a,
+  b, c be the three edges meeting at a point, then we may take the
+  rectangle contained by two of them, say by b and c, as base and the
+  third as altitude. Let V be its volume, V' that of another rectangular
+  parallelepiped which has the edges a', b, c, hence the same base as
+  the first. It follows then easily, from Prop. 25 or 32, that V : V' =
+  a : a'; or in words,
+
+  _Rectangular parallelepipeds on equal bases are proportional to their
+  altitudes._
+
+  If we have two rectangular parallelepipeds, of which the first has the
+  volume V and the edges a, b, c, and the second, the volume V' and the
+  edges a', b', c', we may compare them by aid of two new ones which
+  have respectively the edges a', b, c and a', b', c, and the volumes V1
+  and V2. We then have
+
+    V : V1 = a : a'; V1 : V2 = b : b', V2 : V' = c : c'.
+
+  Compounding these, we have
+
+    V : V' = (a : a')(b : b')(c : c'),
+
+  or
+
+    V    a    b    c
+    -- = -- · -- · --.
+    V'   a'   b'   c'
+
+  Hence, as a special case, making V' equal to the unit cube U on u we
+  get
+
+    V    a    b    c
+    -- = -- · -- · -- = [alpha]·ß·[gamma],
+    U    u    u    u
+
+  where [alpha], ß, [gamma] are the numerical values of a, b, c; that
+  is, _The number of unit cubes in a rectangular parallelepiped_ is
+  equal to the product of the numerical values of its three edges. This
+  is generally expressed by saying the volume of a rectangular
+  parallelepiped is measured by the product of its sides, or by the
+  product of its base into its altitude, which in this case is the same.
+
+  Prop. 31 allows us to extend this to any parallelepipeds, and Props.
+  28 or 40, to triangular prisms.
+
+  _The volume of any parallelepiped, or of any triangular prism, is
+  measured by the product of base and altitude._
+
+  The consideration that any polygonal prism may be divided into a
+  number of triangular prisms, which have the same altitude and the sum
+  of their bases equal to the base of the polygonal prism, shows further
+  that the same holds for any prism whatever.
+
+
+  BOOK XII.
+
+  § 85. In the last part of Book XI. we have learnt how to compare the
+  volumes of parallelepipeds and of prisms. In order to determine the
+  volume of any solid bounded by plane faces we must determine the
+  volume of pyramids, for every such solid may be decomposed into a
+  number of pyramids.
+
+  As every pyramid may again be decomposed into triangular pyramids, it
+  becomes only necessary to determine their volume. This is done by the
+
+  _Theorem._--Every triangular pyramid is equal in volume to one third
+  of a triangular prism having the same base and the same altitude as
+  the pyramid.
+
+  This is an immediate consequence of Euclid's
+
+  Prop. 7. _Every prism having a triangular base may be divided into
+  three pyramids that have triangular bases, and are equal to one
+  another._
+
+  The proof of this theorem is difficult, because the three triangular
+  pyramids into which the prism is divided are by no means equal in
+  shape, and cannot be made to coincide. It has first to be proved that
+  two triangular pyramids have equal volumes, if they have equal bases
+  and equal altitudes. This Euclid does in the following manner. He
+  first shows (Prop. 3) that a triangular pyramid may be divided into
+  four parts, of which two are equal triangular pyramids similar to the
+  whole pyramid, whilst the other two are equal triangular prisms, and
+  further, that these two prisms together are greater than the two
+  pyramids, hence more than half the given pyramid. He next shows (Prop.
+  4) that if two triangular pyramids are given, having equal bases and
+  equal altitudes, and if each be divided as above, then the two
+  triangular prisms in the one are equal to those in the other, and each
+  of the remaining pyramids in the one has its base and altitude equal
+  to the base and altitude of the remaining pyramids in the other. Hence
+  to these pyramids the same process is again applicable. We are thus
+  enabled to cut out of the two given pyramids equal parts, each greater
+  than half the original pyramid. Of the remainder we can again cut out
+  equal parts greater than half these remainders, and so on as far as we
+  like. This process may be continued till the last remainder is smaller
+  than any assignable quantity, however small. It follows, so we should
+  conclude at present, that the two volumes must be equal, for they
+  cannot differ by any assignable quantity.
+
+  To Greek mathematicians this conclusion offers far greater
+  difficulties. They prove elaborately, by a _reductio ad absurdum_,
+  that the volumes cannot be unequal. This proof must be read in the
+  _Elements._ We must, however, state that we have in the above not
+  proved Euclid's Prop. 5, but only a special case of it. Euclid does
+  not suppose that the bases of the two pyramids to be compared are
+  equal, and hence he proves that the volumes are as the bases. The
+  reasoning of the proof becomes clearer in the special case, from which
+  the general one may be easily deduced.
+
+  § 86. Prop. 6 extends the result to pyramids with polygonal bases.
+  From these results follow again the rules at present given for the
+  mensuration of solids, viz. a pyramid is the third part of a
+  triangular prism having the same base and the same altitude. But a
+  triangular prism is equal in volume to a parallelepiped which has the
+  same base and altitude. Hence if B is the base and h the altitude, we
+  have
+
+    Volume of prism   = Bh,
+    Volume of pyramid = 1/3Bh,
+
+  statements which have to be taken in the sense that B means the number
+  of square units in the base, h the number of units of length in the
+  altitude, or that B and h denote the numerical values of base and
+  altitude.
+
+  § 87. A method similar to that used in proving Prop. 5 leads to the
+  following results relating to solids bounded by simple curved
+  surfaces:--
+
+  Prop. 10. _Every cone is the third part of a cylinder which has the
+  same base, and is of an equal altitude with it._
+
+  Prop. 11. _Cones or cylinders of the same altitude are to one another
+  as their bases._
+
+  Prop. 12. _Similar cones or cylinders have to one another the
+  triplicate ratio of that which the diameters of their bases have._
+
+  Prop. 13. _If a cylinder be cut by a plane parallel to its opposite
+  planes or bases, it divides the cylinder into two cylinders, one of
+  which is to the other as the axis of the first to the axis of the
+  other;_ which may also be stated thus:--
+
+  _Cylinders on the same base are proportional to their altitudes._
+
+  Prop. 14. _Cones or cylinders upon equal bases are to one another as
+  their altitudes._
+
+  Prop. 15. _The bases and altitudes of equal cones or cylinders are
+  reciprocally proportional, and if the bases and altitudes be
+  reciprocally proportional, the cones or cylinders are equal to one
+  another._
+
+  These theorems again lead to formulae in mensuration, if we compare a
+  cylinder with a prism having its base and altitude equal to the base
+  and altitude of the cylinder. This may be done by the method of
+  exhaustion. We get, then, the result that their bases are equal, and
+  have, if B denotes the numerical value of the base, and h that of the
+  altitude,
+
+    Volume of cylinder = Bh,
+    Volume of cone     = 1/3Bh.
+
+  § 88. The remaining propositions relate to circles and spheres. Of the
+  sphere only one property is proved, viz.:--
+
+  Prop. 18. _Spheres have to one another the triplicate ratio of that
+  which their diameters have._ The mensuration of the sphere, like that
+  of the circle, the cylinder and the cone, had not been settled in the
+  time of Euclid. It was done by Archimedes.
+
+
+  BOOK XIII.
+
+  § 89. The 13th and last book of Euclid's _Elements_ is devoted to the
+  regular solids (see POLYHEDRON). It is shown that there are five of
+  them, viz.:--
+
+  1. The regular _tetrahedron_, with 4 triangular faces and 4 vertices;
+
+  2. The _cube_, with 8 vertices and 6 square faces;
+
+  3. The _octahedron_, with 6 vertices and 8 triangular faces;
+
+  4. The _dodecahedron_, with 12 pentagonal faces, 3 at each of the
+  20 vertices;
+
+  5. The _icosahedron_, with 20 triangular faces, 5 at each of the
+  12 vertices.
+
+  It is shown how to inscribe these solids in a given sphere, and how to
+  determine the lengths of their edges.
+
+  § 90. The 13th book, and therefore the _Elements_, conclude with the
+  scholium, "that no other regular solid exists besides the five ones
+  enumerated."
+
+  The proof is very simple. Each face is a regular polygon, hence the
+  angles of the faces at any vertex must be angles in equal regular
+  polygons, must be together less than four right angles (XI. 21), and
+  must be three or more in number. Each angle in a regular triangle
+  equals two-thirds of one right angle. Hence it is possible to form a
+  solid angle with three, four or five regular triangles or faces. These
+  give the solid angles of the tetrahedron, the octahedron and the
+  icosahedron. The angle in a square (the regular quadrilateral) equals
+  one right angle. Hence three will form a solid angle, that of the
+  cube, and four will not. The angle in the regular pentagon equals 6/5
+  of a right angle. Hence three of them equal 18/5 (i.e. less than 4)
+  right angles, and form the solid angle of the dodecahedron. Three
+  regular polygons of six or more sides cannot form a solid angle.
+  Therefore no other regular solids are possible.     (O. H.)
+
+
+II. PROJECTIVE GEOMETRY
+
+It is difficult, at the outset, to characterize projective geometry as
+compared with Euclidean. But a few examples will at least indicate the
+practical differences between the two.
+
+In Euclid's _Elements_ almost all propositions refer to the _magnitude_
+of lines, angles, areas or volumes, and therefore to measurement. The
+statement that an angle is right, or that two straight lines are
+parallel, refers to measurement. On the other hand, the fact that a
+straight line does or does not cut a circle is independent of
+measurement, it being dependent only upon the mutual "position" of the
+line and the circle. This difference becomes clearer if we project any
+figure from one plane to another (see PROJECTION). By this the length of
+lines, the magnitude of angles and areas, is altered, so that the
+projection, or shadow, of a square on a plane will not be a square; it
+will, however, be some quadrilateral. Again, the projection of a circle
+will not be a circle, but some other curve more or less resembling a
+circle. But one property may be stated at once--no straight line can cut
+the projection of a circle in more than two points, because no straight
+line can cut a circle in more than two points. There are, then, some
+properties of figures which do not alter by projection, whilst others
+do. To the latter belong nearly all properties relating to measurement,
+at least in the form in which they are generally given. The others are
+said to be projective properties, and their investigation forms the
+subject of projective geometry.
+
+Different as are the kinds of properties investigated in the old and the
+new sciences, the methods followed differ in a still greater degree. In
+Euclid each proposition stands by itself; its connexion with others is
+never indicated; the leading ideas contained in its proof are not
+stated; general principles do not exist. In the modern methods, on the
+other hand, the greatest importance is attached to the leading thoughts
+which pervade the whole; and general principles, which bring whole
+groups of theorems under one aspect, are given rather than separate
+propositions. The whole tendency is towards generalization. A straight
+line is considered as given in its entirety, extending both ways to
+infinity, while Euclid never admits anything but finite quantities. The
+treatment of the infinite is in fact another fundamental difference
+between the two methods: Euclid avoids it; in modern geometry it is
+systematically introduced.
+
+Of the different modern methods of geometry, we shall treat principally
+of the methods of projection and correspondence which have proved to be
+the most powerful. These have become independent of Euclidean Geometry,
+especially through the _Geometrie der Lage_ of V. Staudt and the
+_Ausdehnungslehre_ of Grassmann.
+
+For the sake of brevity we shall presuppose a knowledge of Euclid's
+_Elements_, although we shall use only a few of his propositions.
+
+  § 1. _Geometrical Elements._ We consider space as filled with points,
+  lines and planes, and these we call the elements out of which our
+  figures are to be formed, calling any combination of these elements a
+  "figure."
+
+  By a line we mean a straight line in its entirety, extending both ways
+  to infinity; and by a plane, a plane surface, extending in all
+  directions to infinity.
+
+  We accept the three-dimensional space of experience--the space assumed
+  by Euclid--which has for its properties (among others):--
+
+  Through any two points in space one and only one line may be drawn;
+
+  Through any three points which are not in a line, one and only one
+  plane may be placed;
+
+  The intersection of two planes is a line;
+
+  A line which has two points in common with a plane lies in the plane,
+  hence the intersection of a line and a plane is a single point; and
+
+  Three planes which do not meet in a line have one single point in
+  common.
+
+  These results may be stated differently in the following form:--
+
+  I. A plane is determined--           A point is determined--
+       1. By three points which do         1. By three planes which do
+           not lie in a line;                  not pass through a line;
+       2. By two intersecting lines;       2. By two intersecting lines;
+       3. By a line and a point            3. By a plane and a line
+           which does not lie in it.           which does not lie in it.
+  II. A line is determined--
+       1. By two points;                   2. By two planes.
+
+  It will be observed that not only are planes determined by points, but
+  also points by planes; that therefore the planes may be considered as
+  elements, like points; and also that in any one of the above
+  statements we may interchange the words point and plane, and we obtain
+  again a correct statement, provided that these statements themselves
+  are true. As they stand, we ought, in several cases, to add "if they
+  are not parallel," or some such words, parallel lines and planes being
+  evidently left altogether out of consideration. To correct this we
+  have to reconsider the theory of parallels.
+
+  [Illustration: FIG. 1.]
+
+  § 2. _Parallels. Point at Infinity._--Let us take in a plane a line p
+  (fig. 1), a point S not in this line, and a line q drawn through S.
+  Then this line q will meet the line p in a point A. If we turn the
+  line q about S towards q', its point of intersection with p will move
+  along p towards B, passing, on continued turning, to a greater and
+  greater distance, until it is moved out of our reach. If we turn q
+  still farther, its continuation will meet p, but now at the other side
+  of A. The point of intersection has disappeared to the right and
+  reappeared to the left. There is one intermediate position where q is
+  parallel to p--that is where it does not cut p. In every other
+  position it cuts p in some finite point. If, on the other hand, we
+  move the point A to an infinite distance in p, then the line q which
+  passes through A will be a line which does not cut p at any finite
+  point. Thus we are led to say: _Every_ line through S which joins it
+  to any point at an infinite distance in p is parallel to p. But by
+  Euclid's 12th axiom there is but one line parallel to p through S. The
+  difficulty in which we are thus involved is due to the fact that we
+  try to reason about infinity as if we, with our finite capabilities,
+  could comprehend the infinite. To overcome this difficulty, we may say
+  that all points at infinity in a line _appear_ to us as one, and may
+  be replaced by a single "ideal" point.
+
+  We may therefore now give the following definitions and axiom:--
+
+  _Definition._--Lines which meet at infinity are called parallel.
+
+  _Axiom._--All points at an infinite distance in a line may be
+  considered as one single point.
+
+  _Definition._--This ideal point is called the _point at infinity_ in
+  the line.
+
+  The axiom is equivalent to Euclid's Axiom 12, for it follows from
+  either that through any point only one line may be drawn parallel to a
+  given line.
+
+  This point at infinity in a line is reached whether we move a point in
+  the one or in the opposite direction of a line to infinity. A line
+  thus appears closed by this point, and we speak as if we could move a
+  point along the line from one position A to another B in two ways,
+  either through the point at infinity or through finite points only.
+
+  It must never be forgotten that this point at infinity is ideal; in
+  fact, the whole notion of "infinity" is only a mathematical
+  conception, and owes its introduction (as a method of research) to the
+  working generalizations which it permits.
+
+  § 3. _Line and Plane at Infinity._--Having arrived at the notion of
+  replacing all points at infinity in a line by one ideal point, there
+  is no difficulty in replacing all points at infinity in a plane by one
+  ideal line.
+
+  To make this clear, let us suppose that a line p, which cuts two fixed
+  lines a and b in the points A and B, moves parallel to itself to a
+  greater and greater distance. It will at last cut both a and b at
+  their points at infinity, so that a line which joins the two points at
+  infinity in two intersecting lines lies altogether at infinity. Every
+  other line in the plane will meet it therefore at infinity, and thus
+  it contains all points at infinity in the plane.
+
+  _All points at infinity in a plane lie in a line, which is called the_
+  line at infinity _in the plane._
+
+  It follows that parallel planes must be considered as planes having a
+  common line at infinity, for any other plane cuts them in parallel
+  lines which have a point at infinity in common.
+
+  If we next take two intersecting planes, then the point at infinity in
+  their line of intersection lies in both planes, so that their lines at
+  infinity meet. Hence every line at infinity meets every other line at
+  infinity, and they are therefore all in one plane.
+
+  _All points at infinity in space may be considered as lying in one
+  ideal plane, which is called the_ plane at infinity.
+
+  § 4. _Parallelism._--We have now the following definitions:--
+
+  Parallel lines are lines which meet at infinity;
+
+  Parallel planes are planes which meet at infinity;
+
+  A line is parallel to a plane if it meets it at infinity.
+
+  Theorems like this--Lines (or planes) which are parallel to a third
+  are parallel to each other--follow at once.
+
+  This view of parallels leads therefore to no contradiction of Euclid's
+  _Elements._
+
+  As immediate consequences we get the propositions:--
+
+  Every line meets a plane in one point, or it lies in it;
+
+  Every plane meets every other plane in a line;
+
+  Any two lines in the same plane meet.
+
+  § 5. _Aggregates of Geometrical Elements._--We have called points,
+  lines and planes the elements of geometrical figures. We also say that
+  an element of one kind contains one of the other if it lies in it or
+  passes through it.
+
+  All the elements of one kind which are contained in one or two
+  elements of a different kind form aggregates which have to be
+  enumerated. They are the following:--
+
+  I. Of one dimension.
+
+     1. The _row_, or range, _of points_ formed by all points in a line,
+          which is called its base.
+
+     2. The _flat pencil_ formed by all the lines through a point in a
+          plane. Its base is the point in the plane.
+
+     3. The _axial pencil_ formed by all planes through a line which is
+          called its base or axis.
+
+  II. Of two dimensions.
+
+     1. The field of points and lines--that is, a plane with all its
+          points and all its lines.
+
+     2. The pencil of lines and planes--that is, a point in space with
+          all lines and all planes through it.
+
+  III. Of three dimensions.
+
+     The space of points--that is, all points in space.
+
+     The space of planes--that is, all planes in space.
+
+  IV. Of four dimensions.
+
+     The space of lines, or all lines in space.
+
+  § 6. _Meaning of "Dimensions."_--The word dimension in the above needs
+  explanation. If in a plane we take a row p and a pencil with centre Q,
+  then through every point in p one line in the pencil will pass, and
+  every ray in Q will cut p in one point, so that we are entitled to say
+  a row contains as many points as a flat pencil lines, and, we may add,
+  as an axial pencil planes, because an axial pencil is cut by a plane
+  in a flat pencil.
+
+  The number of elements in the row, in the flat pencil, and in the
+  axial pencil is, of course, infinite and indefinite too, but the same
+  in all. This number may be denoted by [infinity]. Then a plane
+  contains [infinity]² points and as many lines. To see this, take a
+  flat pencil in a plane. It contains [infinity] lines, and each line
+  contains [infinity] points, whilst each point in the plane lies on one
+  of these lines. Similarly, in a plane each line cuts a fixed line in a
+  point. But this line is cut at each point by [infinity] lines and
+  contains [infinity] points; hence there are [infinity]² lines in a
+  plane.
+
+  A pencil in space contains as many lines as a plane contains points
+  and as many planes as a plane contains lines, for any plane cuts the
+  pencil in a field of points and lines. Hence a pencil contains
+  [infinity]² lines and [infinity]² planes. _The field and the pencil
+  are of two dimensions._
+
+  To count the number of points in space we observe that each point lies
+  on some line in a pencil. But the pencil contains [infinity]² lines,
+  and each line [infinity] points; hence space contains [infinity]³
+  points. Each plane cuts any fixed plane in a line. But a plane
+  contains [infinity]² lines, and through each pass [infinity] planes;
+  therefore space contains [infinity]³ planes.
+
+  Hence space contains as many planes as points, but it contains an
+  infinite number of times more lines than points or planes. To count
+  them, notice that every line cuts a fixed plane in one point. But
+  [infinity]² lines pass through each point, and there are [infinity]²
+  points in the plane. Hence there are [infinity]^4 lines in space. _The
+  space of points and planes is of three dimensions, but the space of
+  lines is of four dimensions._
+
+  A field of points or lines contains an infinite number of rows and
+  flat pencils; a pencil contains an infinite number of flat pencils and
+  of axial pencils; space contains a triple infinite number of pencils
+  and of fields, [infinity]^4 rows and axial pencils and [infinity]^5
+  flat pencils--or, in other words, each point is a centre of
+  [infinity]² flat pencils.
+
+  § 7. The above enumeration allows a classification of figures. Figures
+  in a row consist of groups of points only, and figures in the flat or
+  axial pencil consist of groups of lines or planes. In the plane we may
+  draw polygons; and in the pencil or in the point, solid angles, and so
+  on.
+
+  We may also distinguish the different measurements We have--
+
+    In the row, length of segment;
+    In the flat pencil, angles;
+    In the axial pencil, dihedral angles between two planes;
+    In the plane, areas;
+    In the pencil, solid angles;
+    In the space of points or planes, volumes.
+
+
+  SEGMENTS OF A LINE
+
+  § 8. Any two points A and B in space determine on the line through
+  them a finite part, which may be considered as being described by a
+  point moving from A to B. This we shall denote by AB, and distinguish
+  it from BA, which is supposed as being described by a point moving
+  from B to A, and hence in a direction or in a "sense" opposite to AB.
+  Such a finite line, which has a definite sense, we shall call a
+  "segment," so that AB and BA denote different segments, which are said
+  to be equal in length but of opposite sense. The one sense is often
+  called positive and the other negative.
+
+  In introducing the word "sense" for direction in a line, we have the
+  word direction reserved for direction of the line itself, so that
+  different lines have different directions, unless they be parallel,
+  whilst in each line we have a positive and negative sense.
+
+  We may also say, with Clifford, that AB denotes the "step" of going
+  from A to B.
+
+  [Illustration: FIG. 2.]
+
+  § 9. If we have three points A, B, C in a line (fig. 2), the step AB
+  will bring us from A to B, and the step BC from B to C. Hence both
+  steps are equivalent to the one step AC. This is expressed by saying
+  that AC is the "sum" of AB and BC; in symbols--
+
+    AB + BC = AC,
+
+  where account is to be taken of the sense.
+
+  This equation is true whatever be the position of the three points on
+  the line. As a special case we have
+
+    AB + BA = 0, (1)
+
+  and similarly
+
+    AB + BC + CA = 0, (2)
+
+  which again is true for any three points in a line.
+
+  We further write
+
+    AB = -BA.
+
+  where - denotes negative sense.
+
+  We can then, just as in algebra, change subtraction of segments into
+  addition by changing the sense, so that AB - CB is the same as AB +
+  (-CB) or AB + BC. A figure will at once show the truth of this. The
+  sense is, in fact, in every respect equivalent to the "sign" of a
+  number in algebra.
+
+  § 10. Of the many formulae which exist between points in a line we
+  shall have to use only one more, which connects the segments between
+  any four points A, B, C, D in a line. We have
+
+    BC = BD + DC, CA = CD + DA, AB = AD + DB;
+
+  or multiplying these by AD, BD, CD respectively, we get
+
+    BC·AD = BD·AD + DC·AD = BD·AD - CD·AD
+
+    CA·BD = CD·BD + DA·BD = CD·BD - AD·BD
+
+    AB·CD = AD·CD + DB·CD = AD·CD - BD·CD.
+
+  It will be seen that the sum of the right-hand sides vanishes, hence
+  that
+
+    BC·AD + CA·BD + AB·CD = 0 (3)
+
+  for any four points on a line.
+
+  [Illustration: FIG. 3.]
+
+  § 11. If C is any point in the line AB, then we say that C divides the
+  segment AB in the ratio AC/CB, account being taken of the sense of the
+  two segments AC and CB. If C lies between A and B the ratio is
+  positive, as AC and CB have the same sense. But if C lies without the
+  segment AB, i.e. if C divides AB externally, then the ratio is
+  negative. To see how the value of this ratio changes with C, we will
+  move C along the whole line (fig. 3), whilst A and B remain fixed. If
+  C lies at the point A, then AC = 0, hence the ratio AC : CB vanishes.
+  As C moves towards B, AC increases and CB decreases, so that our ratio
+  increases. At the middle point M of AB it assumes the value +1, and
+  then increases till it reaches an infinitely large value, when C
+  arrives at B. On passing beyond B the ratio becomes negative. If C is
+  at P we have AC = AP = AB + BP, hence
+
+    AC   AB   BP     AB
+    -- = -- + -- = - -- - 1.
+    CB   PB   PB     BP
+
+  In the last expression the ratio AB : BP is positive, has its greatest
+  value [infinity] when C coincides with B, and vanishes when BC becomes
+  infinite. Hence, as C moves from B to the right to the point at
+  infinity, the ratio AC : CB varies from -[infinity] to -1.
+
+  If, on the other hand, C is to the left of A, say at Q, we have AC =
+  AQ = AB + BQ = AB - QB, hence AC/CB = AB/QB - 1.
+
+  Here AB < QB, hence the ratio AB : QB is positive and always less than
+  one, so that the whole is negative and < 1. If C is at the point at
+  infinity it is -1, and then increases as C moves to the right, till
+  for C at A we get the ratio = 0. Hence--
+
+  "As C moves along the line from an infinite distance to the left to an
+  infinite distance at the right, the ratio always increases; it starts
+  with the value -1, reaches 0 at A, +1 at M, [infinity] at B, now
+  changes sign to -[infinity], and increases till at an infinite
+  distance it reaches again the value -1. _It assumes therefore all
+  possible values from -[infinity] to +[infinity], and each value only
+  once, so that not only does every position of C determine a definite
+  value of the ratio AC : CB, but also, conversely, to every positive or
+  negative value of this ratio belongs one single point in the line AB._
+
+  [Relations between segments of lines are interesting as showing an
+  application of algebra to geometry. The genesis of such relations
+  from algebraic identities is very simple. For example, if a, b, c, x
+  be any four quantities, then
+
+              a                       b
+    --------------------- + --------------------- +
+    (a - b)(a - c)(x - a)   (b - c)(b - a)(x - b)
+
+                c                       x
+      --------------------- = ---------------------;
+      (c - a)(c - b)(x - c)   (x - a)(x - b)(x - c)
+
+  this may be proved, cumbrously, by multiplying up, or, simply, by
+  decomposing the right-hand member of the identity into partial
+  fractions. Now take a line ABCDX, and let AB = a, AC = b, AD = c, AX =
+  x. Then obviously (a - b) = AB - AC = -BC, paying regard to signs; (a
+  - c) = AB - AD = DB, and so on. Substituting these values in the
+  identity we obtain the following relation connecting the segments
+  formed by five points on a line:--
+
+       AB         AC         AD         AX
+    -------- + -------- + -------- = --------.
+    BC·BD·BX   CD·CB·CX   DB·DC·DX   BX·CX·DX
+
+  Conversely, if a metrical relation be given, its validity may be
+  tested by reducing to an algebraic equation, which is an identity if
+  the relation be true. For example, if ABCDX be five collinear points,
+  prove
+
+    AD·AX   BD·BX   CD·CX
+    ----- + ----- + ----- = 1.
+    AB·AC   BC·BA   CA·CB
+
+  Clearing of fractions by multiplying throughout by AB·BC·CA, we have
+  to prove
+
+    -AD·AX·BC - BD·BX·CA - CD·CX·AB = AB·BC·CA.
+
+  Take A as origin and let AB = a, AC = b, AD = c, AX = x. Substituting
+  for the segments in terms of a, b, c, x, we obtain on simplification
+
+    a²b - ab² = -ab² + a²b, an obvious identity.
+
+  An alternative method of testing a relation is illustrated in the
+  following example:-- If A, B, C, D, E, F be six collinear points, then
+
+      AE·AF      BE·BF      CE·CF      DE·DF
+    -------- + -------- + -------- + -------- = 0.
+    AB·AC·AD   BC·BD·BA   CD·CA·CB   DA·DB·DC
+
+  Clearing of fractions by multiplying throughout by AB·BC·CD·DA, and
+  reducing to a common origin O (calling OA = a, OB = b, &c.), an
+  equation containing the second and lower powers of OA (= a), &c., is
+  obtained. Calling OA = x, it is found that x = b, x = c, x = d are
+  solutions. Hence the quadratic has three roots; consequently it is an
+  identity.
+
+  The relations connecting five points which we have instanced above may
+  be readily deduced from the six-point relation; the first by taking D
+  at infinity, and the second by taking F at infinity, and then making
+  the obvious permutations of the points.]
+
+
+  PROJECTION AND CROSS-RATIOS
+
+  § 12. If we join a point A to a point S, then the point where the line
+  SA cuts a fixed plane [pi] is called the projection of A on the plane
+  [pi] from S as centre of projection. If we have two planes [pi] and
+  [pi]' and a point S, we may project every point A in [pi] to the other
+  plane. If A' is the projection of A, then A is also the projection of
+  A', so that the relations are reciprocal. To every figure in [pi] we
+  get as its projection a corresponding figure in [pi]'.
+
+  We shall determine such properties of figures as remain true for the
+  projection, and which are called projective properties. For this
+  purpose it will be sufficient to consider at first only constructions
+  in one plane.
+
+  [Illustration: FIG. 4.]
+
+  [Illustration: FIG. 5.]
+
+  Let us suppose we have given in a plane two lines p and p' and a
+  centre S (fig. 4); we may then project the points in p from S to p'.
+  Let A', B' ... be the projections of A, B ..., the point at infinity
+  in p which we shall denote by I will be projected into a finite point
+  I' in p', viz. into the point where the parallel to p through S cuts
+  p'. Similarly one point J in p will be projected into the point J' at
+  infinity in p'. This point J is of course the point where the parallel
+  to p' through S cuts p. We thus see that every point in p is projected
+  into a single point in p'.
+
+  Fig. 5 shows that a segment AB will be projected into a segment A'B'
+  which is not equal to it, at least not as a rule; and also that the
+  ratio AC : CB is not equal to the ratio A'C' : C'B' formed by the
+  projections. These ratios will become equal only if p and p' are
+  parallel, for in this case the triangle SAB is similar to the triangle
+  SA'B'. Between three points in a line and their projections there
+  exists therefore in general no relation. But between four points a
+  relation does exist.
+
+  § 13. Let A, B, C, D be four points in p, A', B', C, D' their
+  projections in p', then the ratio of the two ratios AC : CB and AD :
+  DB into which C and D divide the segment AB is equal to the
+  corresponding expression between A', B', C', D'. In symbols we have
+
+    AC   AD   A'C'   A'D'
+    -- : -- = ---- : ----.
+    CB   DB   C'B'   D'B'
+
+  This is easily proved by aid of similar triangles.
+
+  [Illustration: FIG. 6.]
+
+  Through the points A and B on p draw parallels to p', which cut the
+  projecting rays in C2, D2, B2 and A1, C1, D1, as indicated in fig. 6.
+  The two triangles ACC2 and BCC1 will be similar, as will also be the
+  triangles ADD2 and BDD1.
+
+  The proof is left to the reader.
+
+  This result is of fundamental importance.
+
+  The expression AC/CB : AD/DB has been called by Chasles the
+  "anharmonic ratio of the four points A, B, C, D." Professor Clifford
+  proposed the shorter name of "cross-ratio." We shall adopt the latter.
+  We have then the
+
+  FUNDAMENTAL THEOREM.--_The cross-ratio of four points in a line is
+  equal to the cross-ratio of their projections on any other line which
+  lies in the same plane with it._
+
+  § 14. Before we draw conclusions from this result, we must investigate
+  the meaning of a cross-ratio somewhat more fully.
+
+  If four points A, B, C, D are given, and we wish to form their
+  cross-ratio, we have first to divide them into two groups of two, the
+  points in each group being taken in a definite order. Thus, let A, B
+  be the first, C, D the second pair, A and C being the first points in
+  each pair. The cross-ratio is then the ratio AC : CB divided by AD :
+  DB. This will be denoted by (AB, CD), so that
+
+               AC   AD
+    (AB, CD) = -- : --.
+               CB   DB
+
+  This is easily remembered. In order to write it out, make first the
+  two lines for the fractions, and put above and below these the letters
+  A and B in their places, thus, A*/B : A*/B; and then fill up,
+  crosswise, the first by C and the other by D.
+
+  § 15. If we take the points in a different order, the value of the
+  cross-ratio will change. We can do this in twenty-four different ways
+  by forming all permutations of the letters. But of these twenty-four
+  cross-ratios groups of four are equal, so that there are really only
+  six different ones, and these six are reciprocals in pairs.
+
+  We have the following rules:--
+
+  I. If in a cross-ratio the two groups be interchanged, its value
+  remains unaltered, i.e.
+
+    (AB, CD) = (CD, AB) = (BA, DC) = (DC, BA).
+
+  II. If in a cross-ratio the two points belonging to one of the two
+  groups be interchanged, the cross-ratio changes into its reciprocal,
+  i.e.
+
+    (AB, CD) = 1/(AB, DC) = 1/(BA, CD) = 1/(CD, BA) = 1/(DC, AB).
+
+  From I. and II. we see that eight cross-ratios are associated with
+  (AB, CD).
+
+  III. If in a cross-ratio the two middle letters be interchanged, the
+  cross-ratio [alpha] changes into its complement 1 - [alpha], i.e. (AB,
+  CD) = 1 - (AC, BD).
+
+  [§ 16. If [lambda] = (AB, CD), µ = (AC, DB), [nu] = (AD, BC), then
+  [lambda], µ, [nu] and their reciprocals 1/[lambda], 1/µ, 1/[nu] are
+  the values of the total number of twenty-four cross-ratios. Moreover,
+  [lambda], µ, [nu] are connected by the relations
+
+    [lambda] + 1/µ = µ + 1/[nu] = [nu] + 1/[lambda] = -[lambda]µ[nu] = 1;
+
+  this proposition may be proved by substituting for [lambda], µ, [nu]
+  and reducing to a common origin. There are therefore four equations
+  between three unknowns; hence if one cross-ratio be given, the
+  remaining twenty-three are determinate. Moreover, two of the
+  quantities [lambda], µ, [nu] are positive, and the remaining one
+  negative.
+
+  The following scheme shows the twenty-four cross-ratios expressed in
+  terms of [lambda], µ, [nu].]
+
+    +---------+-----------------------+---------------+---------------+
+    |(AB, CD) |                       |               |               |
+    |(BA, DC) |       [lambda]        |     1 - µ     |  1/(1 - [nu]) |
+    |(CD, AB) |                       |               |               |
+    |(DC, BA) |                       |               |               |
+    +---------+-----------------------+---------------+---------------+
+    |(AC, DB) |                       |               |               |
+    |(BD, CA) |   1/(1 - [lambda])    |      1/µ      |([nu] - 1)/[nu]|
+    |(CA, BD) |                       |               |               |
+    |(DB, AC) |                       |               |               |
+    +---------+-----------------------+---------------+---------------+
+    |(AB, DC) |                       |               |               |
+    |(BA, CD) |      1/[lambda]       |   1/(1 - µ)   |   1 - [nu]    |
+    |(CD, BA) |                       |               |               |
+    |(DC, AB) |                       |               |               |
+    +---------+-----------------------+---------------+---------------+
+    |(AD, BC) |                       |               |               |
+    |(BC, AD) |([lambda] - 1)/[lambda]|   µ/(µ - 1)   |     [nu]      |
+    |(CB, DA) |                       |               |               |
+    |(DA, CB) |                       |               |               |
+    +---------+-----------------------+---------------+---------------+
+    |(AC, BD) |                       |               |               |
+    |(BD, AC) |     1 - [lambda]      |       µ       |[nu]/([nu] - 1)|
+    |(CA, DB) |                       |               |               |
+    |(DB, CA) |                       |               |               |
+    +---------+-----------------------+---------------+---------------+
+    |(AD, CB) |                       |               |               |
+    |(BC, DA) |[lambda]/([lambda] - 1)|   (µ - 1)/µ   |    1/[nu]     |
+    |(CB, AD) |                       |               |               |
+    |(DA, BC) |                       |               |               |
+    +---------+-----------------------+---------------+---------------+
+
+  § 17. If one of the points of which a cross-ratio is formed is the
+  point at infinity in the line, the cross-ratio changes into a simple
+  ratio. It is convenient to let the point at infinity occupy the last
+  place in the symbolic expression for the cross-ratio. Thus if I is a
+  point at infinity, we have (AB, CI) = -AC/CB, because AI : IB = -1.
+
+  Every common ratio of three points in a line may thus be expressed as
+  a cross-ratio, by adding the point at infinity to the group of points.
+
+
+  HARMONIC RANGES
+
+  § 18. If the points have special positions, the cross-ratios may have
+  such a value that, of the six different ones, two and two become
+  equal. If the first two shall be equal, we get [lambda] = 1/[lambda],
+  or [lambda]² = 1, [lambda] = ±1.
+
+  If we take [lambda] = +1, we have (AB, CD) = 1, or AC/CB = AD/DB; that
+  is, the points C and D coincide, provided that A and B are different.
+
+  If we take [lambda] = -1, so that (AB, CD) = -1, we have AC/CB =
+  -AD/DB. _Hence C and D divide AB internally and externally in the same
+  ratio._
+
+  The four points are in this case said to be _harmonic points_, and _C
+  and D are said to be harmonic conjugates with regard to A and B._
+
+  But we have also (CD, AB) = -1, so that A and B are harmonic
+  conjugates with regard to C and D.
+
+  The principal property of harmonic points is that their cross-ratio
+  remains unaltered if we interchange the two points belonging to one
+  pair, viz.
+
+    (AB, CD) = (AB, DC) = (BA, CD).
+
+  For four harmonic points the six cross-ratios become equal two and
+  two:
+
+                                       [lambda]
+    [lambda] = -1, 1 - [lambda] = 2, ------------ = ½,
+                                     [lambda] - 1
+
+           1                1            [lambda] - 1
+      = -------- = -1, ------------ = ½, ------------ = 2.
+        [lambda]       1 - [lambda]        [lambda]
+
+  Hence if we get four points whose cross-ratio is 2 or ½, then they are
+  harmonic, but not arranged so that conjugates are paired. If this is
+  the case the cross-ratio = -1.
+
+  § 19. If we equate any two of the above six values of the
+  cross-ratios, we get either [lambda] = 1, 0, [infinity], or [lambda] =
+  -1, 2, ½, or else [lambda] becomes a root of the equation [lambda]² -
+  [lambda] + 1 = 0, that is, an imaginary cube root of -1. In this case
+  the six values become three and three equal, so that only two
+  different values remain. This case, though important in the theory of
+  cubic curves, is for our purposes of no interest, whilst harmonic
+  points are all-important.
+
+  § 20. From the definition of harmonic points, and by aid of § 11, the
+  following properties are easily deduced.
+
+  If C and D are harmonic conjugates with regard to A and B, then one of
+  them lies in, the other without AB; it is impossible to move from A to
+  B without passing either through C or through D; the one blocks the
+  finite way, the other the way through infinity. This is expressed by
+  saying A and B are "separated" by C and D.
+
+  For every position of C there will be one and only one point D which
+  is its harmonic conjugate with regard to any point pair A, B.
+
+  If A and B are different points, and if C coincides with A or B, D
+  does. But if A and B coincide, one of the points C or D, lying between
+  them, coincides with them, and the other may be anywhere in the line.
+  It follows that, "_if of four harmonic conjugates two coincide, then a
+  third coincides with them, and the fourth may be any point in the
+  line_."
+
+  If C is the middle point between A and B, then D is the point at
+  infinity; for AC : CB = +1, hence AD : DB must be equal to -1. _The
+  harmonic conjugate of the point at infinity in a line with regard to
+  two points A, B is the middle point of AB._
+
+  This important property gives a first example how metric properties
+  are connected with projective ones.
+
+  [§ 21. _Harmonic properties of the complete quadrilateral and
+  quadrangle._
+
+  [Illustration: FIG. 7.]
+
+  [Illustration: FIG. 8.]
+
+  A figure formed by four lines in a plane is called a _complete
+  quadrilateral_, or, shorter, a _four-side_. The four sides meet in six
+  points, named the "vertices," which may be joined by three lines
+  (other than the sides), named the "diagonals" or "harmonic lines." The
+  diagonals enclose the "harmonic triangle of the quadrilateral." In
+  fig. 7, A'B'C', B'AC, C'AB, CBA' are the sides, A, A', B, B', C, C'
+  the vertices, AA', BB', CC' the harmonic lines, and [alpha]ß[gamma]
+  the harmonic triangle of the quadrilateral. A figure formed by four
+  coplanar points is named a _complete quadrangle_, or, shorter, a
+  _four-point_. The four points may be joined by six lines, named the
+  "sides," which intersect in three other points, termed the "diagonal
+  or harmonic points." The harmonic points are the vertices of the
+  "harmonic triangle of the complete quadrangle." In fig. 8, AA', BB'
+  are the points, AA', BB', A'B', B'A, AB, BA' are the sides, L, M, N
+  are the diagonal points, and LMN is the harmonic triangle of the
+  quadrangle.
+
+  The harmonic property of the complete quadrilateral is: Any diagonal
+  or harmonic line is harmonically divided by the other two; and of a
+  complete quadrangle: The angle at any harmonic point is divided
+  harmonically by the joins to the other harmonic points. To prove the
+  first theorem, we have to prove (AA', ß[gamma]), (BB',
+  [gamma][alpha]), (CC', ß[alpha]) are harmonic. Consider the
+  cross-ratio (CC', [alpha]ß). Then projecting from A on BB' we have
+  A(CC', [alpha]ß) = A(B'B, [alpha][gamma]). Projecting from A' on BB',
+  A'(CC', [alpha]ß) = A'(BB', [alpha][gamma]). Hence (B'B,
+  [alpha][gamma]) = (BB', [alpha][gamma]), i.e. the cross-ratio (BB',
+  [alpha][gamma]) equals that of its reciprocal; hence the range is
+  harmonic.
+
+  The second theorem states that the pencils L(BA, NM), M(B'A, LN),
+  N(BA, LM) are harmonic. Deferring the subject of harmonic pencils to
+  the next section, it will suffice to state here that any transversal
+  intersects an harmonic pencil in an harmonic range. Consider the
+  pencil L(BA, NM), then it is sufficient to prove (BA', NM') is
+  harmonic. This follows from the previous theorem by considering A'B as
+  a diagonal of the quadrilateral ALB'M.]
+
+  This property of the complete quadrilateral allows the solution of the
+  problem:
+
+  _To construct the harmonic conjugate D to a point C with regard to two
+  given points A and B._
+
+  Through A draw any two lines, and through C one cutting the former two
+  in G and H. Join these points to B, cutting the former two lines in E
+  and F. The point D where EF cuts AB will be the harmonic conjugate
+  required.
+
+  This remarkable construction requires nothing but the drawing of
+  lines, and is therefore independent of measurement. In a similar
+  manner the harmonic conjugate of the line VA for two lines VC, VD is
+  constructed with the aid of the property of the complete quadrangle.
+
+  § 22. _Harmonic Pencils._--The theory of cross-ratios may be extended
+  from points in a row to lines in a flat pencil and to planes in an
+  axial pencil. We have seen (§ 13) that if the lines which join four
+  points A, B, C, D to any point S be cut by any other line in A', B',
+  C', D', then (AB, CD) = (A'B', C'D'). In other words, four lines in a
+  flat pencil are cut by every other line in four points whose
+  cross-ratio is constant.
+
+  _Definition._--By the cross-ratio of four rays in a flat pencil is
+  meant the cross-ratio of the four points in which the rays are cut by
+  any line. If a, b, c, d be the lines, then this cross-ratio is denoted
+  by (ab, cd).
+
+  _Definition._--By the cross-ratio of four planes in an axial pencil is
+  understood the cross-ratio of the four points in which any line cuts
+  the planes, or, what is the same thing, the cross-ratio of the four
+  rays in which any plane cuts the four planes.
+
+  In order that this definition may have a meaning, it has to be proved
+  that all lines cut the pencil in points which have the same
+  cross-ratio. This is seen at once for two intersecting lines, as their
+  plane cuts the axial pencil in a flat pencil, which is itself cut by
+  the two lines. The cross-ratio of the four points on one line is
+  therefore equal to that on the other, and equal to that of the four
+  rays in the flat pencil.
+
+  If two non-intersecting lines p and q cut the four planes in A, B, C,
+  D and A', B', C', D', draw a line r to meet both p and q, and let this
+  line cut the planes in A", B", C", D". Then (AB, CD) = (A'B', C'D'),
+  for each is equal to (A"B", C"D").
+
+  § 23. We may now also extend the notion of harmonic elements, viz.
+
+  _Definition._--Four rays in a flat pencil and four planes in an axial
+  pencil are said to be harmonic if their cross-ratio equals -1, that
+  is, if they are cut by a line in four harmonic points.
+
+  If we understand by a "median line" of a triangle a line which joins a
+  vertex to the middle point of the opposite side, and by a "median
+  line" of a parallelogram a line joining middle points of opposite
+  sides, we get as special cases of the last theorem:
+
+  _The diagonals and median lines of a parallelogram form an harmonic
+  pencil_; and
+
+  _At a vertex of any triangle, the two sides, the median line, and the
+  line parallel to the base form an harmonic pencil._
+
+  Taking the parallelogram a rectangle, or the triangle isosceles, we
+  get:
+
+  _Any two lines and the bisections of their angles form an harmonic
+  pencil._ Or:
+
+  _In an harmonic pencil, if two conjugate rays are perpendicular, then
+  the other two are equally inclined to them_; and, conversely, _if one
+  ray bisects the angle between conjugate rays, it is perpendicular to
+  its conjugate_.
+
+  This connects perpendicularity and bisection of angles with projective
+  properties.
+
+  § 24. We add a few theorems and problems which are easily proved or
+  solved by aid of harmonics.
+
+  An harmonic pencil is cut by a line parallel to one of its rays in
+  three equidistant points.
+
+  Through a given point to draw a line such that the segment determined
+  on it by a given angle is bisected at that point.
+
+  Having given two parallel lines, to bisect on either any given segment
+  without using a pair of compasses.
+
+  Having given in a line a segment and its middle point, to draw through
+  any given point in the plane a line parallel to the given line.
+
+  To draw a line which joins a given point to the intersection of two
+  given lines which meet off the drawing paper (by aid of § 21).
+
+
+  CORRESPONDENCE. HOMOGRAPHIC AND PERSPECTIVE RANGES
+
+  § 25. Two rows, p and p', which are one the projection of the other
+  (as in fig. 5), stand in a definite relation to each other,
+  characterized by the following properties.
+
+  1. _To each point in either corresponds one point in the other_; that
+  is, those points are said to correspond which are projections of one
+  another.
+
+  2. _The cross-ratio of any four points in one equals that of the
+  corresponding points in the other._
+
+  3. _The lines joining corresponding points all pass through the same
+  point._
+
+  If we suppose corresponding points marked, and the rows brought into
+  any other position, then the lines joining corresponding points will
+  no longer meet in a common point, and hence the third of the above
+  properties will not hold any longer; but we have still a
+  correspondence between the points in the two rows possessing the first
+  two properties. Such a correspondence has been called a _one-one
+  correspondence_, whilst the two rows between which such correspondence
+  has been established are said to be _projective_ or _homographic_. Two
+  rows which are each the projection of the other are therefore
+  _projective_. We shall presently see, also, that any two projective
+  rows may always be placed in such a position that one appears as the
+  projection of the other. If they are in such a position the rows are
+  said to be in _perspective position_, or simply to be in
+  _perspective_.
+
+  § 26. The notion of a one-one correspondence between rows may be
+  extended to flat and axial pencils, viz. a flat pencil will be said to
+  be projective to a flat pencil if to each ray in the first corresponds
+  one ray in the second, and if the cross-ratio of four rays in one
+  equals that of the corresponding rays in the second.
+
+  Similarly an axial pencil may be projective to an axial pencil. But a
+  flat pencil may also be projective to an axial pencil, or either
+  pencil may be projective to a row. The definition is the same in each
+  case: there is a one-one correspondence between the elements, and four
+  elements have the same cross-ratio as the corresponding ones.
+
+  § 27. There is also in each case a special position which is called
+  _perspective_, viz.
+
+  1. Two projective rows are perspective if they lie in the same plane,
+  and if the one row is a projection of the other.
+
+  2. Two projective flat pencils are perspective--(1) if they lie in the
+  same plane, and have a row as a common section; (2) if they lie in the
+  same pencil (in space), and are both sections of the same axial
+  pencil; (3) if they are in space and have a row as common section, or
+  are both sections of the same axial pencil, one of the conditions
+  involving the other.
+
+  3. Two projective axial pencils, if their axes meet, and if they have
+  a flat pencil as a common section.
+
+  4. A row and a projective flat pencil, if the row is a section of the
+  pencil, each point lying in its corresponding line.
+
+  5. A row and a projective axial pencil, if the row is a section of the
+  pencil, each point lying in its corresponding line.
+
+  6. A flat and a projective axial pencil, if the former is a section of
+  the other, each ray lying in its corresponding plane.
+
+  That in each case the correspondence established by the position
+  indicated is such as has been called projective follows at once from
+  the definition. It is not so evident that the perspective position may
+  always be obtained. We shall show in § 30 this for the first three
+  cases. First, however, we shall give a few theorems which relate to
+  the general correspondence, not to the perspective position.
+
+  § 28. _Two rows or pencils, flat or axial, which are projective to a
+  third are projective to each other_; this follows at once from the
+  definitions.
+
+  § 29. _If two rows, or two pencils, either flat or axial, or a row and
+  a pencil, be projective, we may assume to any three elements in the
+  one the three corresponding elements in the other, and then the
+  correspondence is uniquely determined._
+
+  For if in two projective rows we assume that the points A, B, C in the
+  first correspond to the given points A', B', C' in the second, then to
+  any fourth point D in the first will correspond a point D' in the
+  second, so that
+
+    (AB, CD) = (A'B', C'D').
+
+  But there is only one point, D', which makes the cross-ratio (A'B',
+  C'D') equal to the given number (AB, CD).
+
+  The same reasoning holds in the other cases.
+
+  § 30. If two rows are perspective, then the lines joining
+  corresponding points all meet in a point, the centre of projection;
+  and the point in which the two bases of the rows intersect as a point
+  in the first row coincides with its corresponding point in the second.
+
+  This follows from the definition. The converse also holds, viz.
+
+  _If two projective rows have such a position that one point in the one
+  coincides with its corresponding point in the other, then they are
+  perspective, that is, the lines joining corresponding points all pass
+  through a common point, and form a flat pencil._
+
+  For let A, B, C, D ... be points in the one, and A', B', C', D' ...
+  the corresponding points in the other row, and let A be made to
+  coincide with its corresponding point A'. Let S be the point where the
+  lines BB' and CC' meet, and let us join S to the point D in the first
+  row. This line will cut the second row in a point D", so that A, B, C,
+  D are projected from S into the points A, B', C', D". The cross-ratio
+  (AB, CD) is therefore equal to (AB', C'D"), and by hypothesis it is
+  equal to (A'B', C'D'). Hence (A'B', C'D") = (A'B', C'D'), that is, D"
+  is the same point as D'.
+
+  § 31. If two projected flat pencils in the same plane are in
+  perspective, then the intersections of corresponding lines form a row,
+  and the line joining the two centres as a line in the first pencil
+  corresponds to the same line as a line in the second. And conversely,
+
+  _If two projective pencils in the same plane, but with different
+  centres, have one line in the one coincident with its corresponding
+  line in the other, then the two pencils are perspective, that is, the
+  intersection of corresponding lines lie in a line._
+
+  The proof is the same as in § 30.
+
+  § 32. If two projective flat pencils in the same point (pencil in
+  space), but not in the same plane, are perspective, then the planes
+  joining corresponding rays all pass through a line (they form an axial
+  pencil), and the line common to the two pencils (in which their planes
+  intersect) corresponds to itself. And conversely:--
+
+  If two flat pencils which have a common centre, but do not lie in a
+  common plane, are placed so that one ray in the one coincides with its
+  corresponding ray in the other, then they are perspective, that is,
+  the planes joining corresponding lines all pass through a line.
+
+  § 33. If two projective axial pencils are perspective, then the
+  intersection of corresponding planes lie in a plane, and the plane
+  common to the two pencils (in which the two axes lie) corresponds to
+  itself. And conversely:--
+
+  If two projective axial pencils are placed in such a position that a
+  plane in the one coincides with its corresponding plane, then the two
+  pencils are perspective, that is, corresponding planes meet in lines
+  which lie in a plane.
+
+  The proof again is the same as in § 30.
+
+  § 34. These theorems relating to perspective position become illusory
+  if the projective rows of pencils have a common base. We then have:--
+
+  In two projective rows on the same line--and also in two projective
+  and concentric flat pencils in the same plane, or in two projective
+  axial pencils with a common axis--every element in the one coincides
+  with its corresponding element in the other as soon as three elements
+  in the one coincide with their corresponding elements in the other.
+
+  _Proof_ (in case of two rows).--Between four elements A, B, C, D and
+  their corresponding elements A', B', C', D' exists the relation (ABCD)
+  = (A'B'C'D'). If now A', B', C' coincide respectively with A, B, C, we
+  get (AB, CD) = (AB, CD'), hence D and D' coincide.
+
+  The last theorem may also be stated thus:--
+
+  In two projective rows or pencils, which have a common base but are
+  not identical, not more than two elements in the one can coincide with
+  their corresponding elements in the other.
+
+  Thus two projective rows on the same line cannot have more than two
+  pairs of coincident points unless every point coincides with its
+  corresponding point.
+
+  It is easy to construct two projective rows on the same line, which
+  have two pairs of corresponding points coincident. Let the points A,
+  B, C as points belonging to the one row correspond to A, B, and C' as
+  points in the second. Then A and B coincide with their corresponding
+  points, but C does not. It is, however, not necessary that two such
+  rows have twice a point coincident with its corresponding point; it is
+  possible that this happens only once or not at all. Of this we shall
+  see examples later.
+
+  [Illustration: FIG. 9.]
+
+  § 35. If two projective rows or pencils are in perspective position,
+  we know at once which element in one corresponds to any given element
+  in the other. If p and q (fig. 9) are two projective rows, so that K
+  corresponds to itself, and if we know that to A and B in p correspond
+  A' and B' in q, then the point S, where AA' meets BB', is the centre
+  of projection, and hence, in order to find the point C' corresponding
+  to C, we have only to join C to S; the point C', where this line cuts
+  q, is the point required.
+
+  [Illustration: FIG. 10.]
+
+  If two flat pencils, S1 and S2, in a plane are perspective (fig. 10),
+  we need only to know two pairs, a, a' and b, b', of corresponding rays
+  in order to find the axis s of projection. This being known, a ray c'
+  in S2, corresponding to a given ray c in S1, is found by joining S2 to
+  the point where c cuts the axis s.
+
+  A similar construction holds in the other cases of perspective
+  figures.
+
+  On this depends the solution of the following general problem.
+
+  § 36. Three pairs of corresponding elements in two projective rows or
+  pencils being given, to determine for any element in one the
+  corresponding element in the other.
+
+  We solve this in the two cases of two projective rows and of two
+  projective flat pencils in a plane.
+
+  _Problem_ I.--Let A, B, C be          _Problem_ II.--Let a, b, c be
+  three points in a row s, A', B',      three rays in a pencil S, a',
+  C' the corresponding points in a      b', c' the corresponding rays in
+  projective row s', both being in      a projective pencil S', both
+  a plane; it is required to find       being in the same plane; it is
+  for any point D in s the              required to find for any ray d
+  corresponding point D' in s'.         in S the corresponding ray d' in
+                                        S'.
+
+  The solution is made to depend on the construction of an auxiliary row
+  or pencil which is perspective to both the given ones. This is found
+  as follows:--
+
+  [Illustration: FIG. 11.]
+
+  _Solution of Problem_ I.--On the line joining two corresponding
+  points, say AA' (fig. 11), take any two points, S and S', as centres
+  of auxiliary pencils. Join the intersection B1 of SB and S'B' to the
+  intersection C1 of SC and S'C' by the line s1. Then a row on s1 will
+  be perspective to s with S as centre of projection, and to s' with S'
+  as centre. To find now the point D' on s' corresponding to a point D
+  on s we have only to determine the point D1, where the line SD cuts
+  s1, and to draw S'D1; the point where this line cuts s' will be the
+  required point D'.
+
+  _Proof._--The rows s and s' are both perspective to the row s1, hence
+  they are projective to one another. To A, B, C, D on s correspond A1,
+  B1, C1, D1 on s1, and to these correspond A', B', C', D' on s'; so
+  that D and D' are corresponding points as required.
+
+  [Illustration: FIG. 12.]
+
+  _Solution of Problem_ II.--Through the intersection A of two
+  corresponding rays a and a' (fig. 12), take two lines, s and s', as
+  bases of auxiliary rows. Let S1 be the point where the line b1, which
+  joins B and B', cuts the line c1, which joins C and C'. Then a pencil
+  S1 will be perspective to S with s as axis of projection. To find the
+  ray d' in S' corresponding to a given ray d in S, cut d by s at D;
+  project this point from S1 to D' on s' and join D' to S'. This will be
+  the required ray.
+
+  _Proof._--That the pencil S1 is perspective to S and also to S'
+  follows from construction. To the lines a1, b1, c1, d1 in S1
+  correspond the lines a, b, c, d in S and the lines a', b', c', d' in
+  S', so that d and d' are corresponding rays.
+
+  In the first solution the two centres, S, S', are _any_ two points on
+  a line joining any two corresponding points, so that the solution of
+  the problem allows of a great many different constructions. _But
+  whatever construction be used, the point D', corresponding to D, must
+  be always the same_, according to the theorem in § 29. This gives rise
+  to a number of theorems, into which, however, we shall not enter. The
+  same remarks hold for the second problem.
+
+  § 37. _Homological Triangles._--As a further application of the
+  theorems about perspective rows and pencils we shall prove the
+  following important theorem.
+
+  _Theorem._--If ABC and A'B'C' (fig. 13) be two triangles, such that
+  the lines AA', BB', CC' meet in a point S, then the intersections of
+  BC and B'C', of CA and C'A', and of AB and A'B' will lie in a line.
+  Such triangles are said to be homological, or in perspective. The
+  triangles are "co-axial" in virtue of the property that the meets of
+  corresponding sides are collinear and copolar, since the lines joining
+  corresponding vertices are concurrent.
+
+  _Proof._--Let a, b, c denote the lines AA', BB', CC', which meet at S.
+  Then these may be taken as bases of projective rows, so that A, A', S
+  on a correspond to B, B', S on b, and to C, C', S on c. As the point S
+  is common to all, any two of these rows will be perspective.
+
+    If S1 be the centre of projection of rows b and c,
+       S2      "          "          "        c and a,
+       S3      "          "          "        a and b,
+
+  and if the line S1S2 cuts a in A1, and b in B1, and c in C1, then A1,
+  B1 will be corresponding points in a and b, both corresponding to C1
+  in c. But a and b are perspective, therefore the line A1B1, that is
+  S1S2, joining corresponding points must pass through the centre of
+  projection S3 of a and b. In other words, S1, S2, S3 lie in a line.
+  This is Desargues' celebrated theorem if we state it thus:--
+
+  [Illustration: FIG. 13.]
+
+  _Theorem of Desargues._--If each of two triangles has one vertex on
+  each of three concurrent lines, then the intersections of
+  corresponding sides lie in a line, those sides being called
+  corresponding which are opposite to vertices on the same line.
+
+  The converse theorem holds also, viz.
+
+  _Theorem._--If the sides of one triangle meet those of another in
+  three points which lie in a line, then the vertices lie on three lines
+  which meet in a point.
+
+  The proof is almost the same as before.
+
+  § 38. _Metrical Relations between Projective Rows._--Every row
+  contains one point which is distinguished from all others, viz. the
+  point at infinity. In two projective rows, to the point I at infinity
+  in one corresponds a point I' in the other, and to the point J' at
+  infinity in the second corresponds a point J in the first. The points
+  I' and J are in general finite. If now A and B are any two points in
+  the one, A', B' the corresponding points in the other row, then
+
+    (AB, JI) = (A'B', J'I'),
+
+  or
+
+    AJ/JB : AI/IB = A'J'/J'B' : A'I'/I'B'.
+
+  But, by § 17,
+
+    AI/IB = A'J'/J'B' = -1;
+
+  therefore the last equation changes into
+
+    AJ·A'I' = BJ·B'I',
+
+  that is to say--
+
+  _Theorem._--The product of the distances of any two corresponding
+  points in two projective rows from the points which correspond to the
+  points at infinity in the other is constant, viz. AJ·A'I' = k.
+  Steiner has called this number k the _Power of the correspondence_.
+
+  [The relation AJ . A'I' = k shows that if J, I' be given then the
+  point A' corresponding to a specified point A is readily found; hence
+  A, A' generate homographic ranges of which I and J' correspond to the
+  points at infinity on the ranges. If we take any two origins O, O', on
+  the ranges and reduce the expression AJ . A'I' = k to its algebraic
+  equivalent, we derive an equation of the form [alpha]xx' + ßx +
+  [gamma]x' + [delta] = 0. Conversely, if a relation of this nature
+  holds, then points corresponding to solutions in x, x' form
+  homographic ranges.]
+
+  § 39. _Similar Rows._--If the points at infinity in two projective
+  rows correspond so that I' and J are at infinity, this result loses
+  its meaning. But if A, B, C be any three points in one, A', B', C' the
+  corresponding ones on the other row, we have
+
+    (AB, CI) = (A'B', C'I'),
+
+  which reduces to
+
+    AC/CB = A'C'/C'B' or AC/A'C' = BC/B'C',
+
+  that is, corresponding segments are proportional. Conversely, if
+  corresponding segments are proportional, then to the point at infinity
+  in one corresponds the point at infinity in the other. If we call such
+  rows _similar_, we may state the result thus--
+
+  _Theorem._--Two projective rows are similar if to the point at
+  infinity in one corresponds the point at infinity in the other, and
+  conversely, if two rows are similar then they are projective, and the
+  points at infinity are corresponding points.
+
+  From this the well-known propositions follow:--
+
+  Two lines are cut proportionally (in similar rows) by a series of
+  parallels. The rows are perspective, with centre of projection at
+  infinity.
+
+  If two similar rows are placed parallel, then the lines joining
+  homologous points pass through a common point.
+
+  § 40. If two flat pencils be projective, then there exists in either,
+  one single pair of lines at right angles to one another, such that the
+  corresponding lines in the other pencil are again at right angles.
+
+  [Illustration: FIG. 14.]
+
+  To prove this, we place the pencils in perspective position (fig. 14)
+  by making one ray coincident with its corresponding ray. Corresponding
+  rays meet then on a line p. And now we draw the circle which has its
+  centre O on p, and which passes through the centres S and S' of the
+  two pencils. This circle cuts p in two points H and K. The two pairs
+  of rays, h, k, and h', k', joining these points to S and S' will be
+  pairs of corresponding rays at right angles. The construction gives in
+  general but one circle, but if the line p is the perpendicular
+  bisector of SS', there exists an infinite number, and _to every right
+  angle in the one pencil corresponds a right angle in the other_.
+
+
+  PRINCIPLE OF DUALITY
+
+  § 41. It has been stated in § 1 that not only points, but also planes
+  and lines, are taken as elements out of which figures are built up. We
+  shall now see that the construction of one figure which possesses
+  certain properties gives rise in many cases to the construction of
+  another figure, by replacing, according to definite rules, elements of
+  one kind by those of another. The new figure thus obtained will then
+  possess properties which may be stated as soon as those of the
+  original figure are known.
+
+  We obtain thus a principle, known as the _principle of duality_ or of
+  _reciprocity_, which enables us to construct to any figure not
+  containing any measurement in its construction a _reciprocal_ figure,
+  as it is called, and to deduce from any theorem a _reciprocal_
+  theorem, for which no further proof is needed.
+
+  It is convenient to print reciprocal propositions on opposite sides of
+  a page broken into two columns, and this plan will occasionally be
+  adopted.
+
+  We begin by repeating in this form a few of our former statements:--
+
+  Two points determine a line.        Two planes determine a line.
+
+  Three points which are not in a     Three planes which do not pass
+  line determine a plane.             through a line determine a point.
+
+  A line and a point without it       A line and a plane not through
+  determine a plane.                  it determine a point.
+
+  Two lines in a plane determine      Two lines through a point
+  a point.                            determine a plane.
+
+  These propositions show that it will be possible, when any figure is
+  given, to construct a second figure by taking planes instead of
+  points, and points instead of planes, but lines where we had lines.
+
+  For instance, if in the first figure we take a plane and three points
+  in it, we have to take in the second figure a point and three planes
+  through it. The three points in the first, together with the three
+  lines joining them two and two, form a triangle; the three planes in
+  the second and their three lines of intersection form a trihedral
+  angle. A triangle and a trihedral angle are therefore reciprocal
+  figures.
+
+  Similarly, to any figure in a plane consisting of points and lines
+  will correspond a figure consisting of planes and lines passing
+  through a point S, and hence belonging to the pencil which has S as
+  centre.
+
+  The figure reciprocal to four points in space which do not lie in a
+  plane will consist of four planes which do not meet in a point. In
+  this case each figure forms a tetrahedron.
+
+  § 42. As other examples we have the following:--
+
+  To a row                   is reciprocal  an axial pencil,
+
+  " a flat pencil                  "        a flat pencil,
+
+  " a field of points and lines    "        a pencil of planes and lines,
+
+  " the space of points            "        the space of planes.
+
+  For the row consists of a line and all the points in it, reciprocal to
+  it therefore will be a line with all planes through it, that is, an
+  axial pencil; and so for the other cases.
+
+  This correspondence of reciprocity breaks down, however, if we take
+  figures which contain measurement in their construction. For instance,
+  there is no figure reciprocal to two planes at _right angles_, because
+  there is no segment in a row which has a magnitude as definite as a
+  right angle.
+
+  We add a few examples of reciprocal propositions which are easily
+  proved.
+
+  _Theorem._--If A, B, C, D are        _Theorem._--If [alpha], ß,
+  any four points in space, and if     [gamma], [delta] are four planes
+  the lines AB and CD meet, then       in space, and if the lines
+  all four points lie in a plane,      [alpha]ß and [gamma][delta] meet,
+  hence also AC and BD, as well        then all four planes lie in a
+  as AD and BC, meet.                  point (pencil), hence also
+                                       [alpha][gamma] and ß[delta], as
+                                       well as [alpha][delta] and
+                                       ß[gamma], meet.
+
+  Theorem.--_If of any number of lines every one meets every other,
+  whilst all do not_
+
+  _lie in a point, then all lie in     _lie in a plane, then all lie in
+  a plane._                            a point (pencil)._
+
+  § 43. Reciprocal figures as explained lie both in space of three
+  dimensions. If the one is confined to a plane (is formed of elements
+  which lie in a plane), then the reciprocal figure is confined to a
+  pencil (is formed of elements which pass through a point).
+
+  But there is also a more special principle of duality, according to
+  which figures are reciprocal which lie both in a plane or both in a
+  pencil. In the plane we take points and lines as reciprocal elements,
+  for they have this fundamental property in common, that two elements
+  of one kind determine one of the other. In the pencil, on the other
+  hand, lines and planes have to be taken as reciprocal, and here it
+  holds again that two lines or planes determine one plane or line.
+
+  Thus, to one plane figure we can construct one reciprocal figure in
+  the plane, and to each one reciprocal figure in a pencil. We mention a
+  few of these. At first we explain a few names:--
+
+  A figure consisting of n points     A figure consisting of n lines
+  in a plane will be called an        in a plane will be called an
+  n-point.                            n-side.
+
+  A figure consisting of n planes     A figure consisting of n lines
+  in a pencil will be called an       in a pencil will be called an
+  n-flat.                             n-edge.
+
+  It will be understood that an n-side is different from a polygon of n
+  sides. The latter has sides of finite length and n vertices, the
+  former has sides all of infinite extension, and every point where two
+  of the sides meet will be a vertex. A similar difference exists
+  between a solid angle and an n-edge or an n-flat. We notice
+  particularly--
+
+  A four-point has six sides, of      A four-side has six vertices, of
+  which two and two are opposite,     which two and two are opposite,
+  and three diagonal points, which    and three diagonals, which join
+  are intersections of opposite       opposite vertices.
+  sides.
+
+  A four-flat has six edges, of       A four-edge has six faces, of
+  which two and two are opposite,     which two and two are opposite,
+  and three diagonal planes, which    and three diagonal edges, which
+  pass through opposite edges.        are intersections of opposite
+                                      faces.
+
+  A four-side is usually called a complete quadrilateral, and a
+  four-point a complete quadrangle. The above notation, however, seems
+  better adapted for the statement of reciprocal propositions.
+
+  § 44.
+
+  If a point moves in a plane it      If a line moves in a plane it
+  describes a plane curve.            envelopes a plane curve (fig. 15).
+
+  If a plane moves in a pencil it     If a line moves in a pencil it
+  envelopes a cone.                   describes a cone.
+
+  A curve thus appears as generated either by points, and then we call
+  it a "locus," or by lines, and then we call it an "envelope." In the
+  same manner a cone, which means here a surface, appears either as the
+  locus of lines passing through a fixed point, the "vertex" of the
+  cone, or as the envelope of planes passing through the same point.
+
+  [Illustration: FIG. 15.]
+
+  To a surface as locus of points corresponds, in the same manner, a
+  surface as envelope of planes; and to a curve in space as locus of
+  points corresponds a developable surface as envelope of planes.
+
+  It will be seen from the above that we may, by aid of the principle of
+  duality, construct for every figure a reciprocal figure, and that to
+  any property of the one a reciprocal property of the other will exist,
+  as long as we consider only properties which depend upon nothing but
+  the positions and intersections of the different elements and not upon
+  measurement.
+
+  For such propositions it will therefore be unnecessary to prove more
+  than one of two reciprocal theorems.
+
+
+  GENERATION OF CURVES AND CONES OF SECOND ORDER OR SECOND CLASS
+
+  § 45. _Conics._--If we have two projective pencils in a plane,
+  corresponding rays will meet, and their point of intersection will
+  constitute some locus which we have to investigate. Reciprocally, if
+  two projective rows in a plane are given, then the lines which join
+  corresponding points will envelope some curve. We prove first:--
+
+  _Theorem._--If two projective       _Theorem._--If two projective
+  flat pencils lie in a plane, but    rows lie in a plane, but are
+  are neither in perspective nor      neither in perspective nor on a
+  concentric, then the locus of       common base, then the envelope
+  intersections of corresponding      of lines joining corresponding
+  rays is a curve of the second       points is a curve of the second
+  order, that is, no line contains    class, that is, through no point
+  more than two points of the         pass more than two of the
+  locus.                              enveloping lines.
+
+  Proof.--We draw any line t.         _Proof._--We take any point T
+  This cuts each of the pencils in    and join it to all points in each
+  a row, so that we have on t two     row. This gives two concentric
+  rows, and these are projective      pencils, which are projective
+  because the pencils are             because the rows are projective.
+  projective. If corresponding rays   If a line joining corresponding
+  of the two pencils meet on the      points in the two rows passes
+  line t, their intersection will     through T, it will be a line in
+  be a point in the one row which     the one pencil which coincides
+  coincides with its corresponding    with its corresponding line in
+  point in the other. But two         the other. But two projective
+  projective rows on the same base    concentric flat pencils in the
+  cannot have more than two           same plane cannot have more than
+  points of one coincident with       two lines of one coincident with
+  their corresponding points in       their corresponding line in the
+  the other (§ 34).                   other (§ 34).
+
+  It will be seen that the proofs are reciprocal, so that the one may be
+  copied from the other by simply interchanging the words point and
+  line, locus and envelope, row and pencil, and so on. We shall
+  therefore in future prove seldom more than one of two reciprocal
+  theorems, and often state one theorem only, the reader being
+  recommended to go through the reciprocal proof by himself, and to
+  supply the reciprocal theorems when not given.
+
+  § 46. We state the theorems in the pencil reciprocal to the last,
+  without proving them:--
+
+  _Theorem._--If two projective       _Theorem._--If two projective
+  flat pencils are concentric, but    axial pencils lie in the same
+  are neither perspective nor         pencil (their axes meet in a
+  coplanar, then the envelope of      point), but are neither perspective
+  the planes joining corresponding    nor co-axial, then the locus
+  rays is a cone of the second        of lines joining corresponding
+  class; that is, no line through     planes is a cone of the second
+  the common centre contains more     order; that is, no plane in the
+  than two of the enveloping          pencil contains more than two
+  planes.
+
+  § 47. Of theorems about cones of second order and cones of second
+  class we shall state only very few. We point out, however, the
+  following connexion between the curves and cones under consideration:
+
+  The lines which join any point      Every plane section of a cone
+  in space to the points on a curve   of the second order is a curve of
+  of the second order form a cone     the second order.
+  of the second order.
+
+  The planes which join any           Every plane section of a cone
+  point in space to the lines         of the second class is a curve of
+  enveloping a curve of the           the second class.
+  second class envelope themselves
+  a cone of the second class.
+
+  By its aid, or by the principle of duality, it will be easy to obtain
+  theorems about them from the theorems about the curves.
+
+  We prove the first. A curve of the second order is generated by two
+  projective pencils. These pencils, when joined to the point in space,
+  give rise to two projective axial pencils, which generate the cone in
+  question as the locus of the lines where corresponding planes meet.
+
+  §48.
+
+  _Theorem._--The curve of second    _Theorem._--The envelope of
+  order which is generated by two    second class which is generated
+  projective flat pencils passes     by two projective rows contains
+  through the centres of the two     the bases of these rows as
+  pencils.                           enveloping lines or tangents.
+
+  _Proof._--If S and S' are the      _Proof._--If s and s' are the
+  two pencils, then to the ray SS'   two rows, then to the point ss'
+  or p' in the pencil S'             or P' as a point in s'
+  corresponds in the pencil S a      corresponds in s a point P,
+  ray p, which is different from     which is not coincident with P',
+  p', for the pencils are not        for the rows are not
+  perspective. But p and p' meet     perspective. But P and P' are
+  at S, so that S is a point on      joined by s, so that s is one of
+  the curve, and similarly S'.       the enveloping lines, and
+                                     similarly s'.
+
+  It follows that every line in one of the two pencils cuts the curve in
+  two points, viz. once at the centre S of the pencil, and once where it
+  cuts its corresponding ray in the other pencil. These two points,
+  however, coincide, if the line is cut by its corresponding line at S
+  itself. The line p in S, which corresponds to the line SS' in S', is
+  therefore the only line through S which has but one point in common
+  with the curve, or which cuts the curve in two coincident points. Such
+  a line is called a _tangent_ to the curve, touching the latter at the
+  point S, which is called the "point of contact."
+
+  In the same manner we get in the reciprocal investigation the result
+  that through every point in one of the rows, say in s, two tangents
+  may be drawn to the curve, the one being s, the other the line joining
+  the point to its corresponding point in s'. There is, however, one
+  point P in s for which these two lines coincide. Such a point in one
+  of the tangents is called the "point of contact" of the tangent. We
+  thus get--
+
+  _Theorem._--To the line joining    _Theorem._--To the point of
+  the centres of the projective      intersection of the bases of two
+  pencils as a line in one pencil    projective rows as a point in
+  corresponds in the other the       one row corresponds in the other
+  tangent at its centre.             the _point of contact_ of its
+                                     base.
+
+  § 49. Two projective pencils are determined if three pairs of
+  corresponding lines are given. Hence if a1, b1, c1 are three lines in
+  a pencil S1, and a2, b2, c2 the corresponding lines in a projective
+  pencil S2, the correspondence and therefore the curve of the second
+  order generated by the points of intersection of corresponding rays is
+  determined. Of this curve we know the two centres S1 and S2, and the
+  three points a1a2, b1b2, c1c2, hence five points in all. This and the
+  reciprocal considerations enable us to solve the following two
+  problems:
+
+  _Problem._--To construct a curve   _Problem._--To construct a curve
+  of the second order, of which      of the second class, of which
+  five points S1, S2, A, B, C are    five tangents u1, u2, a, b, c
+  given.                             are given.
+
+  In order to solve the left-hand problem, we take two of the given
+  points, say S1 and S2, as centres of pencils. These we make projective
+  by taking the rays a1, b1, c1, which join S1 to A, B, C respectively,
+  as corresponding to the rays a2, b2, c2, which join S2 to A, B, C
+  respectively, so that three rays meet their corresponding rays at the
+  given points A, B, C. This determines the correspondence of the
+  pencils which will generate a curve of the second order passing
+  through A, B, C and through the centres S1 and S2, hence through the
+  five given points. To find more points on the curve we have to
+  construct for any ray in S1 the corresponding ray in S2. This has been
+  done in § 36. But we repeat the construction in order to deduce
+  further properties from it. We also solve the right-hand problem. Here
+  we select two, viz. u1, u2 of the five given lines, u1, u2, a, b, c,
+  as bases of two rows, and the points A1, B1, C1 where a, b, c cut u1
+  as corresponding to the points A2, B2, C2 where a, b, c cut u2.
+
+  We get then the following solutions of the two problems:
+
+  _Solution._--Through the point A   _Solution._--In the line a take
+  draw any two lines, u1 and u2      any two points S1 and S2 as
+  (fig. 16), the first u1 to cut     centres of pencils (fig. 17),
+  the pencil S1 in a row AB1C1,      the first S1 (A1B1C1) to project
+  the other u2 to cut the pencil     the row u1, the other S2
+  S2 in a row AB2C2. These two       (A2B2C2) to project the row u2.
+  rows will be perspective, as the   These two pencils will be
+  point A corresponds to itself,     perspective, the line S1A1 being
+  and the centre of projection       the same as the corresponding
+  will be the point S, where the     line S2A2, and the axis of
+  lines B1B2 and C1C2 meet. To       projection will be the line u,
+  find now for any ray d1 in S1      which joins the intersection B
+  its corresponding ray d2 in S2,    of S1B1 and S2B2 to the
+  we determine the point D1 where    intersection C of S1C1 and S2C2.
+  d1 cuts u1, project this point     To find now for any point D1 in
+  from S to D2 on u2 and join S2     u1 the corresponding point D2 in
+  to D2. This will be the required   u2, we draw S1D1 and project the
+  ray d2 which cuts d1 at some       point D where this line cuts u
+  point D on the curve.              from S2 to u2. This will give
+                                     the required point D2, and the
+                                     line d joining D1 to D2 will be
+                                     a new tangent to the curve.
+
+  § 50. These constructions prove, when rightly interpreted, very
+  important properties of the curves in question.
+
+  [Illustration: FIG. 16.]
+
+  If in fig. 16 we draw in the pencil S1 the ray k1 which passes through
+  the auxiliary centre S, it will be found that the corresponding ray k2
+  cuts it on u2. Hence--
+
+  _Theorem._--In the above           _Theorem._--In the above
+  construction the bases of the      construction (fig. 17) the
+  auxiliary rows u1 and u2 cut the   tangents to the curve from the
+  curve where they cut the rays      centres of the auxiliary pencils
+  S2S and S1S respectively.          S1 and S2 are the lines which
+                                     pass through u2u and u1u
+                                     respectively.
+
+  As A is any given point on the curve, and u1 any line through it, we
+  have solved the problems:
+
+  _Problem._--To find the second     _Problem._--To find the second
+  point in which any line through    tangent which can be drawn from
+  a known point on the curve cuts    any point in a given tangent to
+  the curve.                         the curve.
+
+  If we determine in S1 (fig. 16) the ray corresponding to the ray S2S1
+  in S2, we get the tangent at S1. Similarly, we can determine the point
+  of contact of the tangents u1 or u2 in fig. 17.
+
+  [Illustration: FIG. 17.]
+
+  § 51. If five points are given, of which not three are in a line, then
+  we can, as has just been shown, always draw a curve of the second
+  order through them; we select two of the points as centres of
+  projective pencils, and then one such curve is determined. It will be
+  presently shown that we get always the same curve if two other points
+  are taken as centres of pencils, that therefore five points
+  _determine_ one curve of the second order, and reciprocally, that five
+  tangents determine one curve of the second class. Six points taken at
+  random will therefore not lie on a curve of the second order. In order
+  that this may be the case a certain condition has to be satisfied, and
+  this condition is easily obtained from the construction in § 49, fig.
+  16. If we consider the conic determined by the five points A, S1, S2,
+  K, L, then the point D will be on the curve if, and only if, the
+  points on D1, S, D2 be in a line.
+
+  [Illustration: FIG. 18.]
+
+  This may be stated differently if we take AKS1DS2L (figs. 16 and 18)
+  as a hexagon inscribed in the conic, then AK and DS2 will be opposite
+  sides, so will be KS1 and S2L, as well as S1D and LA. The first two
+  meet in D2, the others in S and D1 respectively. We may therefore
+  state the required condition, together with the reciprocal one, as
+  follows:--
+
+  _Pascal's Theorem._--If a hexagon  _Brianchon's Theorem._--If a
+  be inscribed in a curve of the     hexagon be circumscribed about
+  second order, then the             a curve of the second class, then
+  intersectionsof opposite sides     the lines joining opposite vertices
+  are three points in a line.        are three lines meeting in a point.
+
+  These celebrated theorems, which are known by the names of their
+  discoverers, are perhaps the most fruitful in the whole theory of
+  conics. Before we go over to their applications we have to show that
+  we obtain the same curve if we take, instead of S1, S2, any two other
+  points on the curve as centres of projective pencils.
+
+  § 52. We know that the curve depends only upon the correspondence
+  between the pencils S1 and S2, and not upon the special construction
+  used for finding new points on the curve. The point A (fig. 16 or 18),
+  through which the two auxiliary rows u1, u2 were drawn, may therefore
+  be changed to any other point on the curve. Let us now suppose the
+  curve drawn, and keep the points S1, S2, K, L and D, and hence also
+  the point S fixed, whilst we move A along the curve. Then the line AL
+  will describe a pencil about L as centre, and the point D1 a row on
+  S1D perspective to the pencil L. At the same time AK describes a
+  pencil about K and D2 a row perspective to it on S2D. But by Pascal's
+  theorem D1 and D2 will always lie in a line with S, so that the rows
+  described by D1 and D2 are perspective. It follows that the pencils K
+  and L will themselves be projective, corresponding rays meeting on the
+  curve. This proves that we get the same curve whatever pair of the
+  five given points we take as centres of projective pencils. Hence--
+
+  Only one curve of the second       Only one curve of the second
+  order can be drawn which passes    class can be drawn which touches
+  through five given points.         five given lines.
+
+  We have seen that if on a curve of the second order two points
+  coincide at A, the line joining them becomes the tangent at A. If,
+  therefore, a point on the curve and its tangent are given, this will
+  be equivalent to having given two points on the curve. Similarly, if
+  on the curve of second class a tangent and its point of contact are
+  given, this will be equivalent to two given tangents.
+
+  We may therefore extend the last theorem:
+
+  Only one curve of the second       Only one curve of the second
+  order can be drawn, of which       class can be drawn, of which four
+  four points and the tangent at     tangents and the point of contact
+  oneof them, or three points        at one of them, or three tangents
+  and the tangents at two of         and the points of contact at two
+  them, are given.                   of them, are given.
+
+  § 53. At the same time it has been proved:
+
+  If all points on a curve of the    All tangents to a curve of second
+  second order be joined to any      class are cut by any two of
+  two of them, then the two          them in projective rows, those
+  pencils thus formed are            being corresponding points which
+  projective, those rays being       lie on the same tangent. Hence--
+  corresponding which meet on the
+  curve. Hence--
+
+  The cross-ratio of four rays       The cross-ratio of the four
+  joining a point S on a curve of    points in which any tangent u is
+  second order to four fixed         cut by four fixed tangents a, b, c,
+  points A, B, C, D in the curve     d is independent of the position of
+  is independent of the position     u, and is called the cross-ratio of
+  of S, and is called the cross-     the four tangents a, b, c, d.
+  ratio of the four points A, B,
+  C, D.
+
+  If this cross-ratio equals -1     If this cross-ratio equals -1
+  the four points are said to be    the four tangents are said to be
+  four harmonic points.             four harmonic tangents.
+
+  We have seen that a curve of second order, as generated by projective
+  pencils, has at the centre of each pencil one tangent; and further,
+  that any point on the curve may be taken as centre of such pencil.
+  Hence--
+
+  A curve of second order has       A curve of second class has on
+  at every point one tangent.       every tangent a point of contact.
+
+  § 54. We return to Pascal's and Brianchon's theorems and their
+  applications, and shall, as before, state the results both for curves
+  of the second order and curves of the second class, but prove them
+  only for the former.
+
+  Pascal's theorem may be used when five points are given to find more
+  points on the curve, viz. it enables us to find the point where any
+  line through one of the given points cuts the curve again. It is
+  convenient, in making use of Pascal's theorem, to number the points,
+  to indicate the order in which they are to be taken in forming a
+  hexagon, which, by the way, may be done in 60 different ways. It will
+  be seen that 1 2 (leaving out 3) 4 5 are opposite sides, so are 2 3
+  and (leaving out 4) 5 6, and also 3 4 and (leaving out 5) 6 1.
+
+  If the points 1 2 3 4 5 are given, and we want a 6th point on a line
+  drawn through 1, we know all the sides of the hexagon with the
+  exception of 5 6, and this is found by Pascal's theorem.
+
+  If this line should happen to pass through 1, then 6 and 1 coincide,
+  or the line 6 1 is the tangent at 1. And always if two consecutive
+  vertices of the hexagon approach nearer and nearer, then the side
+  joining them will ultimately become a tangent.
+
+  We may therefore consider a pentagon inscribed in a curve of second
+  order and the tangent at one of its vertices as a hexagon, and thus
+  get the theorem:
+
+  Every pentagon inscribed in a      Every pentagon circumscribed
+  curve of second order has the      about a curve of the second class
+  property that the intersections    has the property that the lines
+  of two pairs of non-consecutive    which join two pairs of non-
+  sides lie in a line with the       consecutive vertices meet on that
+  point where the fifth side cuts    line which joins the fifth vertex
+  the tangent at the opposite        to the point of contact of the
+  vertex.                            opposite side.
+
+  This enables us also to solve the following problems.
+
+  Given five points on a curve of    Given five tangents to a curve
+  second order to construct the      of second class to construct the
+  tangent at any one of them.        point of contact of any one of
+                                     them.
+
+  [Illustration: FIG. 19.]
+
+  If two pairs of adjacent vertices coincide, the hexagon becomes a
+  quadrilateral, with tangents at two vertices. These we take to be
+  opposite, and get the following theorems:
+
+  If a quadrilateral be inscribed    If a quadrilateral be circumscribed
+  in a curve of second order, the    about a curve of second
+  intersections of opposite sides,   class, the lines joining opposite
+  and also the intersections of      vertices, and also the lines joining
+  the tangents at opposite           points of contact of opposite
+  vertices, lie in a line (fig.      sides, meet in a point.
+  19).
+
+  [Illustration: FIG. 20.]
+
+  If we consider the hexagon made up of a triangle and the tangents at
+  its vertices, we get--
+
+  If a triangle is inscribed in a    If a triangle be circumscribed
+  curve of the second order, the     about a curve of second class,
+  points in which the sides are      the lines which join the vertices
+  cut by the tangents at the         to the points of contact of the
+  opposite vertices meet in a        opposite sides meet in a point
+  point.                             (fig. 20).
+
+  § 55. Of these theorems, those about the quadrilateral give rise to a
+  number of others. Four points A, B, C, D may in three different ways
+  be formed into a quadrilateral, for we may take them in the order
+  ABCD, or ACBD, or ACDB, so that either of the points B, C, D may be
+  taken as the vertex opposite to A. Accordingly we may apply the
+  theorem in three different ways.
+
+  Let A, B, C, D be four points on a curve of second order (fig. 21),
+  and let us take them as forming a quadrilateral by taking the points
+  in the order ABCD, so that A, C and also B, D are pairs of opposite
+  vertices. Then P, Q will be the points where opposite sides meet, and
+  E, F the intersections of tangents at opposite vertices. The four
+  points P, Q, E, F lie therefore in a line. The quadrilateral ACBD
+  gives us in the same way the four points Q, R, G, H in a line, and the
+  quadrilateral ABDC a line containing the four points R, P, I, K. These
+  three lines form a triangle PQR.
+
+  The relation between the points and lines in this figure may be
+  expressed more clearly if we consider ABCD as a four-point inscribed
+  in a conic, and the tangents at these points as a four-side
+  circumscribed about it,--viz. it will be seen that P, Q, R are the
+  diagonal points of the four-point ABCD, whilst the sides of the
+  triangle PQR are the diagonals of the circumscribing four-side. Hence
+  the theorem--
+
+  _Any four-point on a curve of the second order and the four-side
+  formed by the tangents at these points stand in this relation that the
+  diagonal points of the four-point lie in the diagonals of the
+  four-side._ And conversely,
+
+  _If a four-point and a circumscribed four-side stand in the above
+  relation, then a curve of the second order may be described which
+  passes through the four points and touches there the four sides of
+  these figures._
+
+  That the last part of the theorem is true follows from the fact that
+  the four points A, B, C, D and the line a, as tangent at A, determine
+  a curve of the second order, and the tangents to this curve at the
+  other points B, C, D are given by the construction which leads to fig.
+  21.
+
+  [Illustration: FIG. 21.]
+
+  The theorem reciprocal to the last is--
+
+  _Any four-side circumscribed about a curve of second class and the
+  four-point formed by the points of contact stand in this relation that
+  the diagonals of the four-side pass through the diagonal points of the
+  four-point._ And conversely,
+
+  _If a four-side and an inscribed four-point stand in the above
+  relation, then a curve of the second class may be described which
+  touches the sides of the four-side at the points of the four-point._
+
+  § 56. The four-point and the four-side in the two reciprocal theorems
+  are alike. Hence if we have a four-point ABCD and a four-side abcd
+  related in the manner described, then not only may a curve of the
+  second order be drawn, but also a curve of the second class, which
+  both touch the lines a, b, c, d at the points A, B, C, D.
+
+  The curve of second order is already more than determined by the
+  points A, B, C and the tangents a, b, c at A, B and C. The point D may
+  therefore be _any_ point on this curve, and d any tangent to the
+  curve. On the other hand the curve of the second class is more than
+  determined by the three tangents a, b, c and their points of contact
+  A, B, C, so that d is any tangent to this curve. It follows that every
+  tangent to the curve of second order is a tangent of a curve of the
+  second class having the same point of contact. In other words, the
+  curve of second order is a curve of second class, and _vice versa_.
+  Hence the important theorems--
+
+  _Every curve of second order is    _Every curve of second class is a
+  a curve of second class._          curve of second order._
+
+  The curves of second order and of second class, having thus been
+  proved to be identical, shall henceforth be called by the common name
+  of _Conics_.
+
+  For these curves hold, therefore, all properties which have been
+  proved for curves of second order or of second class. We may therefore
+  now state Pascal's and Brianchon's theorem thus--
+
+  _Pascal's Theorem._--If a hexagon be inscribed in a conic, then the
+  intersections of opposite sides lie in a line.
+
+  _Brianchon's Theorem._--If a hexagon be circumscribed about a conic,
+  then the diagonals forming opposite centres meet in a point.
+
+  § 57. If we suppose in fig. 21 that the point D together with the
+  tangent d moves along the curve, whilst A, B, C and their tangents a,
+  b, c remain fixed, then the ray DA will describe a pencil about A, the
+  point Q a projective row on the fixed line BC, the point F the row b,
+  and the ray EF a pencil about E. But EF passes always through Q. Hence
+  the pencil described by AD is projective to the pencil described by
+  EF, and therefore to the row described by F on b. At the same time the
+  line BD describes a pencil about B projective to that described by AD
+  (§ 53). Therefore the pencil BD and the row F on b are projective.
+  Hence--
+
+  _If on a conic a point A be taken and the tangent a at this point,
+  then the cross-ratio of the four rays which join A to any four points
+  on the curve is equal to the cross-ratio of the points in which the
+  tangents at these points cut the tangent at A._
+
+  § 58. There are theorems about cones of second order and second class
+  in a pencil which are reciprocal to the above, according to § 43. We
+  mention only a few of the more important ones.
+
+  The locus of intersections of corresponding planes in two projective
+  axial pencils whose axes meet is a cone of the second order.
+
+  The envelope of planes which join corresponding lines in two
+  projective flat pencils, not in the same plane, is a cone of the
+  second class.
+
+  Cones of second order and cones of second class are identical.
+
+  Every plane cuts a cone of the second order in a conic.
+
+  _A cone of second order is uniquely determined by five of its edges or
+  by five of its tangent planes, or by four edges and the tangent plane
+  at one of them, &c. &c._
+
+  _Pascal's Theorem._--If a solid angle of six faces be inscribed in a
+  cone of the second order, then the intersections of opposite faces are
+  three lines in a plane.
+
+  _Brianchon's Theorem._--If a solid angle of six edges be circumscribed
+  about a cone of the second order, then the planes through opposite
+  edges meet in a line.
+
+  Each of the other theorems about conics may be stated for cones of the
+  second order.
+
+  § 59. _Projective Definitions of the Conics._--We now consider the
+  shape of the conics. We know that any line in the plane of the conic,
+  and hence that the line at infinity, either has no point in common
+  with the curve, or one (counting for two coincident points) or two
+  distinct points. If the line at infinity has no point on the curve the
+  latter is altogether finite, and is called an _Ellipse_ (fig. 21). If
+  the line at infinity has only one point in common with the conic, the
+  latter extends to infinity, and has the line at infinity a tangent. It
+  is called a _Parabola_ (fig. 22). If, lastly, the line at infinity
+  cuts the curve in two points, it consists of two separate parts which
+  each extend in two branches to the points at infinity where they meet.
+  The curve is in this case called an _Hyperbola_ (see fig. 20). The
+  tangents at the two points at infinity are finite because the line at
+  infinity is not a tangent. They are called _Asymptotes_. The branches
+  of the hyperbola approach these lines indefinitely as a point on the
+  curves moves to infinity.
+
+  [Illustration: FIG. 22.]
+
+  § 60. That the circle belongs to the curves of the second order is
+  seen at once if we state in a slightly different form the theorem that
+  in a circle all angles at the circumference standing upon the same arc
+  are equal. If two points S1, S2 on a circle be joined to any other two
+  points A and B on the circle, then the angle included by the rays S1A
+  and S1B is equal to that between the rays S2A and S2B, so that as A
+  moves along the circumference the rays S1A and S2A describe equal and
+  therefore projective pencils. The circle can thus be generated by two
+  projective pencils, and is a curve of the second order.
+
+  If we join a point in space to all points on a circle, we get a
+  (circular) cone of the second order (§ 43). Every plane section of
+  this cone is a conic. This conic will be an ellipse, a parabola, or an
+  hyperbola, according as the line at infinity in the plane has no, one
+  or two points in common with the conic in which the plane at infinity
+  cuts the cone. It follows that our curves of second order may be
+  obtained as sections of a circular cone, and that they are identical
+  with the "Conic Sections" of the Greek mathematicians.
+
+  § 61. Any two tangents to a parabola are cut by all others in
+  projective rows; but the line at infinity being one of the tangents,
+  the points at infinity on the rows are corresponding points, and the
+  rows therefore similar. Hence the theorem--
+
+  _The tangents to a parabola cut each other proportionally._
+
+
+  POLE AND POLAR
+
+  § 62. We return once again to fig. 21, which we obtained in § 55.
+
+  If a four-side be circumscribed about and a four-point inscribed in a
+  conic, so that the vertices of the second are the points of contact of
+  the sides of the first, then the triangle formed by the diagonals of
+  the first is the same as that formed by the diagonal points of the
+  other.
+
+  Such a triangle will be called a _polar-triangle_ of the conic, so
+  that PQR in fig. 21 is a polar-triangle. It has the property that on
+  the side p opposite P meet the tangents at A and B, and also those at
+  C and D. From the harmonic properties of four-points and four-sides it
+  follows further that the points L, M, where it cuts the lines AB and
+  CD, are harmonic conjugates with regard to AB and CD respectively.
+
+  If the point P is given, and we draw a line through it, cutting the
+  conic in A and B, then the point Q harmonic conjugate to P with regard
+  to AB, and the point H where the tangents at A and B meet, are
+  determined. But they lie both on p, and therefore this line is
+  determined. If we now draw a second line through P, cutting the conic
+  in C and D, then the point M harmonic conjugate to P with regard to
+  CD, and the point G where the tangents at C and D meet, must also lie
+  on p. As the first line through P already determines p, the second may
+  be any line through P. Now every two lines through P determine a
+  four-point ABCD on the conic, and therefore a polar-triangle which has
+  one vertex at P and its opposite side at p. This result, together with
+  its reciprocal, gives the theorems--
+
+  _All polar-triangles which have one vertex in common have also the
+  opposite side in common._
+
+  _All polar-triangles which have one side in common have also the
+  opposite vertex in common._
+
+  § 63. To any point P in the plane of, but not on, a conic corresponds
+  thus one line p as the side opposite to P in all polar-triangles which
+  have one vertex at P, and reciprocally to every line p corresponds one
+  point P as the vertex opposite to p in all triangles which have p as
+  one side.
+
+  We call the line p the _polar_ of P, and the point P the _pole_ of the
+  line p with regard to the conic.
+
+  If a point lies on the conic, we call the tangent at that point its
+  polar; and reciprocally we call the point of contact the pole of
+  tangent.
+
+  § 64. From these definitions and former results follow--
+
+  The polar of any point P not       The pole of any line p not a
+  on the conic is a line p, which    tangent to the conic is a point
+  has the following properties:--    P, which has the following
+                                     properties:--
+
+  1. On every line through P         1. Of all lines through a point
+  which cuts the conic, the polar    on p from which two tangents
+  of P contains the harmonic         may be drawn to the conic, the
+  conjugate  of P with regard to     pole P contains the line which is
+  those points on the conic.         harmonic conjugate to p, with
+                                     regard to the two tangents.
+
+  2. If tangents can be drawn        2. If p cuts the conic, the
+  from P, their points of contact    tangents at the intersections
+  lie on p.                          meet at P.
+
+  3. Tangents drawn at the           3. The point of contact of
+  points where any line through P    tangents drawn from any point
+  cuts the conic meet on p; and      on p to the conic lie in a line
+  conversely,                        with P; and conversely,
+
+  4. If from any point on p,         4. Tangents drawn at points
+  tangents be drawn, their points    where any line through P cuts the
+  of contact will lie in a line      conic meet on p.
+  with P.
+
+  5. Any four-point on the conic     5. Any four-side circumscribed
+  which has one diagonal point at    about a conic which has one
+  P has the other two lying on p.    diagonal on p has the other two
+                                     meeting at P.
+
+  The truth of 2 follows from 1. If T be a point where p cuts the conic,
+  then one of the points where PT cuts the conic, and which are harmonic
+  conjugates with regard to PT, coincides with T; hence the other
+  does--that is, PT touches the curve at T.
+
+  That 4 is true follows thus: If we draw from a point H on the polar
+  one tangent a to the conic, join its point of contact A to the pole P,
+  determine the second point of intersection B of this line with the
+  conic, and draw the tangent at B, it will pass through H, and will
+  therefore be the second tangent which may be drawn from H to the
+  curve.
+
+  § 65. The second property of the polar or pole gives rise to the
+  theorem--
+
+  From a point in the plane of a     A line in the plane of a conic
+  conic, two, one or no tangents     has two, one or no points in
+  may be drawn to the conic,         common with the conic, according
+  as its polar has two,              as two, one or no tangents
+  one, or no points in common        can be drawn from its pole to the
+  with the curve.                    conic.
+
+  Of any point in the plane of a conic we say that it was _without_, on
+  or _within_ the curve according as two, one or no tangents to the
+  curve pass through it. The points on the conic separate those within
+  the conic from those without. That this is true for a circle is known
+  from elementary geometry. That it also holds for other conics follows
+  from the fact that every conic may be considered as the projection of
+  a circle, which will be proved later on.
+
+  The fifth property of pole and polar stated in § 64 shows how to find
+  the polar of any point and the pole of any line by aid of the
+  straight-edge only. Practically it is often convenient to draw three
+  secants through the pole, and to determine only one of the diagonal
+  points for two of the four-points formed by pairs of these lines and
+  the conic (fig. 22).
+
+  These constructions also solve the problem--
+
+  From a point without a conic, to draw the two tangents to the conic by
+  aid of the straight-edge only.
+
+  For we need only draw the polar of the point in order to find the
+  points of contact.
+
+  § 66. The property of a polar-triangle may now be stated thus--
+
+  In a polar-triangle each side is the polar of the opposite vertex, and
+  each vertex is the pole of the opposite side.
+
+  [Illustration: FIG. 23.]
+
+  If P is one vertex of a polar-triangle, then the other vertices, Q and
+  R, lie on the polar p of P. One of these vertices we may choose
+  arbitrarily. For if from any point Q on the polar a secant be drawn
+  cutting the conic in A and D (fig. 23), and if the lines joining these
+  points to P cut the conic again at B and C, then the line BC will pass
+  through Q. Hence P and Q are two of the vertices on the polar-triangle
+  which is determined by the four-point ABCD. The third vertex R lies
+  also on the line p. It follows, therefore, also--
+
+  _If Q is a point on the polar of P, then P is a point on the polar of
+  Q_; and reciprocally,
+
+  _If q is a line through the pole of p, then p is a line through the
+  pole of q._
+
+  This is a very important theorem. It may also be stated thus--
+
+  _If a point moves along a line describing a row, its polar turns about
+  the pole of the line describing a pencil._
+
+  _This pencil is projective to the row, so that the cross-ratio of four
+  poles in a row equals the cross-ratio of its four polars, which pass
+  through the pole of the row._
+
+  To prove the last part, let us suppose that P, A and B in fig. 23
+  remain fixed, whilst Q moves along the polar p of P. This will make CD
+  turn about P and move R along p, whilst QD and RD describe projective
+  pencils about A and B. Hence Q and R describe projective rows, and
+  hence PR, which is the polar of Q, describes a pencil projective to
+  either.
+
+  § 67. Two points, of which one, and therefore each, lies on the polar
+  of the other, are said to be _conjugate with regard to the conic_; and
+  two lines, of which one, and therefore each, passes through the pole
+  of the other, are said to be _conjugate with regard to the conic_.
+  Hence all points conjugate to a point P lie on the polar of P; all
+  lines conjugate to a line p pass through the pole of p.
+
+  If the line joining two conjugate poles cuts the conic, then the poles
+  are harmonic conjugates with regard to the points of intersection;
+  hence one lies within the other without the conic, and all points
+  conjugate to a point within a conic lie without it.
+
+  Of a polar-triangle any two vertices are conjugate poles, any two
+  sides conjugate lines. If, therefore, one side cuts a conic, then one
+  of the two vertices which lie on this side is within and the other
+  without the conic. The vertex opposite this side lies also without,
+  for it is the pole of a line which cuts the curve. In this case
+  therefore one vertex lies within, the other two without. If, on the
+  other hand, we begin with a side which does not cut the conic, then
+  its pole lies within and the other vertices without. Hence--
+
+  Every polar-triangle has one and only one vertex within the conic.
+
+  We add, without a proof, the theorem--
+
+  The four points in which a conic is cut by two conjugate polars are
+  four harmonic points in the conic.
+
+  § 68. If two conics intersect in four points (they cannot have more
+  points in common, § 52), there exists one and only one four-point
+  which is inscribed in both, and therefore one polar-triangle common to
+  both.
+
+  _Theorem._--Two conics which intersect in four points have always one
+  and only one common polar-triangle; and reciprocally,
+
+  Two conics which have four common tangents have always one and only
+  one common polar-triangle.
+
+
+  DIAMETERS AND AXES OF CONICS
+
+  § 69. _Diameters._--The theorems about the harmonic properties of
+  poles and polars contain, as special cases, a number of important
+  metrical properties of conics. These are obtained if either the pole
+  or the polar is moved to infinity,--it being remembered that the
+  harmonic conjugate to a point at infinity, with regard to two points
+  A, B, is the middle point of the segment AB. The most important
+  properties are stated in the following theorems:--
+
+  _The middle points of parallel chords of a conic lie in a line--viz.
+  on the polar to the point at infinity on the parallel chords._
+
+  This line is called a _diameter_.
+
+  _The polar of every point at infinity is a diameter._
+
+  _The tangents at the end points of a diameter are parallel, and are
+  parallel to the chords bisected by the diameter._
+
+  _All diameters pass through a common point, the pole of the line at
+  infinity._
+
+  _All diameters of a parabola are parallel_, the pole to the line at
+  infinity being the point where the curve touches the line at
+  infinity.
+
+  In case of the ellipse and hyperbola, the pole to the line at infinity
+  is a finite point called the _centre_ of the curve.
+
+  _A centre of a conic bisects every chord through it._
+
+  _The centre of an ellipse is within the curve_, for the line at
+  infinity does not cut the ellipse.
+
+  _The centre of an hyperbola is without the curve_, because the line at
+  infinity cuts the curve. Hence also--
+
+  _From the centre of an hyperbola two tangents can be drawn to the
+  curve which have their point of contact at infinity._ These are called
+  _Asymptotes_ (§ 59).
+
+  _To construct a diameter_ of a conic, draw two parallel chords and
+  join their middle points.
+
+  _To find the centre_ of a conic, draw two diameters; their
+  intersection will be the centre.
+
+  § 70. _Conjugate Diameters._--A polar-triangle with one vertex at the
+  centre will have the opposite side at infinity. The other two sides
+  pass through the centre, and are called _conjugate diameters_, each
+  being the polar of the point at infinity on the other.
+
+  _Of two conjugate diameters each bisects the chords parallel to the
+  other, and if one cuts the curve, the tangents at its ends are
+  parallel to the other diameter._
+
+  Further--
+
+  _Every parallelogram inscribed in a conic has its sides parallel to
+  two conjugate diameters_; and
+
+  _Every parallelogram circumscribed about a conic has as diagonals two
+  conjugate diameters._
+
+  This will be seen by considering the parallelogram in the first case
+  as an inscribed four-point, in the other as a circumscribed four-side,
+  and determining in each case the corresponding polar-triangle. The
+  first may also be enunciated thus--
+
+  _The lines which join any point on an ellipse or an hyperbola to the
+  ends of a diameter are parallel to two conjugate diameters._
+
+  § 71. _If every diameter is perpendicular to its conjugate the conic
+  is a circle._
+
+  For the lines which join the ends of a diameter to any point on the
+  curve include a right angle.
+
+  _A conic which has more than one pair of conjugate diameters at right
+  angles to each other is a circle._
+
+  [Illustration: FIG. 24.]
+
+  Let AA' and BB' (fig. 24) be one pair of conjugate diameters at right
+  angles to each other, CC and DD' a second pair. If we draw through the
+  end point A of one diameter a chord AP parallel to DD', and join P to
+  A', then PA and PA' are, according to § 70, parallel to two conjugate
+  diameters. But PA is parallel to DD', hence PA' is parallel to CC, and
+  therefore PA and PA' are perpendicular. If we further draw the
+  tangents to the conic at A and A', these will be perpendicular to AA',
+  they being parallel to the conjugate diameter BB'. We know thus five
+  points on the conic, viz. the points A and A' with their tangents, and
+  the point P. Through these a circle may be drawn having AA' as
+  diameter; and as through five points one conic only can be drawn, this
+  circle must coincide with the given conic.
+
+  § 72. _Axes._--Conjugate diameters perpendicular to each other are
+  called _axes_, and the points where they cut the curve _vertices_ of
+  the conic.
+
+  In a circle every diameter is an axis, every point on it is a vertex;
+  and any two lines at right angles to each other may be taken as a pair
+  of axes of any circle which has its centre at their intersection.
+
+  [Illustration: FIG. 25.]
+
+  If we describe on a diameter AB of an ellipse or hyperbola a circle
+  concentric to the conic, it will cut the latter in A and B (fig. 25).
+  Each of the semicircles in which it is divided by AB will be partly
+  within, partly without the curve, and must cut the latter therefore
+  again in a point. The circle and the conic have thus four points A, B,
+  C, D, and therefore one polar-triangle, in common (§ 68). Of this the
+  centre is one vertex, for the line at infinity is the polar to this
+  point, both with regard to the circle and the other conic. The other
+  two sides are conjugate diameters of both, hence perpendicular to each
+  other. This gives--
+
+  An ellipse as well as an hyperbola has one pair of axes.
+
+  This reasoning shows at the same time _how to construct the axis of an
+  ellipse or of an hyperbola_.
+
+  _A parabola has one axis_, if we define an axis as a diameter
+  perpendicular to the chords which it bisects. It is easily
+  constructed. The line which bisects any two parallel chords is a
+  diameter. Chords perpendicular to it will be bisected by a parallel
+  diameter, and this is the axis.
+
+  § 73. The first part of the right-hand theorem in § 64 may be stated
+  thus: any two conjugate lines through a point P without a conic are
+  harmonic conjugates with regard to the two tangents that may be drawn
+  from P to the conic.
+
+  If we take instead of P the centre C of an hyperbola, then the
+  conjugate lines become conjugate diameters, and the tangents
+  asymptotes. Hence--
+
+  _Any two conjugate diameters of an hyperbola are harmonic conjugates
+  with regard to the asymptotes._
+
+  As the axes are conjugate diameters at right angles to one another, it
+  follows (§ 23)--
+
+  _The axes of an hyperbola bisect the angles between the asymptotes._
+
+  [Illustration: FIG. 26.]
+
+  Let O be the centre of the hyperbola (fig. 26), t any secant which
+  cuts the hyperbola in C, D and the asymptotes in E, F, then the line
+  OM which bisects the chord CD is a diameter conjugate to the diameter
+  OK which is parallel to the secant t, so that OK and OM are harmonic
+  with regard to the asymptotes. The point M therefore bisects EF. But
+  by construction M bisects CD. It follows that DF = EC, and ED = CF; or
+
+  _On any secant of an hyperbola the segments between the curve and the
+  asymptotes are equal._
+
+  If the chord is changed into a tangent, this gives--
+
+  _The segment between the asymptotes on any tangent to an hyperbola is
+  bisected by the point of contact._
+
+  The first part allows a simple solution of the problem to find any
+  number of points on an hyperbola, of which the asymptotes and one
+  point are given. This is equivalent to three points and the tangents
+  at two of them. This construction requires measurement.
+
+  § 74. For the parabola, too, follow some metrical properties. A
+  diameter PM (fig. 27) bisects every chord conjugate to it, and the
+  pole P of such a chord BC lies on the diameter. But a diameter cuts
+  the parabola once at infinity. Hence--
+
+  _The segment PM which joins the middle point M of a chord of a
+  parabola to the pole P of the chord is bisected by the parabola at A._
+
+  § 75. Two asymptotes and any two tangents to an hyperbola may be
+  considered as a quadrilateral circumscribed about the hyperbola. But
+  in such a quadrilateral the intersections of the diagonals and the
+  points of contact of opposite sides lie in a line (§ 54). If therefore
+  DEFG (fig. 28) is such a quadrilateral, then the diagonals DF and GE
+  will meet on the line which joins the points of contact of the
+  asymptotes, that is, on the line at infinity; hence they are parallel.
+  From this the following theorem is a simple deduction:
+
+  _All triangles formed by a tangent and the asymptotes of an hyperbola
+  are equal in area._
+
+  [Illustration: FIG. 27.]
+
+  [Illustration: FIG. 28.]
+
+  If we draw at a point P (fig. 28) on an hyperbola a tangent, the part
+  HK between the asymptotes is bisected at P. The parallelogram PQOQ'
+  formed by the asymptotes and lines parallel to them through P will be
+  half the triangle OHK, and will therefore be constant. If we now take
+  the asymptotes OX and OY as oblique axes of co-ordinates, the lines OQ
+  and QP will be the co-ordinates of P, and will satisfy the equation xy
+  = const. = a².
+
+  _For the asymptotes as axes of co-ordinates the equation of the
+  hyperbola is xy = const._
+
+
+  INVOLUTION
+
+  [Illustration: FIG. 29.]
+
+  § 76. If we have two projective rows, ABC on u and A'B'C' on u', and
+  place their bases on the same line, then each point in this line
+  counts twice, once as a point in the row u and once as a point in the
+  row u'. In fig. 29 we denote the points as points in the one row by
+  letters above the line A, B, C ..., and as points in the second row by
+  A', B', C' ... below the line. Let now A and B' be the same point,
+  then to A will correspond a point A' in the second, and to B' a point
+  B in the first row. In general these points A' and B will be
+  different. It may, however, happen that they coincide. Then the
+  correspondence is a peculiar one, as the following theorem shows:
+
+  _If two projective rows lie on the same base, and if it happens that
+  to one point in the base the same point corresponds, whether we
+  consider the point as belonging to the first or to the second row,
+  then the same will happen for every point in the base--that is to say,
+  to every point in the line corresponds the same point in the first as
+  in the second row._
+
+  [Illustration: FIG. 30.]
+
+  In order to determine the correspondence, we may assume three pairs of
+  corresponding points in two projective rows. Let then A', B', C', in
+  fig. 30, correspond to A, B, C, so that A and B', and also B and A',
+  denote the same point. Let us further denote the point C' when
+  considered as a point in the first row by D; then it is to be proved
+  that the point D', which corresponds to D, is the same point as C. We
+  know that the cross-ratio of four points is equal to that of the
+  corresponding row. Hence
+
+    (AB, CD) = (A'B', C'D')
+
+  but replacing the dashed letters by those undashed ones which denote
+  the same points, the second cross-ratio equals (BA, DD'), which,
+  according to § 15, equals (AB, D'D); so that the equation becomes
+
+    (AB, CD) = (AB, D'D).
+
+  This requires that C and D' coincide.
+
+  § 77. Two projective rows on the same base, which have the above
+  property, that to every point, whether it be considered as a point in
+  the one or in the other row, corresponds the same point, are said to
+  be in _involution_, or to form an _involution_ of points on the line.
+
+  We mention, but without proving it, that any two projective rows may
+  be placed so as to form an involution.
+
+  An involution may be said to consist of a row of pairs of points, to
+  every point A corresponding a point A', and to A' again the point A.
+  These points are said to be conjugate, or, better, one point is termed
+  the "mate" of the other.
+
+  From the definition, according to which an involution may be
+  considered as made up of two projective rows, follow at once the
+  following important properties:
+
+  1. The cross-ratio of four points equals that of the four conjugate
+  points.
+
+  2. If we call a point which coincides with its mate a "focus" or
+  "double point" of the involution, we may say: An involution has either
+  two foci, or one, or none, and is called respectively a hyperbolic,
+  parabolic or elliptic involution (§ 34).
+
+  3. In an hyperbolic involution any two conjugate points are harmonic
+  conjugates with regard to the two foci.
+
+  For if A, A' be two conjugate points, F1, F2 the two foci, then to the
+  points F1, F2, A, A' in the one row correspond the points F1, F2, A',
+  A in the other, each focus corresponding to itself. Hence (F1F2, AA')
+  = (F1F2, A'A)--that is, we may interchange the two points AA' without
+  altering the value of the cross-ratio, which is the characteristic
+  property of harmonic conjugates (§ 18).
+
+  4. The point conjugate to the point at infinity is called the "centre"
+  of the involution. Every involution has a centre, unless the point at
+  infinity be a focus, in which case we may say that the centre is at
+  infinity.
+
+  In an hyperbolic involution the centre is the middle point between the
+  foci.
+
+  5. The product of the distances of two conjugate points A, A' from the
+  centre O is constant: OA . OA' = c.
+
+  For let A, A' and B, B' be two pairs of conjugate points, the centre,
+  I the point at infinity, then
+
+    (AB, OI) = (A'B', IO),
+
+  or
+
+    OA . OA' = OB . OB'.
+
+  In order to determine the distances of the foci from the centre, we
+  write F for A and A' and get
+
+  OF² = c; OF = ±[root]c.
+
+  Hence if c is positive OF is real, and has two values, equal and
+  opposite. The involution is hyperbolic.
+
+  If c = 0, OF = 0, and the two foci both coincide with the centre. If c
+  is negative, [root]c becomes imaginary, and there are no foci. Hence
+  we may write--
+
+    In an hyperbolic involution, OA·OA' = k²,
+    In a parabolic involution, OA·OA' = 0,
+    In an elliptic involution, OA·OA' = -k².
+
+  From these expressions it follows that conjugate points A, A' in an
+  hyperbolic involution lie on the same side of the centre, and in an
+  elliptic involution on opposite sides of the centre, and that in a
+  parabolic involution one coincides with the centre.
+
+  In the first case, for instance, OA·OA' is positive; hence OA and OA'
+  have the same sign.
+
+  It also follows that two segments, AA' and BB', between pairs of
+  conjugate points have the following positions: in an hyperbolic
+  involution they lie either one altogether within or altogether without
+  each other; in a parabolic involution they have one point in common;
+  and in an elliptic involution they overlap, each being partly within
+  and partly without the other.
+
+  _Proof._--We have OA·OA' = OB·OB' = k² in case of an hyperbolic
+  involution. Let A and B be the points in each pair which are nearer to
+  the centre O. If now A, A' and B, B' lie on the same side of O, and if
+  B is nearer to O than A, so that OB < OA, then OB' > OA'; hence B' lies
+  farther away from O than A', or the segment AA' lies within BB'. And so
+  on for the other cases.
+
+  6. An involution is determined--
+
+    ([alpha]) By two pairs of conjugate points. Hence also
+    (ß) By one pair of conjugate points and the centre;
+    ([gamma]) By the two foci;
+    ([delta]) By one focus and one pair of conjugate points;
+    ([epsilon]) By one focus and the centre.
+
+  7. The condition that A, B, C and A', B', C' may form an involution
+  may be written in one of the forms--
+
+     (AB, CC') = (A'B', C'C),
+
+  or (AB, CA') = (A'B', C'A),
+
+  or (AB, C'A') = (A'B', CA),
+
+  for each expresses that in the two projective rows in which A, B, C
+  and A', B', C' are conjugate points two conjugate elements may be
+  interchanged.
+
+  8. Any three pairs. A, A', B, B', C, C', of conjugate points are
+  connected by the relations:
+
+    AB'·BC'·CA'   AB'·BC·C'A'   AB·B'C'·CA'   AB·B'C·C'A'
+    ----------- = ----------- = ----------- = ----------- = -1.
+    A'B·B'C·C'A   A'B·B'C'·CA   A'B'·BC·C'A   A'B'·BC'·CA
+
+  These relations readily follow by working out the relations in (7)
+  (above).
+
+  § 78. _Involution of a quadrangle.--The sides of any four-point are
+  cut by any line in six points in involution, opposite sides being cut
+  in conjugate points._
+
+  Let A1B1C1D1 (fig. 31) be the four-point. If its sides be cut by the
+  line p in the points A, A', B, B', C, C', if further, C1D1 cuts the
+  line A1B1 in C2, and if we project the row A1B1C2C to p once from D1
+  and once from C1, we get (A'B', C'C) = (BA, C'C).
+
+  Interchanging in the last cross-ratio the letters in each pair we get
+  (A'B', C'C) = (AB, CC'). Hence by § 77 (7) the points are in
+  involution.
+
+  The theorem may also be stated thus:
+
+  _The three points in which any line cuts the sides of a triangle and
+  the projections, from any point in the plane, of the vertices of the
+  triangle on to the same line are six points in involution._
+
+  [Illustration: FIG. 31.]
+
+  Or again--
+
+  The projections from any point on to any line of the six vertices of a
+  four-side are six points in involution, the projections of opposite
+  vertices being conjugate points.
+
+  This property gives a simple means to construct, by aid of the
+  straight edge only, in an involution of which two pairs of conjugate
+  points are given, to any point its conjugate.
+
+  § 79. _Pencils in Involution._--The theory of involution may at once
+  be extended from the row to the flat and the axial pencil--viz. we say
+  that there is an involution in a flat or in an axial pencil if any
+  line cuts the pencil in an involution of points. An involution in a
+  pencil consists of pairs of conjugate rays or planes; it has two, one
+  or no _focal rays_ (double lines) or _planes_, but nothing
+  corresponding to a centre.
+
+  An involution in a flat pencil contains always one, and in general
+  only one, pair of conjugate rays which are perpendicular to one
+  another. For in two projective flat pencils exist always two
+  corresponding right angles (§ 40).
+
+  Each involution in an axial pencil contains in the same manner one
+  pair of conjugate planes at right angles to one another.
+
+  As a rule, there exists but one pair of conjugate lines or planes at
+  right angles to each other. But it is possible that there are more,
+  and then there is an infinite number of such pairs. An involution in a
+  flat pencil, in which every ray is perpendicular to its conjugate ray,
+  is said to be _circular_. That such involution is possible is easily
+  seen thus: if in two concentric flat pencils each ray on one is made
+  to correspond to that ray on the other which is perpendicular to it,
+  then the two pencils are projective, for if we turn the one pencil
+  through a right angle each ray in one coincides with its corresponding
+  ray in the other. But these two projective pencils are in involution.
+
+  A circular involution has no focal rays, because no ray in a pencil
+  coincides with the ray perpendicular to it.
+
+  § 80. _Every elliptical involution in a row may be considered as a
+  section of a circular involution._
+
+  In an elliptical involution any two segments AA' and BB' lie partly
+  within and partly without each other (fig. 32). Hence two circles
+  described on AA' and BB' as diameters will intersect in two points E
+  and E'. The line EE' cuts the base of the involution at a point O,
+  which has the property that OA·OA' = OB . OB', for each is equal to
+  OE . OE'. The point O is therefore the centre of the involution. If we
+  wish to construct to any point C the conjugate point C', we may draw
+  the circle through CEE'. This will cut the base in the required point
+  C' for OC·OC' = OA·OA'. But EC and EC' are at right angles. Hence the
+  involution which is obtained by joining E or E' to the points in the
+  given involution is circular. This may also be expressed thus:
+
+  [Illustration: FIG. 32.]
+
+  _Every elliptical involution has the property that there are two
+  definite points in the plane from which any two conjugate points are
+  seen under a right angle._
+
+  At the same time the following problem has been solved:
+
+  To determine the centre and also the point corresponding to any given
+  point in an elliptical involution of which two pairs of conjugate
+  points are given.
+
+  § 81. _Involution Range on a Conic._--By the aid of § 53, the points
+  on a conic may be made to correspond to those on a line, so that the
+  row of points on the conic is projective to a row of points on a line.
+  We may also have two projective rows on the same conic, and these will
+  be in involution as soon as one point on the conic has the same point
+  corresponding to it all the same to whatever row it belongs. An
+  involution of points on a conic will have the property (as follows
+  from its definition, and from § 53) that the lines which join
+  conjugate points of the involution to any point on the conic are
+  conjugate lines of an involution in a pencil, and that a fixed tangent
+  is cut by the tangents at conjugate points on the conic in points
+  which are again conjugate points of an involution on the fixed
+  tangent. For such involution on a conic the following theorem holds:
+
+  _The lines which join corresponding points in an involution on a conic
+  all pass through a fixed point; and reciprocally, the points of
+  intersection of conjugate lines in an involution among tangents to a
+  conic lie on a line._
+
+  [Illustration: FIG. 33]
+
+  We prove the first part only. The involution is determined by two
+  pairs of conjugate points, say by A, A' and B, B' (fig. 33). Let AA'
+  and BB' meet in P. If we join the points in involution to any point on
+  the conic, and the conjugate points to another point on the conic, we
+  obtain two projective pencils. We take A and A' as centres of these
+  pencils, so that the pencils A(A'BB') and A'(AB'B) are projective, and
+  in perspective position, because AA' corresponds to A'A. Hence
+  corresponding rays meet in a line, of which two points are found by
+  joining AB' to A'B and AB to A'B'. It follows that the _axis_ of
+  perspective is the polar of the point P, where AA' and BB' meet. If we
+  now wish to construct to any other point C on the conic the
+  corresponding point C', we join C to A' and the point where this line
+  cuts p to A. The latter line cuts the conic again in C'. But we know
+  from the theory of pole and polar that the line CC' passes through P.
+  The point of concurrence is called the "pole of the involution," and
+  the line of collinearity of the meets is called the "axis of the
+  involution."
+
+
+  INVOLUTION DETERMINED BY A CONIC ON A LINE.--FOCI
+
+  § 82. The polars, with regard to a conic, of points in a row p form a
+  pencil P projective to the row (§ 66). This pencil cuts the base of
+  the row p in a projective row.
+
+  If A is a point in the given row, A' the point where the polar of A
+  cuts p, then A and A' will be corresponding points. If we take A' a
+  point in the first row, then the polar of A' will pass through A, so
+  that A corresponds to A'--in other words, the rows are in involution.
+  The conjugate points in this involution are conjugate points with
+  regard to the conic. Conjugate points coincide only if the polar of a
+  point A passes through A--that is, if A lies on the conic. Hence--
+
+  _A conic determines on every line in its plane an involution, in which
+  those points are conjugate which are also conjugate with regard to the
+  conic._
+
+  _If the line cuts the conic the involution is hyperbolic, the points
+  of intersection being the foci._
+
+  _If the line touches the conic the involution is parabolic, the two
+  foci coinciding at the point of contact._
+
+  _If the line does not cut the conic the involution is elliptic, having
+  no foci._
+
+  If, on the other hand, we take a point P in the plane of a conic, we
+  get to each line a through P one conjugate line which joins P to the
+  pole of a. These pairs of conjugate lines through P form an involution
+  in the pencil at P. The focal rays of this involution are the tangents
+  drawn from P to the conic. This gives the theorem reciprocal to the
+  last, viz:--
+
+  _A conic determines in every pencil in its plane an involution,
+  corresponding lines being conjugate lines with regard to the conic._
+
+  _If the point is without the conic the involution is hyperbolic, the
+  tangents from the points being the focal rays._
+
+  _If the point lies on the conic the involution is parabolic, the
+  tangent at the point counting for coincident focal rays._
+
+  _If the point is within the conic the involution is elliptic, having
+  no focal rays._
+
+  It will further be seen that the involution determined by a conic on
+  any line p is a section of the involution, which is determined by the
+  conic at the pole P of p.
+
+  § 83. _Foci._--The centre of a pencil in which the conic determines a
+  circular involution is called a "focus" of the conic.
+
+  In other words, a focus is such a point that every line through it is
+  perpendicular to its conjugate line. The polar to a focus is called a
+  _directrix_ of the conic.
+
+  From the definition it follows that _every focus lies on an axis_, for
+  the line joining a focus to the centre of the conic is a diameter to
+  which the conjugate lines are perpendicular; and _every line joining
+  two foci is an axis_, for the perpendiculars to this line through the
+  foci are conjugate to it. These conjugate lines pass through the pole
+  of the line, the pole lies therefore at infinity, and the line is a
+  diameter, hence by the last property an axis.
+
+  It follows that all _foci lie on one axis_, for no line joining a
+  point in one axis to a point in the other can be an axis.
+
+  As the conic determines in the pencil which has its centre at a focus
+  a circular involution, no tangents can be drawn from the focus to the
+  conic. Hence _each focus lies within a conic_; and _a directrix does
+  not cut the conic_.
+
+  Further properties are found by the following considerations:
+
+  § 84. Through a point P one line p can be drawn, which is with regard
+  to a given conic conjugate to a given line q, viz. that line which
+  joins the point P to the pole of the line q. If the line q is made to
+  describe a pencil about a point Q, then the line p will describe a
+  pencil about P. These two pencils will be projective, for the line p
+  passes through the pole of q, and whilst q describes the pencil Q, its
+  pole describes a projective row, and this row is perspective to the
+  pencil P.
+
+  We now take the point P on an axis of the conic, draw any line p
+  through it, and from the pole of p draw a perpendicular q to p. Let q
+  cut the axis in Q. Then, in the pencils of conjugate lines, which have
+  their centres at P and Q, the lines p and q are conjugate lines at
+  right angles to one another. Besides, to the axis as a ray in either
+  pencil will correspond in the other the perpendicular to the axis (§
+  72). The conic generated by the intersection of corresponding lines in
+  the two pencils is therefore the circle on PQ as diameter, _so that
+  every line in P is perpendicular to its corresponding line in Q_.
+
+  To every point P on an axis of a conic corresponds thus a point Q,
+  such that conjugate lines through P and Q are perpendicular.
+
+  We shall show that these _point-pairs_ P, Q _form an involution_. To
+  do this let us move P along the axis, and with it the line p, keeping
+  the latter parallel to itself. Then P describes a row, p a perspective
+  pencil (of parallels), and the pole of p a projective row. At the same
+  time the line q describes a pencil of parallels perpendicular to p,
+  and perspective to the row formed by the pole of p. The point Q,
+  therefore, where q cuts the axis, describes a row projective to the
+  row of points P. The two points P and Q describe thus two projective
+  rows on the axis; and not only does P as a point in the first row
+  correspond to Q, but also Q as a point in the first corresponds to P.
+  The two rows therefore form an involution. _The centre of this
+  involution, it is easily seen, is the centre of the conic._
+
+  _A focus of this involution has the property that any two conjugate
+  lines through it are perpendicular; hence, it is a focus to the
+  conic._
+
+  Such involution exists on each axis. But only one of these can have
+  foci, because all foci lie on the same axis. The involution on one of
+  the axes is elliptic, and appears (§ 80) therefore as the section of
+  two circular involutions in two pencils whose centres lie in the other
+  axis. These centres are foci, hence the one axis contains two foci,
+  the other axis none; _or every central conic has two foci which lie on
+  one axis equidistant from the centre_.
+
+  The axis which contains the foci is called the _principal axis_; in
+  case of an hyperbola it is the axis which cuts the curve, because the
+  foci lie within the conic.
+
+  In case of the parabola there is but one axis. The involution on this
+  axis has its centre at infinity. One focus is therefore at infinity,
+  the one focus only is finite. _A parabola has only one focus._
+
+  [Illustration: FIG. 34.]
+
+  § 85. If through any point P (fig. 34) on a conic the tangent PT and
+  the normal PN (i.e. the perpendicular to the tangent through the point
+  of contact) be drawn, these will be conjugate lines with regard to the
+  conic, and at right angles to each other. They will therefore cut the
+  principal axis in two points, which are conjugate in the involution
+  considered in § 84; hence they are harmonic conjugates with regard to
+  the foci. If therefore the two foci F1 and F2 be joined to P, these
+  lines will be harmonic with regard to the tangent and normal. As the
+  latter are perpendicular, they will bisect the angles between the
+  other pair. Hence--
+
+  _The lines joining any point on a conic to the two foci are equally
+  inclined to the tangent and normal at that point._
+
+  In case of the parabola this becomes--
+
+  _The line joining any point on a parabola to the focus and the
+  diameter through the point, are equally inclined to the tangent and
+  normal at that point._
+
+  From the definition of a focus it follows that--
+
+  _The segment of a tangent between the directrix and the point of
+  contact is seen from the focus belonging to the directrix under a
+  right angle_, because the lines joining the focus to the ends of this
+  segment are conjugate with regard to the conic, and therefore
+  perpendicular.
+
+  With equal ease the following theorem is proved:
+
+  _The two lines which join the points of contact of two tangents each
+  to one focus, but not both to the same, are seen from the intersection
+  of the tangents under equal angles._
+
+  § 86. Other focal properties of a conic are obtained by the following
+  considerations:
+
+  [Illustration: FIG. 35.]
+
+  Let F (fig. 35) be a focus to a conic, f the corresponding directrix,
+  A and B the points of contact of two tangents meeting at T, and P the
+  point where the line AB cuts the directrix. Then TF will be the polar
+  of P (because polars of F and T meet at P). Hence TF and PF are
+  conjugate lines through a focus, and therefore perpendicular. They are
+  further harmonic conjugates with regard to FA and FB (§§ 64 and 13),
+  so that they bisect the angles formed by these lines. This by the way
+  proves--
+
+  _The segments between the point of intersection of two tangents to a
+  conic and their points of contact are seen from a focus under equal
+  angles._
+
+  If we next draw through A and B lines parallel to TF, then the points
+  A1, B1 where these cut the directrix will be harmonic conjugates with
+  regard to P and the point where FT cuts the directrix. The lines FT
+  and FP bisect therefore also the angles between FA1 and FB1. From this
+  it follows easily that the triangles FAA1 and FBB1 are equiangular,
+  and therefore similar, so that FA : AA1 = FB : BB1.
+
+  The triangles AA1A2 and BB1B2 formed by drawing perpendiculars from A
+  and B to the directrix are also similar, so that AA1 : AA2 = = BB1 :
+  BB2. This, combined with the above proportion, gives FA : AA2 = FB :
+  BB2. Hence the theorem:
+
+  _The ratio of the distances of any point on a conic from a focus and
+  the corresponding directrix is constant._
+
+  To determine this ratio we consider its value for a vertex on the
+  principal axis. In an ellipse the focus lies between the two vertices
+  on this axis, hence the focus is nearer to a vertex than to the
+  corresponding directrix. Similarly, in an hyperbola a vertex is nearer
+  to the directrix than to the focus. In a parabola the vertex lies
+  halfway between directrix and focus.
+
+  It follows in an ellipse the ratio between the distance of a point
+  from the focus to that from the directrix is less than unity, in the
+  parabola it equals unity, and in the hyperbola it is greater than
+  unity.
+
+  It is here the same which focus we take, because the two foci lie
+  symmetrical to the axis of the conic. If now P is any point on the
+  conic having the distances r1 and r2 from the foci and the distances
+  d1 and d2 from the corresponding directrices, then r1/d1 = r2/d2 =
+  e, where e is constant. Hence also r1 ± r2 / d1 ± d2 = e.
+
+  In the ellipse, which lies between the directrices, d1 + d2 is
+  constant, therefore also r1 +r2. In the hyperbola on the other hand d1
+  - d2 is constant, equal to the distance between the directrices,
+  therefore in this case r1 - r2 is constant.
+
+  If we call the distances of a point on a conic from the focus its
+  focal distances we have the theorem:
+
+  _In an ellipse the sum of the focal distances is constant; and in an
+  hyperbola the difference of the focal distances is constant._
+
+  _This constant sum or difference equals in both cases the length of
+  the principal axis._
+
+
+  PENCIL OF CONICS
+
+  § 87. Through four points A, B, C, D in a plane, of which no three lie
+  in a line, an infinite number of conics may be drawn, viz. through
+  these four points and any fifth one single conic. This system of
+  conics is called a pencil of conics. Similarly, all conics touching
+  four fixed lines form a system such that any fifth tangent determines
+  one and only one conic. We have here the theorems:
+
+  The pairs of points in which        The pairs of tangents which
+  any line is cut by a system of      can be drawn from a point to
+  conics through four fixed points    a system of conics touching four
+  are in involution.                  fixed lines are in involution.
+
+  [Illustration: FIG. 36.]
+
+  We prove the first theorem only. Let ABCD (fig. 36) be the four-point,
+  then any line t will cut two opposite sides AC, BD in the points E,
+  E', the pair AD, BC in points F, F', and any conic of the system in M,
+  N, and we have A(CD, MN) = B(CD, MN).
+
+  If we cut these pencils by t we get
+
+     (EF, MN) = (F'E', MN)
+
+  or (EF, MN) = (E'F', NM).
+
+  But this is, according to § 77 (7), the condition that M, N are
+  corresponding points in the involution determined by the point pairs
+  E, E', F, F' in which the line t cuts pairs of opposite sides of the
+  four-point ABCD. This involution is independent of the particular
+  conic chosen.
+
+  § 88. There follow several important theorems:
+
+  _Through four points two, one, or no conics may be drawn which touch
+  any given line, according as the involution determined by the given
+  four-point on the line has real, coincident or imaginary foci._
+
+  _Two, one, or no conics may be drawn which touch four given lines and
+  pass through a given point, according as the involution determined by
+  the given four-side at the point has real, coincident or imaginary
+  focal rays._
+
+  For the conic through four points which touches a given line has its
+  point of contact at a focus of the involution determined by the
+  four-point on the line.
+
+  As a special case we get, by taking the line at infinity:
+
+  _Through four points of which none is at infinity either two or no
+  parabolas may be drawn._
+
+  The problem of drawing a conic through four points and touching a
+  given line is solved by determining the points of contact on the line,
+  that is, by determining the foci of the involution in which the line
+  cuts the sides of the four-point. The corresponding remark holds for
+  the problem of drawing the conics which touch four lines and pass
+  through a given point.
+
+
+  RULED QUADRIC SURFACES
+
+  § 89. We have considered hitherto projective rows which lie in the
+  same plane, in which case lines joining corresponding points envelop a
+  conic. We shall now consider projective rows whose bases do not meet.
+  In this case, corresponding points will be joined by lines which do
+  not lie in a plane, but on some surface, which like every surface
+  generated by lines is called a _ruled_ surface. This surface clearly
+  contains the bases of the two rows.
+
+  If the points in either row be joined to the base of the other, we
+  obtain two axial pencils which are also projective, those planes being
+  corresponding which pass through corresponding points in the given
+  rows. If A', A be two corresponding points, [alpha], [alpha]' the
+  planes in the axial pencils passing through them, then AA' will be the
+  line of intersection of the corresponding planes [alpha], [alpha]' and
+  also the line joining corresponding points in the rows.
+
+  If we cut the whole figure by a plane this will cut the axial pencils
+  in two projective flat pencils, and the curve of the second order
+  generated by these will be the curve in which the plane cuts the
+  surface. Hence
+
+  _The locus of lines joining corresponding points in two projective
+  rows which do not lie in the same plane is a surface which contains
+  the bases of the rows, and which can also be generated by the lines of
+  intersection of corresponding planes in two projective axial pencils.
+  This surface is cut by every plane in a curve of the second order,
+  hence either in a conic or in a line-pair. No line which does not lie
+  altogether on the surface can have more than two points in common with
+  the surface, which is therefore said to be of the second order or is
+  called a ruled quadric surface._
+
+  That no line which does not lie on the surface can cut the surface in
+  more than two points is seen at once if a plane be drawn through the
+  line, for this will cut the surface in a conic. It follows also that a
+  line which contains more than two points of the surface lies
+  altogether on the surface.
+
+  § 90. Through any point in space one line can always be drawn cutting
+  two given lines which do not themselves meet.
+
+  If therefore three lines in space be given of which no two meet, then
+  through every point in either one line may be drawn cutting the other
+  two.
+
+  _If a line moves so that it always cuts three given lines of which no
+  two meet, then it generates a ruled quadric surface._
+
+  Let a, b, c be the given lines, and p, q, r ... lines cutting them in
+  the points A, A', A" ...; B, B', B" ...; C, C', C" ... respectively;
+  then the planes through a containing p, q, r, and the planes through b
+  containing the same lines, may be taken as corresponding planes in two
+  axial pencils which are projective, because both pencils cut the line
+  c in the same row, C, C', C" ...; the surface can therefore be
+  generated by projective axial pencils.
+
+  Of the lines p, q, r ... no two can meet, for otherwise the lines a,
+  b, c which cut them would also lie in their plane. There is a single
+  infinite number of them, for one passes through each point of a. These
+  lines are said to form a set of lines on the surface.
+
+  If now three of the lines p, q, r be taken, then every line d cutting
+  them will have three points in common with the surface, and will
+  therefore lie altogether on it. This gives rise to a second set of
+  lines on the surface. From what has been said the theorem follows:
+
+  _A ruled quadric surface contains two sets of straight lines. Every
+  line of one set cuts every line of the other, but no two lines of the
+  same set meet._
+
+  _Any two lines of the same set may be taken as bases of two projective
+  rows, or of two projective pencils which generate the surface. They
+  are cut by the lines of the other set in two projective rows._
+
+  The plane at infinity like every other plane cuts the surface either
+  in a conic proper or in a line-pair. In the first case the surface is
+  called an _Hyperboloid of one sheet_, in the second an _Hyperbolic
+  Paraboloid_.
+
+  The latter may be generated by a line cutting three lines of which one
+  lies at infinity, that is, cutting two lines and remaining parallel to
+  a given plane.
+
+
+  QUADRIC SURFACES
+
+  § 91. The conics, the cones of the second order, and the ruled quadric
+  surfaces complete the figures which can be generated by projective
+  rows or flat and axial pencils, that is, by those aggregates of
+  elements which are of one dimension (§§ 5, 6). We shall now consider
+  the simpler figures which are generated by aggregates of two
+  dimensions. The space at our disposal will not, however, allow us to
+  do more than indicate a few of the results.
+
+  § 92. We establish a correspondence between the lines and planes in
+  pencils in space, or reciprocally between the points and lines in two
+  or more planes, but consider principally pencils.
+
+  In two pencils we may either make planes correspond to planes and
+  lines to lines, or else planes to lines and lines to planes. If hereby
+  the condition be satisfied that to a flat, or axial, pencil
+  corresponds in the first case a projective flat, or axial, pencil, and
+  in the second a projective axial, or flat, pencil, the pencils are
+  said to be _projective_ in the first case and _reciprocal_ in the
+  second.
+
+  For instance, two pencils which join two points S1 and S2 to the
+  different points and lines in a given plane [pi] are projective (and
+  in perspective position), if those lines and planes be taken as
+  corresponding which meet the plane [pi] in the same point or in the
+  same line. In this case every plane through both centres S1 and S2 of
+  the two pencils will correspond to itself. If these pencils are
+  brought into any other position they will be projective (but not
+  perspective).
+
+  _The correspondence between two projective pencils is uniquely
+  determined, if to four rays (or planes) in the one the corresponding
+  rays (or planes) in the other are given, provided that no three rays
+  of either set lie in a plane._
+
+  Let a, b, c, d be four rays in the one, a', b', c', d' the
+  corresponding rays in the other pencil. We shall show that we can find
+  for every ray e in the first a single corresponding ray e' in the
+  second. To the axial pencil a (b, c, d ...) formed by the planes which
+  join a to b, c, d ..., respectively corresponds the axial pencil a'
+  (b', c', d' ... ), and this correspondence is determined. Hence, the
+  plane a'e' which corresponds to the plane ae is determined. Similarly
+  the plane b'e' may be found and both together determine the ray e'.
+
+  Similarly the correspondence between two reciprocal pencils is
+  determined if for four rays in the one the corresponding planes in the
+  other are given.
+
+  § 93. We may now combine--
+
+  1. Two reciprocal pencils.
+
+    Each ray cuts its corresponding plane in a point, the locus of these
+    points is a quadric surface.
+
+  2. Two projective pencils.
+
+    Each plane cuts its corresponding plane in a line, but a ray as a
+    rule does not cut its corresponding ray. The locus of points where a
+    ray cuts its corresponding ray is a twisted cubic. The lines where a
+    plane cuts its corresponding plane are secants.
+
+  3. Three projective pencils.
+
+    The locus of intersection of corresponding planes is a cubic
+    surface.
+
+  Of these we consider only the first two cases.
+
+  § 94. If two pencils are reciprocal, then to a plane in either
+  corresponds a line in the other, to a flat pencil an axial pencil, and
+  so on. Every line cuts its corresponding plane in a point. If S1 and
+  S2 be the centres of the two pencils, and P be a point where a line a1
+  in the first cuts its corresponding plane [alpha]2, _then the line b2
+  in the pencil S2 which passes through P will meet its corresponding
+  plane ß1 in P_. For b2 is a line in the plane [alpha]2. The
+  corresponding plane ß1 must therefore pass through the line a1, hence
+  through P.
+
+  The points in which the lines in S1 cut the planes corresponding to
+  them in S2 are therefore the same as the points in which the lines in
+  S2 cut the planes corresponding to them in S1.
+
+  _The locus of these points is a surface which is cut by a plane in a
+  conic or in a line-pair and by a line in not more than two points
+  unless it lies altogether on the surface. The surface itself is
+  therefore called a quadric surface, or a surface of the second order._
+
+  To prove this we consider any line p in space.
+
+  The flat pencil in S1 which lies in the plane drawn through p and the
+  corresponding axial pencil in S2 determine on p two projective rows,
+  and those points in these which coincide with their corresponding
+  points lie on the surface. But there exist only two, or one, or no
+  such points, unless every point coincides with its corresponding
+  point. In the latter case the line lies altogether on the surface.
+
+  This proves also that a plane cuts the surface in a curve of the
+  second order, as no line can have more than two points in common with
+  it. To show that this is a curve of the same kind as those considered
+  before, we have to show that it can be generated by projective flat
+  pencils. We prove first that this is true for any plane through the
+  centre of one of the pencils, and afterwards that every point on the
+  surface may be taken as the centre of such pencil. Let then [alpha]1
+  be a plane through S1. To the flat pencil in S1 which it contains
+  corresponds in S2 a projective axial pencil with axis a2 and this cuts
+  [alpha]1 in a second flat pencil. These two flat pencils in [alpha]1
+  are projective, and, in general, neither concentric nor perspective.
+  They generate therefore a conic. But if the line a2 passes through S1
+  the pencils will have S1 as common centre, and may therefore have two,
+  or one, or no lines united with their corresponding lines. The section
+  of the surface by the plane [alpha]1 will be accordingly a line-pair
+  or a single line, or else the plane [alpha]1 will have only the point
+  S1 in common with the surface.
+
+  Every line l1 through S1 cuts the surface in two points, viz. first in
+  S1 and then at the point where it cuts its corresponding plane. If now
+  the corresponding plane passes through S1, as in the case just
+  considered, then the two points where l1 cuts the surface coincide at
+  S1, and the line is called a tangent to the surface with S1 as point
+  of contact. Hence if l1 be a tangent, it lies in that plane [tau]1
+  which corresponds to the line S2S1 as a line in the pencil S2. The
+  section of this plane has just been considered. It follows that--
+
+  _All tangents to quadric surface at the centre of one of the
+  reciprocal pencils lie in a plane which is called the tangent plane to
+  the surface at that point as point of contact._
+
+  _To the line joining the centres of the two pencils as a line in one
+  corresponds in the other the tangent plane at its centre._
+
+  _The tangent plane to a quadric surface either cuts the surface in two
+  lines, or it has only a single line, or else only a single point in
+  common with the surface._
+
+  _In the first case the point of contact is said to be hyperbolic, in
+  the second parabolic, in the third elliptic._
+
+  § 95. It remains to be proved that every point S on the surface may be
+  taken as centre of one of the pencils which generate the surface. Let
+  S be any point on the surface [Phi]' generated by the reciprocal
+  pencils S1 and S2. We have to establish a reciprocal correspondence
+  between the pencils S and S1, so that the surface generated by them is
+  identical with [Phi]. To do this we draw two planes [alpha]1 and ß1
+  through S1, cutting the surface [Phi] in two conics which we also
+  denote by [alpha]1 and ß1. These conics meet at S1, and at some other
+  point T where the line of intersection of [alpha]1 and ß1 cuts the
+  surface.
+
+  In the pencil S we draw some plane [sigma] which passes through T, but
+  not through S1 or S2. It will cut the two conics first at T, and
+  therefore each at some other point which we call A and B respectively.
+  These we join to S by lines a and b, and now establish the required
+  correspondence between the pencils S1 and S as follows:--To S1T shall
+  correspond the plane [sigma], to the plane [alpha]1 the line a, and to
+  ß1 the line b, hence to the flat pencil in [alpha]1 the axial pencil
+  a. These pencils are made projective by aid of the conic in [alpha]1.
+
+  In the same manner the flat pencil in ß1 is made projective to the
+  axial pencil b by aid of the conic in ß1, corresponding elements being
+  those which meet on the conic. This determines the correspondence, for
+  we know for more than four rays in S1 the corresponding planes in S.
+  The two pencils S and S1 thus made reciprocal generate a quadric
+  surface [Phi]', which passes through the point S and through the two
+  conics [alpha]1 and ß1.
+
+  The two surfaces [Phi] and [Phi]' have therefore the points S and S1
+  and the conics [alpha]1 and ß1 in common. To show that they are
+  identical, we draw a plane through S and S2, cutting each of the
+  conics [alpha]1 and ß1 in two points, which will always be possible.
+  This plane cuts [Phi] and [Phi]' in two conics which have the point S
+  and the points where it cuts [alpha]1 and ß1 in common, that is five
+  points in all. The conics therefore coincide.
+
+  This proves that all those points P on [Phi]' lie on [Phi] which have
+  the property that the plane SS2P cuts the conics [alpha]1, ß1 in two
+  points each. If the plane SS2P has not this property, then we draw a
+  plane SS1P. This cuts each surface in a conic, and these conics have
+  in common the points S, S1, one point on each of the conics [alpha]1,
+  ß1, and one point on one of the conics through S and S2 which lie on
+  both surfaces, hence five points. They are therefore coincident, and
+  our theorem is proved.
+
+  § 96. The following propositions follow:--
+
+  _A quadric surface has at every point a tangent plane._
+
+  _Every plane section of a quadric surface is a conic or a line-pair._
+
+  _Every line which has three points in common with a quadric surface
+  lies on the surface._
+
+  _Every conic which has five points in common with a quadric surface
+  lies on the surface._
+
+  _Through two conics which lie in different planes, but have two points
+  in common, and through one external point always one quadric surface
+  may be drawn._
+
+  § 97. _Every plane which cuts a quadric surface in a line-pair is a
+  tangent plane._ For every line in this plane through the centre of the
+  line-pair (the point of intersection of the two lines) cuts the
+  surface in two coincident points and is therefore a tangent to the
+  surface, _the centre of the line-pair being the point of contact_.
+
+  _If a quadric surface contains a line, then every plane through this
+  line cuts the surface in a line-pair (or in two coincident lines)._
+  For this plane cannot cut the surface in a conic. Hence:--
+
+  _If a quadric surface contains one line p then it contains an infinite
+  number of lines, and through every point Q on the surface, one line q
+  can be drawn which cuts p._ For the plane through the point Q and the
+  line p cuts the surface in a line-pair which must pass through Q and
+  of which p is one line.
+
+  _No two such lines q on the surface can meet_. For as both meet p
+  their plane would contain p and therefore cut the surface in a
+  triangle.
+
+  _Every line which cuts three lines q will be on the surface_; for it
+  has three points in common with it.
+
+  _Hence the quadric surfaces which contain lines are the same as the
+  ruled quadric surfaces considered in_ §§ 89-93, but with one important
+  exception. In the last investigation we have left out of consideration
+  the possibility of a plane having only one line (two coincident lines)
+  in common with a quadric surface.
+
+  § 98. To investigate this case we suppose first that there is one
+  point A on the surface through which two different lines a, b can be
+  drawn, which lie altogether on the surface.
+
+  If P is any other point on the surface which lies neither on a nor b,
+  then the plane through P and a will cut the surface in a second line
+  a' which passes through P and which cuts a. Similarly there is a line
+  b' through P which cuts b. These two lines a' and b' _may_ coincide,
+  but then they must coincide with PA.
+
+  If this happens for one point P, it happens for every other point Q.
+  For if two different lines could be drawn through Q, then by the same
+  reasoning the line PQ would be altogether on the surface, hence two
+  lines would be drawn through P against the assumption. From this
+  follows:--
+
+  _If there is one point on a quadric surface through which one, but
+  only one, line can be drawn on the surface, then through every point
+  one line can be drawn, and all these lines meet in a point. The
+  surface is a cone of the second order_.
+
+  _If through one point on a quadric surface, two, and only two, lines
+  can be drawn on the surface, then through every point two lines may be
+  drawn, and the surface is ruled quadric surface._
+
+  _If through one point on a quadric surface no line on the surface can
+  be drawn, then the surface contains no lines._
+
+  Using the definitions at the end of § 95, we may also say:--
+
+  _On a quadric surface the points are all hyperbolic, or all parabolic,
+  or all elliptic._
+
+  As an example of a quadric surface with elliptical points, we mention
+  the sphere which may be generated by two reciprocal pencils, where to
+  each line in one corresponds the plane perpendicular to it in the
+  other.
+
+  § 99. _Poles and Polar Planes._--The theory of poles and polars with
+  regard to a conic is easily extended to quadric surfaces.
+
+  Let P be a point in space not on the surface, which we suppose not to
+  be a cone. On every line through P which cuts the surface in two
+  points we determine the harmonic conjugate Q of P with regard to the
+  points of intersection. Through one of these lines we draw two planes
+  [alpha] and ß. The locus of the points Q in [alpha] is a line a, the
+  polar of P with regard to the conic in which [alpha] cuts the surface.
+  Similarly the locus of points Q in ß is a line b. This cuts a, because
+  the line of intersection of [alpha] and ß contains but one point Q.
+  The locus of all points Q therefore is a plane. _This plane is called
+  the polar plane of the point P, with regard to the quadric surface. If
+  P lies on the surface we take the tangent plane of P as its polar._
+
+  The following propositions hold:--
+
+  1. _Every point has a polar plane_, which is constructed by drawing
+  the polars of the point with regard to the conics in which two planes
+  through the point cut the surface.
+
+  2. _If Q is a point in the polar of P, then P is a point in the polar
+  of Q_, because this is true with regard to the conic in which a plane
+  through PQ cuts the surface.
+
+  3. _Every plane is the polar plane of one point, which is called the
+  Pole of the plane._
+
+  The pole to a plane is found by constructing the polar planes of three
+  points in the plane. Their intersection will be the pole.
+
+  4. _The points in which the polar plane of P cuts the surface are
+  points of contact of tangents drawn from P to the surface_, as is
+  easily seen. Hence:--
+
+  5. _The tangents drawn from a point P to a quadric surface form a cone
+  of the second order_, for the polar plane of P cuts it in a conic.
+
+  6. _If the pole describes a line a, its polar plane will turn about
+  another line a'_, as follows from 2. _These lines a and a' are said to
+  be conjugate with regard to the surface._
+
+  § 100. The pole of the line at infinity is called the _centre_ of the
+  surface. If it lies at the infinity, the plane at infinity is a
+  tangent plane, and the surface is called a _paraboloid_.
+
+  _The polar plane to any point at infinity passes through the centre,
+  and is called a diametrical plane._
+
+  _A line through the centre is called a diameter. It is bisected at the
+  centre. The line conjugate to it lies at infinity._
+
+  _If a point moves along a diameter its polar plane turns about the
+  conjugate line at infinity_; that is, _it moves parallel to itself,
+  its centre moving on the first line._
+
+  _The middle points of parallel chords lie in a plane_, viz. in the
+  polar plane of the point at infinity through which the chords are
+  drawn.
+
+  _The centres of parallel sections lie in a diameter which is a line
+  conjugate to the line at infinity in which the planes meet._
+
+
+  TWISTED CUBICS
+
+  § 101. If two pencils with centres S1 and S2 are made projective, then
+  to a ray in one corresponds a ray in the other, to a plane a plane, to
+  a flat or axial pencil a projective flat or axial pencil, and so on.
+
+  There is a double infinite number of lines in a pencil. We shall see
+  that a single infinite number of lines in one pencil meets its
+  corresponding ray, and that the points of intersection form a curve in
+  space.
+
+  Of the double infinite number of planes in the pencils each will meet
+  its corresponding plane. This gives a system of a double infinite
+  number of lines in space. We know (§ 5) that there is a quadruple
+  infinite number of lines in space. From among these we may select
+  those which satisfy one or more given conditions. The systems of lines
+  thus obtained were first systematically investigated and classified by
+  Plücker, in his _Geometrie des Raumes_. He uses the following names:--
+
+  A _treble infinite_ number of lines, that is, all lines which satisfy
+  one condition, are said to form a _complex of lines_; e.g. all lines
+  cutting a given line, or all lines touching a surface.
+
+  A _double infinite_ number of lines, that is, all lines which satisfy
+  two conditions, or which are common to two complexes, are said to form
+  a _congruence of lines_; e.g. all lines in a plane, or all lines
+  cutting two curves, or all lines cutting a given curve twice.
+
+  A _single infinite_ number of lines, that is, all lines which satisfy
+  three conditions, or which belong to three complexes, form a _ruled
+  surface_; e.g. one set of lines on a ruled quadric surface, or
+  developable surfaces which are formed by the tangents to a curve.
+
+  It follows that all lines in which corresponding planes in two
+  projective pencils meet form a congruence. We shall see this
+  congruence consists of all lines which cut a twisted cubic twice, or
+  of all _secants_ to a twisted cubic.
+
+  § 102. Let l1 be the line S1S2 as a line in the pencil S1. To it
+  corresponds a line l2 in S2. _At each of the centres two corresponding
+  lines meet._ The two axial pencils with l1 and l2 as axes are
+  projective, and, as, their axes meet at S2, the intersections of
+  corresponding planes form a cone of the second order (§ 58), with S2
+  as centre. If [pi]1 and [pi]2 be corresponding planes, then their
+  intersection will be a line p2 which passes through S2. Corresponding
+  to it in S1 will be a line p1 which lies in the plane [pi]1, and which
+  therefore meets p2 at some point P. Conversely, if p2 be any line in
+  S2 which meets its corresponding line p1 at a point P, then to the
+  plane l2p2 will correspond the plane l1p1, that is, the plane S1S2P.
+  These planes intersect in p2, so that p2 is a line on the quadric cone
+  generated by the axial pencils l1 and l2. Hence:--
+
+  _All lines in one pencil which meet their corresponding lines in the
+  other form a cone of the second order which has its centre at the
+  centre of the first pencil, and passes through the centre of the
+  second._
+
+  From this follows that the points in which corresponding rays meet lie
+  on two cones of the second order which have the ray joining their
+  centres in common, and form therefore, together with the line S1S2 or
+  l1, the intersection of these cones. Any plane cuts each of the cones
+  in a conic. These two conics have necessarily that point in common in
+  which it cuts the line l1, and therefore besides either one or three
+  other points. It follows that the curve is of the third order as a
+  plane may cut it in three, but not in more than three, points.
+  Hence:--
+
+  _The locus of points in which corresponding lines on two projective
+  pencils meet is a curve of the third order or a "twisted cubic" k,
+  which passes through the centres of the pencils, and which appears as
+  the intersection of two cones of the second order, which have one line
+  in common._
+
+  _A line belonging to the congruence determined by the pencils is a
+  secant of the cubic; it has two, or one, or no points in common with
+  this cubic, and is called accordingly a secant proper, a tangent, or a
+  secant improper of the cubic._ A secant improper may be considered, to
+  use the language of coordinate geometry, as a secant with imaginary
+  points of intersection.
+
+  § 103. If a1 and a2 be any two corresponding lines in the two pencils,
+  then corresponding planes in the axial pencils having a1 and a2 as
+  axes generate a ruled quadric surface. If P be any point on the cubic
+  k, and if p1, p2 be the corresponding rays in S1 and S2 which meet at
+  P, then to the plane a1p1 in S1 corresponds a2p2 in S2. These
+  therefore meet in a line through P.
+
+  This may be stated thus:--
+
+  _Those secants of the cubic which cut a ray a1, drawn through the
+  centre S1 of one pencil, form a ruled quadric surface which passes
+  through both centres, and which contains the twisted cubic k. Of such
+  surfaces an infinite number exists. Every ray through S1 or S2 which
+  is not a secant determines one of them._
+
+  If, however, the rays a1 and a2 are secants meeting at A, then the
+  ruled quadric surface becomes a cone of the second order, having A as
+  centre. Or _all lines of the congruence which pass through a point on
+  the twisted cubic k form a cone of the second order_. In other words,
+  the projection of a twisted cubic from any point in the curve on to
+  any plane is a conic.
+
+  If a1 is not a secant, but made to pass through any point Q in space,
+  the ruled quadric surface determined by a1 will pass through Q. _There
+  will therefore be one line of the congruence passing through Q, and
+  only one._ For if two such lines pass through Q, then the lines S1Q
+  and S2Q will be corresponding lines; hence Q will be a point on the
+  cubic k, and an infinite number of secants will pass through it.
+  Hence:--
+
+  _Through every point in space not on the twisted cubic one and only
+  one secant to the cubic can be drawn._
+
+  § 104. The fact that all the secants through a point on the cubic form
+  a quadric cone shows that the centres of the projective pencils
+  generating the cubic are not distinguished from any other points on
+  the cubic. If we take any two points S, S' on the cubic, and draw the
+  secants through each of them, we obtain two quadric cones, which have
+  the line SS' in common, and which intersect besides along the cubic.
+  If we make these two pencils having S and S' as centres projective by
+  taking four rays on the one cone as corresponding to the four rays on
+  the other which meet the first on the cubic, the correspondence is
+  determined. These two pencils will generate a cubic, and the two cones
+  of secants having S and S' as centres will be identical with the above
+  cones, for each has five rays in common with one of the first, viz.
+  the line SS' and the four lines determined for the correspondence;
+  therefore these two cones intersect in the original cubic. This gives
+  the theorem:--
+
+  _On a twisted cubic any two points may be taken as centres of
+  projective pencils which generate the cubic, corresponding planes
+  being those which meet on the same secant._
+
+  Of the two projective pencils at S and S' we may keep the first fixed,
+  and move the centre of the other along the curve. The pencils will
+  hereby remain projective, and a plane [alpha] in S will be cut by its
+  corresponding plane [alpha]' always in the same secant a. Whilst S'
+  moves along the curve the plane [alpha]' will turn about a, describing
+  an axial pencil.
+
+  AUTHORITIES.--In this article we have given a purely geometrical
+  theory of conics, cones of the second order, quadric surfaces, &c. In
+  doing so we have followed, to a great extent, Reye's _Geometrie der
+  Lage_, and to this excellent work those readers are referred who wish
+  for a more exhaustive treatment of the subject. Other works especially
+  valuable as showing the development of the subject are: Monge,
+  _Géométrie descriptive_: Carnot, _Géométrie de position_ (1803),
+  containing a theory of transversals; Poncelet's great work _Traité des
+  propriétés projectives des figures_ (1822); Möbins, _Barycentrischer
+  Calcul_ (1826); Steiner, _Abhängigkeit geometrischer Gestalten_
+  (1832), containing the first full discussion of the projective
+  relations between rows, pencils, &c.; Von Staudt, _Geometrie der Lage_
+  (1847) and _Beiträge zur Geometrie der Lage_ (1856-1860), in which a
+  system of geometry is built up from the beginning without any
+  reference to number, so that ultimately a number itself gets a
+  geometrical definition, and in which imaginary elements are
+  systematically introduced into pure geometry; Chasles, _Aperēu
+  historique_ (1837), in which the author gives a brilliant account of
+  the progress of modern geometrical methods, pointing out the
+  advantages of the different purely geometrical methods as compared
+  with the analytical ones, but without taking as much account of the
+  German as of the French authors; Id., _Rapport sur les progrčs de la
+  géométrie_ (1870), a continuation of the _Aperēu_; Id., _Traité de
+  géométrie supérieure_ (1852); Cremona, _Introduzione ad una teoria
+  geometrica delle curve piane_ (1862) and its continuation _Preliminari
+  di una teoria geometrica delle superficie_ (German translations by
+  Curtze). As more elementary books, we mention: Cremona, _Elements of
+  Projective Geometry_, translated from the Italian by C. Leudesdorf
+  (2nd ed., 1894); J.W. Russell, _Pure Geometry_ (2nd ed., 1905).
+       (O. H.)
+
+
+III. DESCRIPTIVE GEOMETRY
+
+This branch of geometry is concerned with the methods for representing
+solids and other figures in three dimensions by drawings in one plane.
+The most important method is that which was invented by Monge towards
+the end of the 18th century. It is based on parallel projections to a
+plane by rays perpendicular to the plane. Such a projection is called
+orthographic (see PROJECTION, § 18). If the plane is horizontal the
+projection is called the plan of the figure, and if the plane is
+vertical the elevation. In Monge's method a figure is represented by its
+plan and elevation. It is therefore often called drawing in plan and
+elevation, and sometimes simply orthographic projection.
+
+  § 1. We suppose then that we have two planes, one horizontal, the
+  other vertical, and these we call the planes of plan and of elevation
+  respectively, or the horizontal and the vertical plane, and denote
+  them by the letters [pi]1 and [pi]2. Their line of intersection is
+  called the axis, and will be denoted by xy.
+
+  If the surface of the drawing paper is taken as the plane of the plan,
+  then the vertical plane will be the plane perpendicular to it through
+  the axis xy. To bring this also into the plane of the drawing paper we
+  turn it about the axis till it coincides with the horizontal plane.
+  This process of turning one plane down till it coincides with another
+  is called _rabatting_ one to the other. Of course there is no
+  necessity to have one of the two planes horizontal, but even when this
+  is not the case it is convenient to retain the above names.
+
+  [Illustration: FIG. 37.]
+
+  [Illustration: FIG. 38.]
+
+  The whole arrangement will be better understood by referring to fig.
+  37. A point A in space is there projected by the perpendicular AA1 and
+  AA2 to the planes [pi]1 and [pi]2 so that A1 and A2 are the horizontal
+  and vertical projections of A.
+
+  If we remember that a line is perpendicular to a plane that is
+  perpendicular to every line in the plane if only it is perpendicular
+  to any two intersecting lines in the plane, we see that the axis which
+  is perpendicular both to AA1 and to AA2 is also perpendicular to A1A0
+  and to A2A0 because these four lines are all in the same plane. Hence,
+  if the plane [pi]2 be turned about the axis till it coincides with the
+  plane [pi]1, then A2A0 will be the continuation of A1A0. This position
+  of the planes is represented in fig. 38, in which the line A1A2 is
+  perpendicular to the axis x.
+
+  Conversely any two points A1, A2 in a line perpendicular to the axis
+  will be the projections of some point in space when the plane [pi]2 is
+  turned about the axis till it is perpendicular to the plane [pi]1,
+  because in this position the two perpendiculars to the planes [pi]1
+  and [pi]2 through the points A1 and A2 will be in a plane and
+  therefore meet at some point A.
+
+  _Representation of Points._--We have thus the following method of
+  representing in a single plane the position of points in space:--_we
+  take in the plane a line xy as the axis, and then any pair of points
+  A1, A2 in the plane on a line perpendicular to the axis represent a
+  point A in space_. If the line A1A2 cuts the axis at A0, and if at A1
+  a perpendicular be erected to the plane, then the point A will be in
+  it at a height A1A = A0A2 above the plane. This gives the position of
+  the point A relative to the plane [pi]1. In the same way, if in a
+  perpendicular to [pi]2 through A2 a point A be taken such that A2A =
+  A0A1, then this will give the point A relative to the plane [pi]2.
+
+  [Illustration: FIG. 39.]
+
+  § 2. The two planes [pi]1, [pi]2 in their original position divide
+  space into four parts. These are called the four quadrants. We suppose
+  that the plane [pi]2 is turned as indicated in fig. 37, so that the
+  point P comes to Q and R to S, then the quadrant in which the point A
+  lies is called the first, and we say that in the first quadrant a
+  point lies above the horizontal and in front of the vertical plane.
+  Now we go round the axis in the sense in which the plane [pi]2 is
+  turned and come in succession to the second, third and fourth
+  quadrant. In the second a point lies above the plane of the plan and
+  behind the plane of elevation, and so on. In fig. 39, which represents
+  a side view of the planes in fig. 37 the quadrants are marked, and in
+  each a point with its projection is taken. Fig. 38 shows how these are
+  represented when the plane [pi]2 is turned down. We see that
+
+  _A point lies in the first quadrant if the plan lies below, the
+  elevation above the axis; in the second if plan and elevation both lie
+  above; in the third if the plan lies above, the elevation below; in
+  the fourth if plan and elevation both lie below the axis._
+
+  _If a point lies in the horizontal plane_, its elevation lies in the
+  axis and the plan coincides with the point itself. _If a point lies in
+  the vertical plane_, its plan lies in the axis and the elevation
+  coincides with the point itself. _If a point lies in the axis_, both
+  its plan and elevation lie in the axis and coincide with it.
+
+  Of each of these propositions, which will easily be seen to be true,
+  the converse holds also.
+
+  § 3. _Representation of a Plane._--As we are thus enabled to represent
+  points in a plane, we can represent any finite figure by representing
+  its separate points. It is, however, not possible to represent a plane
+  in this way, for the projections of its points completely cover the
+  planes [pi]1 and [pi]2, and no plane would appear different from any
+  other. But any plane [alpha] cuts each of the planes [pi]1, [pi]2 in a
+  line. These are called the traces of the plane. They cut each other in
+  the axis at the point where the latter cuts the plane [alpha].
+
+  _A plane is determined by its two traces, which are two lines that
+  meet on the axis_, and, conversely, _any two lines which meet on the
+  axis determine a plane_.
+
+  _If the plane is parallel to the axis its traces are parallel to the
+  axis._ Of these one may be at infinity; then the plane will cut one of
+  the planes of projection at infinity and will be parallel to it. Thus
+  a plane parallel to the horizontal plane of the plan has only one
+  finite trace, viz. that with the plane of elevation.
+
+  [Illustration: FIG. 40.]
+
+  _If the plane passes through the axis both its traces coincide with
+  the axis._ This is the only case in which the representation of the
+  plane by its two traces fails. A third plane of projection is
+  therefore introduced, which is best taken perpendicular to the other
+  two. We call it simply the third plane and denote it by [pi]3. As it
+  is perpendicular to [pi]1, it may be taken as the plane of elevation,
+  its line of intersection [gamma] with [pi]1 being the axis, and be
+  turned down to coincide with [pi]1. This is represented in fig. 40. OC
+  is the axis xy whilst OA and OB are the traces of the third plane.
+  They lie in one line [gamma]. The plane is rabatted about [gamma] to
+  the horizontal plane. A plane [alpha] through the axis xy will then
+  show in it a trace [alpha]3. In fig. 40 the lines OC and OP will thus
+  be the traces of a plane through the axis xy, which makes an angle POQ
+  with the horizontal plane.
+
+  We can also find the trace which any other plane makes with [pi]3. In
+  rabatting the plane [pi]3 its trace OB with the plane [pi]2 will come
+  to the position OD. Hence a plane ß having the traces CA and CB will
+  have with the third plane the trace ß3, or AD if OD = OB.
+
+  It also follows immediately that--
+
+  _If a plane [alpha] is perpendicular to the horizontal plane, then
+  every point in it has its horizontal projection in the horizontal
+  trace of the plane_, as all the rays projecting these points lie in
+  the plane itself.
+
+  _Any plane which is perpendicular to the horizontal plane has its
+  vertical trace perpendicular to the axis._
+
+  _Any plane which is perpendicular to the vertical plane has its
+  horizontal trace perpendicular to the axis and the vertical
+  projections of all points in the plane lie in this trace._
+
+  § 4. _Representation of a Line._--A line is determined either by two
+  points in it or by two planes through it. We get accordingly two
+  representations of it either by projections or by traces.
+
+  First.--_A line a is represented by its projections a1 and a2 on the
+  two planes [pi]1 and [pi]2._ These may be any two lines, for, bringing
+  the planes [pi]1, [pi]2 into their original position, the planes
+  through these lines perpendicular to [pi]1 and [pi]2 respectively will
+  intersect in some line a which has a1, a2 as its projections.
+
+  Secondly.--_A line a is represented by its traces--that is, by the
+  points in which it cuts the two planes [pi]1, [pi]2._ Any two points
+  may be taken as the traces of a line in space, for it is determined
+  when the planes are in their original position as the line joining the
+  two traces. This representation becomes undetermined if the two traces
+  coincide in the axis. In this case we again use a third plane, or else
+  the projections of the line.
+
+  The fact that there are different methods of representing points and
+  planes, and hence two methods of representing lines, suggests the
+  principle of duality (section ii., _Projective Geometry_, § 41). It is
+  worth while to keep this in mind. It is also worth remembering that
+  traces of planes or lines always lie in the planes or lines which they
+  represent. Projections do not as a rule do this excepting when the
+  point or line projected lies in one of the planes of projection.
+
+  Having now shown how to represent points, planes and lines, we have to
+  state the conditions which must hold in order that these elements may
+  lie one in the other, or else that the figure formed by them may
+  possess certain metrical properties. It will be found that the former
+  are very much simpler than the latter.
+
+  Before we do this, however, we shall explain the notation used; for it
+  is of great importance to have a systematic notation. We shall denote
+  points in space by capitals A, B, C; planes in space by Greek letters
+  [alpha], ß, [gamma]; lines in space by small letters a, b, c;
+  horizontal projections by suffixes 1, like A1, a1; vertical
+  projections by suffixes 2, like A2, a2; traces by single and double
+  dashes [alpha]' [alpha]", a', a". Hence P1 will be the horizontal
+  projection of a point P in space; a line a will have the projections
+  a1, a2 and the traces a' and a"; a plane [alpha] has the traces
+  [alpha]' and [alpha]".
+
+  § 5. _If a point lies in a line, the projections of the point lie in
+  the projections of the line._
+
+  _If a line lies in a plane, the traces of the line lie in the traces
+  of the plane._
+
+  These propositions follow at once from the definitions of the
+  projections and of the traces.
+
+  If a point lies in two lines its projections must lie in the
+  projections of both. Hence
+
+  _If two lines, given by their projections, intersect, the intersection
+  of their planes and the intersection of their elevations must lie in a
+  line perpendicular to the axis_, because they must be the projections
+  of the point common to the two lines.
+
+  Similarly--_If two lines given by their traces lie in the same plane
+  or intersect, then the lines joining their horizontal and vertical
+  traces respectively must meet on the axis_, because they must be the
+  traces of the plane through them.
+
+  § 6. _To find the projections of a line which joins two points A, B
+  given by their projections A1, A2 and B1, B2_, we join A1, B1 and A2,
+  B2; these will be the projections required. For example, the traces of
+  a line are two points in the line whose projections are known or at
+  all events easily found. They are the traces themselves and the feet
+  of the perpendiculars from them to the axis.
+
+  Hence _if a' a" (fig. 41) are the traces of a line a, and if the
+  perpendiculars from them cut the axis in P and Q respectively, then
+  the line a'Q will be the horizontal and a"P the vertical projection of
+  the line_.
+
+  [Illustration: FIG. 41.]
+
+  Conversely, if the projections a1, a2 of a line are given, and if
+  these cut the axis in Q and P respectively, then _the perpendiculars
+  Pa' and Qa" to the axis drawn through these points cut the projections
+  a1 and a2 in the traces a' and a"_.
+
+  _To find the line of intersection of two planes_, we observe that this
+  line lies in both planes; its traces must therefore lie in the traces
+  of both. Hence the points where the horizontal traces of the given
+  planes meet will be the horizontal, and the point where the vertical
+  traces meet the vertical trace of the line required.
+
+  § 7. _To decide whether a point A, given by its projections, lies in a
+  plane [alpha], given by its traces_, we draw a line p by joining A to
+  some point in the plane [alpha] and determine its traces. If these lie
+  in the traces of the plane, then the line, and therefore the point A,
+  lies in the plane; otherwise not. This is conveniently done by joining
+  A1 to some point p' in the trace [alpha]'; this gives p1; and the
+  point where the perpendicular from p' to the axis cuts the latter we
+  join to A2; this gives p2. If the vertical trace of this line lies in
+  the vertical trace of the plane, then, and then only, does the line p,
+  and with it the point A, lie in the plane [alpha].
+
+  § 8. _Parallel planes have parallel traces_, because parallel planes
+  are cut by any plane, hence also by [pi]1 and by [pi]2, in parallel
+  lines.
+
+  _Parallel lines have parallel projections_, because points at infinity
+  are projected to infinity.
+
+  _If a line is parallel to a plane, then lines through the traces of
+  the line and parallel to the traces of the plane must meet on the
+  axis_, because these lines are the traces of a plane parallel to the
+  given plane.
+
+  § 9. _To draw a plane through two intersecting lines or through two
+  parallel lines_, we determine the traces of the lines; the lines
+  joining their horizontal and vertical traces respectively will be the
+  horizontal and vertical traces of the plane. They will meet, at a
+  finite point or at infinity, on the axis if the lines do intersect.
+
+  _To draw a plane through a line and a point without the line_, we join
+  the given point to any point in the line and determine the plane
+  through this and the given line.
+
+  _To draw a plane through three points which are not in a line_, we
+  draw two of the lines which each join two of the given points and draw
+  the plane through them. If the traces of all three lines AB, BC, CA be
+  found, these must lie in two lines which meet on the axis.
+
+  § 10. We have in the last example got more points, or can easily get
+  more points, than are necessary for the determination of the figure
+  required--in this case the traces of the plane. This will happen in a
+  great many constructions and is of considerable importance. It may
+  happen that some of the points or lines obtained are not convenient in
+  the actual construction. The horizontal traces of the lines AB and AC
+  may, for instance, fall very near together, in which case the line
+  joining them is not well defined. Or, one or both of them may fall
+  beyond the drawing paper, so that they are practically non-existent
+  for the construction. In this case the traces of the line BC may be
+  used. Or, if the vertical traces of AB and AC are both in convenient
+  position, so that the vertical trace of the required plane is found
+  and one of the horizontal traces is got, then we may join the latter
+  to the point where the vertical trace cuts the axis.
+
+  The draughtsman must remember that the lines which he draws are not
+  mathematical lines without thickness, and therefore every drawing is
+  affected by some errors. It is therefore very desirable to be able
+  constantly to check the latter. Such checks always present themselves
+  when the same result can be obtained by different constructions, or
+  when, as in the above case, some lines must meet on the axis, or if
+  three points must lie in a line. A careful draughtsman will always
+  avail himself of these checks.
+
+  § 11. _To draw a plane through a given point parallel to a given plane
+  [alpha]_, we draw through the point two lines which are parallel to
+  the plane [alpha], and determine the plane through them; or, as we
+  know that the traces of the required plane are parallel to those of
+  the given one (§ 8), we need only draw one line l through the point
+  parallel to the plane and find one of its traces, say the vertical
+  trace l"; a line through this parallel to the vertical trace of
+  [alpha] will be the vertical trace ß" of the required plane ß, and a
+  line parallel to the horizontal trace of [alpha] meeting ß" on the
+  axis will be the horizontal trace ß'.
+
+  [Illustration: FIG. 42.]
+
+  Let A1 A2 (fig. 42) be the given point, [alpha]' [alpha]" the given
+  plane, a line l1 through A1, parallel to [alpha]' and a horizontal
+  line l2 through A2 will be the projections of a line l through A
+  parallel to the plane, because the horizontal plane through this line
+  will cut the plane [alpha] in a line c which has its horizontal
+  projection c1 parallel to [alpha]'.
+
+  § 12. We now come to the metrical properties of figures.
+
+  _A line is perpendicular to a plane if the projections of the line are
+  perpendicular to the traces of the plane._ We prove it for the
+  horizontal projection. If a line p is perpendicular to a plane
+  [alpha], every plane through p is perpendicular to [alpha]; hence also
+  the vertical plane which projects the line p to p1. As this plane is
+  perpendicular both to the horizontal plane and to the plane [alpha],
+  it is also perpendicular to their intersection--that is, to the
+  horizontal trace of [alpha]. It follows that every line in this
+  projecting plane, therefore also p1, the plan of p, is perpendicular
+  to the horizontal trace of [alpha].
+
+  _To draw a plane through a given point A perpendicular to a given line
+  p_, we first draw through some point O in the axis lines [gamma]',
+  [gamma]" perpendicular respectively to the projections p1 and p2 of
+  the given line. These will be the traces of a plane [gamma] which is
+  perpendicular to the given line. We next draw through the given point
+  A a plane parallel to the plane [gamma]; this will be the plane
+  required.
+
+  Other metrical properties depend on the determination of the real size
+  or shape of a figure.
+
+  In general the projection of a figure differs both in size and shape
+  from the figure itself. But figures in a plane parallel to a plane of
+  projection will be identical with their projections, and will thus be
+  given in their true dimensions. In other cases there is the problem,
+  constantly recurring, either to find the true shape and size of a
+  plane figure when plan and elevation are given, or, conversely, to
+  find the latter from the known true shape of the figure itself. To do
+  this, the plane is turned about one of its traces till it is laid down
+  into that plane of projection to which the trace belongs. This is
+  technically called rabatting the plane respectively into the plane of
+  the plan or the elevation. As there is no difference in the treatment
+  of the two cases, we shall consider only the case of rabatting a plane
+  [alpha] into the plane of the plan. The plan of the figure is a
+  parallel (orthographic) projection of the figure itself. The results
+  of parallel projection (see PROJECTION, §§ 17 and 18) may therefore
+  now be used. The trace [alpha]' will hereby take the place of what
+  formerly was called the axis of projection. Hence we see that
+  corresponding points in the plan and in the rabatted plane are joined
+  by lines which are perpendicular to the trace [alpha]' and that
+  corresponding lines meet on this trace. We also see that the
+  correspondence is completely determined if we know for one point or
+  one line in the plan the corresponding point or line in the rabatted
+  plane.
+
+  Before, however, we treat of this we consider some special cases.
+
+  § 13. _To determine the distance between two points A, B given by
+  their projections A1, B1 and A2, B2, or, in other words, to determine
+  the true length of a line the plan and elevation of which are given._
+
+  [Illustration: FIG. 43.]
+
+  _Solution._--The two points A, B in space lie vertically above their
+  plans A1, B1 (fig. 43) and A1A = A0A2, B1B = B0B2. The four points A,
+  B, A1, B1 therefore form a plane quadrilateral on the base A1B1 and
+  having right angles at the base. This plane we rabatt about A1B1 by
+  drawing A1A and B1B perpendicular to A1B1 and making A1A = A0A2, B1B =
+  B0B2. Then AB will give the length required.
+
+  The construction might have been performed in the elevation by making
+  A2A = A0A1 and B2B = B0B1 on lines perpendicular to A2B2. Of course AB
+  must have the same length in both cases.
+
+  This figure may be turned into a model. Cut the paper along A1A, AB
+  and BB1, and fold the piece A1ABB1 over along A1B1 till it stands
+  upright at right angles to the horizontal plane. The points A, B will
+  then be in their true position in space relative to [pi]1. Similarly
+  if B2BAA2 be cut out and turned along A2B2 through a right angle we
+  shall get AB in its true position relative to the plane [pi]2. Lastly
+  we fold the whole plane of the paper along the axis x till the plane
+  [pi]2 is at right angles to [pi]1. In this position the two sets of
+  points AB will coincide if the drawing has been accurate.
+
+  Models of this kind can be made in many cases and their construction
+  cannot be too highly recommended in order to realize orthographic
+  projection.
+
+  § 14. _To find the angle between two given lines a, b of which the
+  projections a1, b1 and a2, b2 are given._
+
+  [Illustration: FIG. 44.]
+
+  _Solution._--Let a1, b1 (fig. 44) meet in P1, a2, b2 in T, then if the
+  line P1T is not perpendicular to the axis the two lines will not meet.
+  In this case we draw a line parallel to b to meet the line a. This is
+  easiest done by drawing first the line P1P2 perpendicular to the axis
+  to meet a2 in P2, and then drawing through P2 a line c2 parallel to
+  b2; then b1, c2 will be the projections of a line c which is parallel
+  to b and meets a in P. The plane [alpha] which these two lines
+  determine we rabatt to the plan. We determine the traces a' and c' of
+  the lines a and c; then a'c' is the trace [alpha]' of their plane. On
+  rabatting the point P comes to a point S on the line P1Q perpendicular
+  to a'c', so that QS = QP. But QP is the hypotenuse of a triangle PP1Q
+  with a right angle P1. This we construct by making QR = P0P2; then P1R
+  = PQ. The lines a'S and c'S will therefore include angles equal to
+  those made by the given lines. It is to be remembered that two lines
+  include two angles which are supplementary. Which of these is to be
+  taken in any special case depends upon the circumstances.
+
+  _To determine the angle between a line and a plane_, we draw through
+  any point in the line a perpendicular to the plane (§ 12) and
+  determine the angle between it and the given line. The complement of
+  this angle is the required one.
+
+  _To determine the angle between two planes_, we draw through any point
+  two lines perpendicular to the two planes and determine the angle
+  between the latter as above.
+
+  In special cases it is simpler to determine at once the angle between
+  the two planes by taking a plane section perpendicular to the
+  intersection of the two planes and rabatt this. This is especially the
+  case if one of the planes is the horizontal or vertical plane of
+  projection.
+
+  Thus in fig. 45 the angle P1QR is the angle which the plane [alpha]
+  makes with the horizontal plane.
+
+  § 15. We return to the general case of rabatting a plane [alpha] of
+  which the traces [alpha]' [alpha]" are given.
+
+  [Illustration: FIG. 45.]
+
+  Here it will be convenient to determine first the position which the
+  trace [alpha]"--which is a line in [alpha]--assumes when rabatted.
+  Points in this line coincide with their elevations. Hence it is given
+  in its true dimension, and we can measure off along it the true
+  distance between two points in it. If therefore (fig. 45) P is any
+  point in [alpha]" originally coincident with its elevation P2, and if
+  O is the point where [alpha]" cuts the axis xy, so that O is also in
+  [alpha]', then the point P will after rabatting the plane assume such
+  a position that OP = OP2. At the same time the plan is an orthographic
+  projection of the plane [alpha]. Hence the line joining P to the plan
+  P1 will after rabatting be perpendicular to [alpha]'. But P1 is known;
+  it is the foot of the perpendicular from P2 to the axis xy. We draw
+  therefore, to find P, from P1 a perpendicular P1Q to [alpha]' and find
+  on it a point P such that OP = OP2. Then the line OP will be the
+  position of [alpha]" when rabatted. This line corresponds therefore to
+  the plan of [alpha]"--that is, to the axis xy, corresponding points on
+  these lines being those which lie on a perpendicular to [alpha]'.
+
+  We have thus one pair of corresponding lines and can now find for any
+  point B1 in the plan the corresponding point B in the rabatted plane.
+  We draw a line through B1, say B1P1, cutting [alpha]' in C. To it
+  corresponds the line CP, and the point where this is cut by the
+  projecting ray through B1, perpendicular to [alpha]', is the required
+  point B.
+
+  Similarly any figure in the rabatted plane can be found when the plan
+  is known; but this is usually found in a different manner without any
+  reference to the general theory of parallel projection. As this method
+  and the reasoning employed for it have their peculiar advantages, we
+  give it also.
+
+  Supposing the planes [pi]1 and [pi]2 to be in their positions in space
+  perpendicular to each other, we take a section of the whole figure by
+  a plane perpendicular to the trace [alpha]' about which we are going
+  to rabatt the plane [alpha]. Let this section pass through the point Q
+  in [alpha]'. Its traces will then be the lines QP1 and P1P2 (fig. 9).
+  These will be at right angles, and will therefore, together with the
+  section QP2 of the plane [alpha], form a right-angled triangle QP1P2
+  with the right angle at P1, and having the sides P1Q and P1P2 which
+  both are given in their true lengths. This triangle we rabatt about
+  its base P1Q, making P1R = P1P2. The line QR will then give the true
+  length of the line QP in space. If now the plane [alpha] be turned
+  about [alpha]' the point P will describe a circle about Q as centre
+  with radius QP = QR, in a plane perpendicular to the trace [alpha]'.
+  Hence when the plane [alpha] has been rabatted into the horizontal
+  plane the point P will lie in the perpendicular P1Q to [alpha]', so
+  that QP = QR.
+
+  If A1 is the plan of a point A in the plane [alpha], and if A1 lies in
+  QP1, then the point A will lie vertically above A1 in the line QP. On
+  turning down the triangle QP1P2, the point A will come to A0, the line
+  A1A0 being perpendicular to QP1. Hence A will be a point in QP such
+  that QA = QA0.
+
+  If B1 is the plan of another point, but such that A1B1 is parallel to
+  [alpha]', then the corresponding line AB will also be parallel to
+  [alpha]'. Hence, if through A a line AB be drawn parallel to [alpha]',
+  and B1B perpendicular to [alpha]', then their intersection gives the
+  point B. Thus of any point given in plan the real position in the
+  plane [alpha], when rabatted, can be found by this second method. This
+  is the one most generally given in books on geometrical drawing. The
+  first method explained is, however, in most cases preferable as it
+  gives the draughtsman a greater variety of constructions. It requires
+  a somewhat greater amount of theoretical knowledge.
+
+  If instead of our knowing the plan of a figure the latter is itself
+  given, then the process of finding the plan is the reverse of the
+  above and needs little explanation. We give an example.
+
+  § 16. _It is required to draw the plan and elevation of a polygon of
+  which the real shape and position in a given plane [alpha] are known._
+
+  We first rabatt the plane [alpha] (fig. 46) as before so that P1 comes
+  to P, hence OP1 to OP. Let the given polygon in [alpha] be the figure
+  ABCDE. We project, not the vertices, but the sides. To project the
+  line AB, we produce it to cut [alpha]' in F and OP in G, and draw GG1
+  perpendicular to [alpha]'; then G1 corresponds to G, therefore FG1 to
+  FG. In the same manner we might project all the other sides, at least
+  those which cut OF and OP in convenient points. It will be best,
+  however, first to produce all the sides to cut OP and [alpha]' and
+  then to draw all the projecting rays through A, B, C ... perpendicular
+  to [alpha]', and in the same direction the lines G, G1, &c. By drawing
+  FG we get the points A1, B1 on the projecting ray through A and B. We
+  then join B to the point M where BC produced meets the trace [alpha]'.
+  This gives C1. So we go on till we have found E1. The line A1 E1 must
+  then meet AE in [alpha]', and this gives a check. If one of the sides
+  cuts [alpha]' or OP beyond the drawing paper this method fails, but
+  then we may easily find the projection of some other line, say of a
+  diagonal, or directly the projection of a point, by the former
+  methods. The diagonals may also serve to check the drawing, for two
+  corresponding diagonals must meet in the trace [alpha]'.
+
+  [Illustration: FIG. 46.]
+
+  Having got the plan we easily find the elevation. The elevation of G
+  is above G1 in [alpha]", and that of F is at F2 in the axis. This
+  gives the elevation F2G2 of FG and in it we get A2B2 in the verticals
+  through A1 and B1. As a check we have OG = OG2. Similarly the
+  elevation of the other sides and vertices are found.
+
+  § 17. We proceed to give some applications of the above principles to
+  the representation of solids and of the solution of problems connected
+  with them.
+
+  _Of a pyramid are given its base, the length of the perpendicular from
+  the vertex to the base, and the point where this perpendicular cuts
+  the base; it is required first to develop the whole surface of the
+  pyramid into one plane, and second to determine its section by a plane
+  which cuts the plane of the base in a given line and makes a given
+  angle with it._
+
+  1. As the planes of projection are not given we can take them as we
+  like, and we select them in such a manner that the solution becomes as
+  simple as possible. We take the plane of the base as the horizontal
+  plane and the vertical plane perpendicular to the plane of the
+  section. Let then (fig. 47) ABCD be the base of the pyramid, V1 the
+  plan of the vertex, then the elevations of A, B, C, D will be in the
+  axis at A2, B2, C2, D2, and the vertex at some point V2 above V1 at a
+  known distance from the axis. The lines V1A, V1B, &c., will be the
+  plans and the lines V2A2, V2B2, &c., the elevations of the edges of
+  the pyramid, of which thus plan and elevation are known.
+
+  We develop the surface into the plane of the base by turning each
+  lateral face about its lower edge into the horizontal plane by the
+  method used in § 14. If one face has been turned down, say ABV to ABP,
+  then the point Q to which the vertex of the next face BCV comes can be
+  got more simply by finding on the line V1Q perpendicular to BC the
+  point Q such that BQ = BP, for these lines represent the same edge BV
+  of the pyramid. Next R is found by making CR = CQ, and so on till we
+  have got the last vertex--in this case S. The fact that AS must equal
+  AP gives a convenient check.
+
+  2. The plane [alpha] whose section we have to determine has its
+  horizontal trace given perpendicular to the axis, and its vertical
+  trace makes the given angle with the axis. This determines it. To find
+  the section of the pyramid by this plane there are two methods
+  applicable: we find the sections of the plane either with the faces or
+  with the edges of the pyramid. We use the latter.
+
+  As the plane [alpha] is perpendicular to the vertical plane, the trace
+  [alpha]" contains the projection of every figure in it; the points
+  E2, F2, G2, H2 where this trace cuts the elevations of the edges will
+  therefore be the elevations of the points where the edges cut [alpha].
+  From these we find the plans E1, F1, G1, H1, and by joining them the
+  plan of the section. If from E1, F1 lines be drawn perpendicular to
+  AB, these will determine the points E, F on the developed face in
+  which the plane [alpha] cuts it; hence also the line EF. Similarly on
+  the other faces. Of course BF must be the same length on BP and on BQ.
+  If the plane [alpha] be rabatted to the plan, we get the real shape of
+  the section as shown in the figure in EFGH. This is done easily by
+  making F0F = OF2, &c. If the figure representing the development of
+  the pyramid, or better a copy of it, is cut out, and if the lateral
+  faces be bent along the lines AB, BC, &c., we get a model of the
+  pyramid with the section marked on its faces. This may be placed on
+  its plan ABCD and the plane of elevation bent about the axis x. The
+  pyramid stands then in front of its elevations. If next the plane
+  [alpha] with a hole cut out representing the true section be bent
+  along the trace [alpha]' till its edge coincides with [alpha]", the
+  edges of the hole ought to coincide with the lines EF, FG, &c., on the
+  faces.
+
+  § 18. Polyhedra like the pyramid in § 17 are represented by the
+  projections of their edges and vertices. But solids bounded by curved
+  surfaces, or surfaces themselves, cannot be thus represented.
+
+  For a surface we may use, as in case of the plane, its traces--that
+  is, the curves in which it cuts the planes of projection. We may also
+  project points and curves on the surface. A ray cuts the surface
+  generally in more than one point; hence it will happen that some of
+  the rays touch the surface, if two of these points coincide. The
+  points of contact of these rays will form some curve on the surface,
+  and this will appear from the centre of projection as the boundary of
+  the surface or of part of the surface. The outlines of all surfaces of
+  solids which we see about us are formed by the points at which rays
+  through our eye touch the surface. The projections of these contours
+  are therefore best adapted to give an idea of the shape of a surface.
+
+  [Illustration: FIG. 47.]
+
+  Thus the tangents drawn from any finite centre to a sphere form a
+  right circular cone, and this will be cut by any plane in a conic. It
+  is often called the projection of a sphere, but it is better called
+  the contour-line of the sphere, as it is the boundary of the
+  projections of all points on the sphere.
+
+  If the centre is at infinity the tangent cone becomes a right circular
+  cylinder touching the sphere along a great circle, and if the
+  projection is, as in our case, orthographic, then the section of this
+  cone by a plane of projection will be a circle equal to the great
+  circle of the sphere. We get such a circle in the plan and another in
+  the elevation, their centres being plan and elevation of the centre of
+  the sphere.
+
+  Similarly the rays touching a cone of the second order will lie in two
+  planes which pass through the vertex of the cone, the contour-line of
+  the projection of the cone consists therefore of two lines meeting in
+  the projection of the vertex. These may, however, be invisible if no
+  real tangent rays can be drawn from the centre of projection; and this
+  happens when the ray projecting the centre of the vertex lies within
+  the cone. In this case the traces of the cone are of importance. Thus
+  in representing a cone of revolution with a vertical axis we get in
+  the plan a circular trace of the surface whose centre is the plan of
+  the vertex of the cone, and in the elevation the contour, consisting
+  of a pair of lines intersecting in the elevation of the vertex of the
+  cone. The circle in the plan and the pair of lines in the elevation do
+  not determine the surface, for an infinite number of surfaces might be
+  conceived which pass through the circular trace and touch two planes
+  through the contour lines in the vertical plane. The surface becomes
+  only completely defined if we write down to the figure that it shall
+  represent a cone. The same holds for all surfaces. Even a plane is
+  fully represented by its traces only under the silent understanding
+  that the traces are those of a plane.
+
+  § 19. Some of the simpler problems connected with the representation
+  of surfaces are the determination of plane sections and of the curves
+  of intersection of two such surfaces. The former is constantly used in
+  nearly all problems concerning surfaces. Its solution depends of
+  course on the nature of the surface.
+
+  To determine the curve of intersection of two surfaces, we take a
+  plane and determine its section with each of the two surfaces,
+  rabatting this plane if necessary. This gives two curves which lie in
+  the same plane and whose intersections will give us points on both
+  surfaces. It must here be remembered that two curves in space do not
+  necessarily intersect, hence that the points in which their
+  projections intersect are not necessarily the projections of points
+  common to the two curves. This will, however, be the case if the two
+  curves lie in a common plane. By taking then a number of plane
+  sections of the surfaces we can get as many points on their curve of
+  intersection as we like. These planes have, of course, to be selected
+  in such a way that the sections are curves as simple as the case
+  permits of, and such that they can be easily and accurately drawn.
+  Thus when possible the sections should be straight lines or circles.
+  This not only saves time in drawing but determines all points on the
+  sections, and therefore also the points where the two curves meet,
+  with equal accuracy.
+
+  § 20. We give a few examples how these sections have to be selected. A
+  cone is cut by every plane through the vertex in lines, and if it is a
+  cone of revolution by planes perpendicular to the axis in circles.
+
+  A cylinder is cut by every plane parallel to the axis in lines, and if
+  it is a cylinder of revolution by planes perpendicular to the axis in
+  circles.
+
+  A sphere is cut by every plane in a circle.
+
+  Hence in case of two cones situated anywhere in space we take sections
+  through both vertices. These will cut both cones in lines. Similarly
+  in case of two cylinders we may take sections parallel to the axis of
+  both. In case of a sphere and a cone of revolution with vertical axis,
+  horizontal sections will cut both surfaces in circles whose plans are
+  circles and whose elevations are lines, whilst vertical sections
+  through the vertex of the cone cut the latter in lines and the sphere
+  in circles. To avoid drawing the projections of these circles, which
+  would in general be ellipses, we rabatt the plane and then draw the
+  circles in their real shape. And so on in other cases.
+
+  Special attention should in all cases be paid to those points in which
+  the tangents to the projection of the curve of intersection are
+  parallel or perpendicular to the axis x, or where these projections
+  touch the contour of one of the surfaces.     (O. H.)
+
+
+IV. ANALYTICAL GEOMETRY
+
+1. In the name _geometry_ there is a lasting record that the science had
+its origin in the knowledge that two distances may be compared by
+measurement, and in the idea that measurement must be effectual in the
+dissociation of different directions as well as in the comparison of
+distances in the same direction. The distance from an observer's eye of
+an object seen would be specified as soon as it was ascertained that a
+rod, straight to the eye and of length taken as known, could be given
+the direction of the line of vision, and had to be moved along it a
+certain number of times through lengths equal to its own in order to
+reach the object from the eye. Moreover, if a field had for two of its
+boundaries lines straight to the eye, one running from south to north
+and the other from west to east, the position of a point in the field
+would be specified if the rod, when directed west, had to be shifted
+from the point one observed number of times westward to meet the former
+boundary, and also, when directed south, had to be shifted another
+observed number of times southward to meet the latter. Comparison by
+measurement, the beginning of geometry, involved counting, the basis of
+arithmetic; and the science of number was marked out from the first as
+of geometrical importance.
+
+But the arithmetic of the ancients was inadequate as a science of
+number. Though a length might be recognized as known when measurement
+certified that it was so many times a standard length, it was not every
+length which could be thus specified in terms of the same standard
+length, even by an arithmetic enriched with the notion of fractional
+number. The idea of possible incommensurability of lengths was
+introduced into Europe by Pythagoras; and the corresponding idea of
+irrationality of number was absent from a crude arithmetic, while there
+were great practical difficulties in the way of its introduction. Hence
+perhaps it arose that, till comparatively modern times, appeal to
+arithmetical aid in geometrical reasoning was in all possible ways
+restrained. Geometry figured rather as the helper of the more difficult
+science of arithmetic.
+
+2. It was reserved for algebra to remove the disabilities of arithmetic,
+and to restore the earliest ideas of the land-measurer to the position
+of controlling ideas in geometrical investigation. This unified science
+of pure number made comparatively little headway in the hands of the
+ancients, but began to receive due attention shortly after the revival
+of learning. It expresses whole classes of arithmetical facts in single
+statements, gives to arithmetical laws the form of equations involving
+symbols which may mean any known or sought numbers, and provides
+processes which enable us to analyse the information given by an
+equation and derive from that equation other equations, which express
+laws that are in effect consequences or causes of a law started from,
+but differ greatly from it in form. Above all, for present purposes, it
+deals not only with integral and fractional number, but with number
+regarded as capable of continuous growth, just as distance is capable of
+continuous growth. The difficulty of the arithmetical expression of
+irrational number, a difficulty considered by the modern school of
+analysts to have been at length surmounted (see FUNCTION), is not vital
+to it. It can call the ratio of the diagonal of a square to a side, for
+instance, or that of the circumference of a circle to a diameter, a
+number, and let a or x denote that number, just as properly as it may
+allow either letter to denote any rational number which may be greater
+or less than the ratio in question by a difference less than any minute
+one we choose to assign.
+
+Counting only, and not the counting of objects, is of the essence of
+arithmetic, and of algebra. But it is lawful to count objects, and in
+particular to count equal lengths by measure. The widened idea is that
+even when a or x is an irrational number we may speak of a or x unit
+lengths by measure. We may give concrete interpretation to an
+algebraical equation by allowing its terms all to mean numbers of times
+the same unit length, or the same unit area, or &c. and in any equation
+lawfully derived from the first by algebraical processes we may do the
+same. Descartes in his _Géométrie_ (1637) was the first to systematize
+the application of this principle to the inherent first notions of
+geometry; and the methods which he instituted have become the most
+potent methods of all in geometrical research. It is hardly too much to
+say that, when known facts as to a geometrical figure have once been
+expressed in algebraical terms, all strictly consequential facts as to
+the figure can be deduced by almost mechanical processes. Some may well
+be unexpected consequences; and in obtaining those of which there has
+been suggestion beforehand the often bewildering labour of constant
+attention to the figure is obviated. These are the methods of what is
+now called _analytical_, or sometimes _algebraical_, _geometry_.
+
+3. The modern use of the term "analytical" in geometry has obscured, but
+not made obsolete, an earlier use, one as old as Plato. There is nothing
+algebraical in this analysis, as distinguished from synthesis, of the
+Greeks, and of the expositors of pure geometry. It has reference to an
+order of ideas in demonstration, or, more frequently, in discovering
+means to effect the geometrical construction of a figure with an
+assigned special property. We have to suppose hypothetically that the
+construction has been performed, drawing a rough figure which exhibits
+it as nearly as is practicable. We then analyse or critically examine
+the figure, treated as correct, and ascertain other properties which it
+can only possess in association with the one in question. Presently one
+of these properties will often be found which is of such a character
+that the construction of a figure possessing it is simple. The means of
+effecting synthetically a construction such as was desired is thus
+brought to light by what Plato called _analysis_. Or again, being asked
+to prove a theorem A, we ascertain that it must be true if another
+theorem B is, that B must be if C is, and so on, thus eventually finding
+that the theorem A is the consequence, through a chain of
+intermediaries, of a theorem Z of which the establishment is easy. This
+geometrical analysis is not the subject of the present article; but in
+the reasoning from form to form of an equation or system of equations,
+with the object of basing the algebraical proof of a geometrical fact on
+other facts of a more obvious character, the same logic is utilized, and
+the name "analytical geometry" is thus in part explained.
+
+4. In algebra real positive number was alone at first dealt with, and in
+geometry actual signless distance. But in algebra it became of
+importance to say that every equation of the first degree has a root,
+and the notion of negative number was introduced. The negative unit had
+to be defined as what can be added to the positive unit and produce the
+sum zero. The corresponding notion was readily at hand in geometry,
+where it was clear that a unit distance can be measured to the left or
+down from the farther end of a unit distance already measured to the
+right or up from a point O, with the result of reaching O again. Thus,
+to give full interpretation in geometry to the algebraically negative,
+it was only necessary to associate distinctness of sign with
+oppositeness of direction. Later it was discovered that algebraical
+reasoning would be much facilitated, and that conclusions as to the real
+would retain all their soundness, if a pair of imaginary units ±[root]-1
+of what might be called number were allowed to be contemplated, the pair
+being defined, though not separately, by the two properties of having
+the real sum 0 and the real product 1. Only in these two real
+combinations do they enter in conclusions as to the real. An advantage
+gained was that every quadratic equation, and not some quadratics only,
+could be spoken of as having two roots. These admissions of new units
+into algebra were final, as it admitted of proof that all equations of
+degrees higher than two have the full numbers of roots possible for
+their respective degrees in any case, and that every root has a value
+included in the form a + b [root]-1, with a, b, real. The corresponding
+enrichment could be given to geometry, with corresponding advantages and
+the same absence of danger, and this was done. On a line of measurement
+of distance we contemplate as existing, not only an infinite continuum
+of points at real distances from an origin of measurement O, but a
+doubly infinite continuum of points, all but the singly infinite
+continuum of real ones imaginary, and imaginary in conjugate pairs, a
+conjugate pair being at imaginary distances from O, which have a real
+arithmetic and a real geometric mean. To geometry enriched with this
+conception all algebra has its application.
+
+5. Actual geometry is one, two or three-dimensional, i.e. lineal, plane
+or solid. In one-dimensional geometry positions and measurements in a
+single line only are admitted. Now descriptive constructions for points
+in a line are impossible without going out of the line. It has therefore
+been held that there is a sense in which no science of geometry strictly
+confined to one dimension exists. But an algebra of one variable can be
+applied to the study of distances along a line measured from a chosen
+point on it, so that the idea of construction as distinct from
+measurement is not essential to a one-dimensional geometry aided by
+algebra. In geometry of two dimensions, the flat of the land-measurer,
+the passage from one point O to any other point, can be effected by two
+successive marches, one east or west and one north or south, and, as
+will be seen, an algebra of two variables suffices for geometrical
+exploitation. In geometry of three dimensions, that of space, any point
+can be reached from a chosen one by three marches, one east or west, one
+north or south, and one up or down; and we shall see that an algebra of
+three variables is all that is necessary. With three dimensions actual
+geometry stops; but algebra can supply any number of variables. Four or
+more variables have been used in ways analogous to those in which one,
+two and three variables are used for the purposes of one, two and
+three-dimensional geometry, and the results have been expressed in
+quasi-geometrical language on the supposition that a higher space can be
+conceived of, though not realized, in which four independent directions
+exist, such that no succession of marches along three of them can effect
+the same displacement of a point as a march along the fourth; and
+similarly for higher numbers than four. Thus analytical, though not
+actual, geometries exist for four and more dimensions. They are in fact
+algebras furnished with nomenclature of a geometrical cast, suggested by
+convenient forms of expression which actual geometry has, in return for
+benefits received, conferred on algebras of one, two and three
+variables.
+
+We will confine ourselves to the dimensions of actual geometry, and will
+devote no space to the one-dimensional, except incidentally as existing
+within the two-dimensional. The analytical method will now be explained
+for the cases of two and three dimensions in succession. The form of it
+originated by Descartes, and thence known as Cartesian, will alone be
+considered in much detail.
+
+
+  I. _Plane Analytical Geometry._
+
+  [Illustration: FIG. 48.]
+
+  [Illustration: FIG. 49.]
+
+  6. _Coordinates._--It is assumed that the points, lines and figures
+  considered lie in one and the same plane, which plane therefore need
+  not be in any way referred to. In the plane a point O, and two lines
+  x'Ox, y'Oy, intersecting in O, are taken once for all, and regarded as
+  fixed. O is called the origin, and x'Ox, y'Oy the axes of x and y
+  respectively. Other positions in the plane are specified in relation
+  to this fixed origin and these fixed axes. From any point P we suppose
+  PM drawn parallel to the axis of y to meet the axis of x in M, and may
+  also suppose PN drawn parallel to the axis of x to meet the axis of y
+  in N, so that OMPN is a parallelogram. The position of P is determined
+  when we know OM ( = NP) and MP ( = ON). If OM is x times the unit of a
+  scale of measurement chosen at pleasure, and MP is y times the unit,
+  so that x and y have numerical values, we call x and y the (Cartesian)
+  coordinates of P. To distinguish them we often speak of y as the
+  ordinate, and of x as the abscissa.
+
+  It is necessary to attend to signs; x has one sign or the other
+  according as the point P is on one side or the other of the axis of y,
+  and y one sign or the other according as P is on one side or the other
+  of the axis of x. Using the letters N, E, S, W, as in a map, and
+  considering the plane as divided into four quadrants by the axes, the
+  signs are usually taken to be:
+
+    x   y   For quadrant
+
+    +   +      N  E
+    +   -      S  E
+    -   +      N  W
+    -   -      S  W
+
+  A point is referred to as the point (a, b), when its coordinates are x
+  = a, y = b. A point may be fixed, or it may be variable, i.e. be
+  regarded for the time being as free to move in the plane. The
+  coordinates (x, y) of a variable point are algebraic variables, and
+  are said to be "current coordinates."
+
+  The axes of x and y are usually (as in fig. 48) taken at right angles
+  to one another, and we then speak of them as rectangular axes, and of
+  x and y as "rectangular coordinates" of a point P; OMPN is then a
+  rectangle. Sometimes, however, it is convenient to use axes which are
+  oblique to one another, so that (as in fig. 49) the angle xOy between
+  their positive directions is some known angle [omega] distinct from a
+  right angle, and OMPN is always an oblique parallelogram with given
+  angles; and we then speak of x and y as "oblique coordinates." The
+  coordinates are as a rule taken to be rectangular in what follows.
+
+  7. _Equations and loci._ If (x, y) is the point P, and if we are given
+  that x = 0, we are told that, in fig. 48 or fig. 49, the point M lies
+  at O, whatever value y may have, i.e. we are told the one fact that P
+  lies on the axis of y. Conversely, if P lies anywhere on the axis of
+  y, we have always OM = 0, i.e. x = 0. Thus the equation x = 0 is one
+  satisfied by the coordinates (x, y) of every point in the axis of y,
+  and not by those of any other point. We say that x = 0 is the equation
+  of the axis of y, and that the axis of y is the locus represented by
+  the equation x = 0. Similarly y = 0 is the equation of the axis of x.
+  An equation x = a, where a is a constant, expresses that P lies on a
+  parallel to the axis of y through a point M on the axis of x such that
+  OM = a. Every line parallel to the axis of y has an equation of this
+  form. Similarly, every line parallel to the axis of x has an equation
+  of the form y = b, where b is some definite constant.
+
+  These are simple cases of the fact that a single equation in the
+  current coordinates of a variable point (x, y) imposes one limitation
+  on the freedom of that point to vary. The coordinates of a point taken
+  at random in the plane will, as a rule, not satisfy the equation, but
+  infinitely many points, and in most cases infinitely many real ones,
+  have coordinates which do satisfy it, and these points are exactly
+  those which lie upon some locus of one dimension, a straight line or
+  more frequently a curve, which is said to be represented by the
+  equation. Take, for instance, the equation y = mx, where m is a given
+  constant. It is satisfied by the coordinates of every point P, which
+  is such that, in fig. 48, the distance MP, with its proper sign, is m
+  times the distance OM, with its proper sign, i.e. by the coordinates
+  of every point in the straight line through O which we arrive at by
+  making a line, originally coincident with x'Ox, revolve about O in the
+  direction opposite to that of the hands of a watch through an angle of
+  which m is the tangent, and by those of no other points. That line is
+  the locus which it represents. Take, more generally, the equation y =
+  [phi](x), where [phi](x) is any given non-ambiguous function of x.
+  Choosing any point M on x'Ox in fig. 1, and giving to x the value of
+  the numerical measure of OM, the equation determines a single
+  corresponding y, and so determines a single point P on the line
+  through M parallel to y'Oy. This is one point whose coordinates
+  satisfy the equation. Now let M move from the extreme left to the
+  extreme right of the line x'Ox, regarded as extended both ways as far
+  as we like, i.e. let x take all real values from -[oo] to [oo]. With
+  every value goes a point P, as above, on the parallel to y'Oy through
+  the corresponding M; and we thus find that there is a path from the
+  extreme left to the extreme right of the figure, all points P along
+  which are distinguished from other points by the exceptional property
+  of satisfying the equation by their coordinates. This path is a locus;
+  and the equation y = [phi](x) represents it. More generally still,
+  take an equation f(x, y) = 0 which involves both x and y under a
+  functional form. Any particular value given to x in it produces from
+  it an equation for the determination of a value or values of y, which
+  go with that value of x in specifying a point or points (x, y), of
+  which the coordinates satisfy the equation f(x, y) = 0. Here again, as
+  x takes all values, the point or points describe a path or paths,
+  which constitute a locus represented by the equation. Except when y
+  enters to the first degree only in f(x, y), it is not to be expected
+  that all the values of y, determined as going with a chosen value of
+  x, will be necessarily real; indeed it is not uncommon for all to be
+  imaginary for some ranges of values of x. The locus may largely
+  consist of continua of imaginary points; but the real parts of it
+  constitute a real curve or real curves. Note that we have to allow x
+  to admit of all imaginary, as well as of all real, values, in order to
+  obtain all imaginary parts of the locus.
+
+  A locus or curve may be algebraically specified in another way; viz.
+  we may be given two equations x = f([theta]), y = F([theta]), which
+  express the coordinates of any point of it as two functions of the
+  same variable parameter [theta] to which all values are open. As
+  [theta] takes all values in turn, the point (x, y) traverses the
+  curve.
+
+  It is a good exercise to trace a number of curves, taken as defined by
+  the equations which represent them. This, in simple cases, can be done
+  approximately by plotting the values of y given by the equation of a
+  curve as going with a considerable number of values of x, and
+  connecting the various points (x, y) thus obtained. But methods exist
+  for diminishing the labour of this tentative process.
+
+  Another problem, which will be more attended to here, is that of
+  determining the equations of curves of known interest, taken as
+  defined by geometrical properties. It is not a matter for surprise
+  that the curves which have been most and longest studied geometrically
+  are among those represented by equations of the simplest character.
+
+  8. _The Straight Line._--This is the simplest type of locus. Also the
+  simplest type of equation in x and y is Ax + By + C = 0, one of the
+  first degree. Here the coefficients A, B, C are constants. They are,
+  like the current coordinates, x, y, numerical. But, in giving
+  interpretation to such an equation, we must of course refer to numbers
+  Ax, By, C of unit magnitudes of the same kind, of units of counting
+  for instance, or unit lengths or unit squares. It will now be seen
+  that every straight line has an equation of the first degree, and that
+  every equation of the first degree represents a straight line.
+
+  [Illustration: FIG. 50.]
+
+  It has been seen (§ 7) that lines parallel to the axes have equations
+  of the first degree, free from one of the variables. Take now a
+  straight line ABC inclined to both axes. Let it make a given angle
+  [alpha] with the positive direction of the axis of x, i.e. in fig. 50
+  let this be the angle through which Ax must be revolved
+  counter-clockwise about A in order to be made coincident with the
+  line. Let C, of coordinates (h, k), be a fixed point on the line, and
+  P(x, y) any other point upon it. Draw the ordinates CD, PM of C and P,
+  and let the parallel to the axis of x through C meet PM, produced if
+  necessary, in R. The right-angled triangle CRP tells us that, with the
+  signs appropriate to their directions attached to CR and RP,
+
+    RP = CR tan [alpha], i.e. MP - DC = (OM - OD) tan [alpha],
+
+  and this gives that
+
+    y - k = tan [alpha] (x - h),
+
+  an equation of the first degree satisfied by x and y. No point not on
+  the line satisfies the same equation; for the line from C to any point
+  off the line would make with CR some angle ß different from [alpha],
+  and the point in question would satisfy an equation y - k = tan ß(x -
+  h), which is inconsistent with the above equation.
+
+  The equation of the line may also be written y = mx + b, where m = tan
+  [alpha], and b = k - h tan [alpha]. Here b is the value obtained for y
+  from the equation when 0 is put for x, i.e. it is the numerical
+  measure, with proper sign, of OB, the intercept made by the line on
+  the axis of y, measured from the origin. For different straight lines,
+  m and b may have any constant values we like.
+
+  Now the general equation of the first degree Ax + By + C = 0 may be
+  written y = -(A/B)x - C/B, unless B = 0, in which case it represents a
+  line parallel to the axis of y; and -A/B, -C/B are values which can be
+  given to m and b, so that every equation of the first degree
+  represents a straight line. It is important to notice that the general
+  equation, which in appearance contains three constants A, B, C, in
+  effect depends on two only, the ratios of two of them to the third. In
+  virtue of this last remark, we see that two distinct conditions
+  suffice to determine a straight line. For instance, it is easy from
+  the above to see that
+
+    x    y
+    -- + -- = 1
+    a    b
+
+  is the equation of a straight line determined by the two conditions
+  that it makes intercepts OA, OB on the two axes, of which a and b are
+  the numerical measures with proper signs: note that in fig. 50 a is
+  negative. Again,
+
+             y2 - y1
+    y - y1 = ------- (x - x1),
+             x2 - x1
+
+  i.e.
+
+    (y1 - y2)x - (x1 - x2)y + x1y2 - x2y1 = 0,
+
+  represents the line determined by the data that it passes through two
+  given points (x1, y1) and (x2, y2). To prove this find m in the
+  equation y - y1 = m(x - x1) of a line through (x1, y1), from the
+  condition that (x2, y2) lies on the line.
+
+  In this paragraph the coordinates have been assumed rectangular. Had
+  they been oblique, the doctrine of similar triangles would have given
+  the same results, except that in the forms of equation y - k = m(x -
+  h), y = mx + b, we should not have had m = tan [alpha].
+
+  9. _The Circle._--It is easy to write down the equation of a given
+  circle. Let (h, k) be its given centre C, and [rho] the numerical
+  measure of its given radius. Take P (x, y) any point on its
+  circumference, and construct the triangle CRP, in fig. 50 as above.
+  The fact that this is right-angled tells us that
+
+    CR² + RP² = CP²,
+
+  and this at once gives the equation
+
+    (x - h)² + (y - k)² = [rho]².
+
+  A point not upon the circumference of the particular circle is at some
+  distance from (h, k) different from [rho], and satisfies an equation
+  inconsistent with this one; which accordingly represents the
+  circumference, or, as we say, the circle.
+
+  The equation is of the form
+
+    x² + y² + 2Ax + 2By + C = 0.
+
+  Conversely every equation of this form represents a circle: we have
+  only to take -A, -B, A² + B² - C for h, k, [rho]² respectively, to
+  obtain its centre and radius. But this statement must appear too
+  unrestricted. Ought we not to require A² + B² - C to be positive?
+  Certainly, if by circle we are only to mean the visible round
+  circumference of the geometrical definition. Yet, analytically, we
+  contemplate altogether imaginary circles, for which [rho]² is
+  negative, and circles, for which [rho] = 0, with all their reality
+  condensed into their centres. Even when [rho]² is positive, so that a
+  visible round circumference exists, we do not regard this as
+  constituting the whole of the circle. Giving to x any value whatever
+  in (x - h)² + (y - k)² = [rho]², we obtain two values of y, real,
+  coincident or imaginary, each of which goes with the abscissa x as the
+  ordinate of a point, real or imaginary, on what is represented by the
+  equation of the circle.
+
+  The doctrine of the imaginary on a circle, and in geometry generally,
+  is of purely algebraical inception; but it has been in its entirety
+  accepted by modern pure geometers, and signal success has attended the
+  efforts of those who, like K.G.C. von Staudt, have striven to base its
+  conclusions on principles not at all algebraical in form, though of
+  course cognate to those adopted in introducing the imaginary into
+  algebra.
+
+  A circle with its centre at the origin has an equation x² + y² =
+  [rho]².
+
+  In oblique coordinates the general equation of a circle is x² + 2xy
+  cos [omega] + y² + 2Ax + 2By + C = 0.
+
+  10. The conic sections are the next simplest loci; and it will be seen
+  later that they are the loci represented by equations of the second
+  degree. Circles are particular cases of conic sections; and they have
+  just been seen to have for their equations a particular class of
+  equations of the second degree. Another particular class of such
+  equations is that included in the form (Ax + By + C)(A'x + B'y + C') =
+  0, which represents two straight lines, because the product on the
+  left vanishes if, and only if, one of the two factors does, i.e. if,
+  and only if, (x, y) lies on one or other of two straight lines. The
+  condition that ax² + 2hxy + by² + 2gx + 2fy + c = 0, which is often
+  written (a, b, c, f, g, h)(x, y, I)² = 0, takes this form is abc +
+  2fgh-af²-bg² - ch² = 0. Note that the two lines may, in particular
+  cases, be parallel or coincident.
+
+  Any equation like F1(x, y) F2(x, y) ... F_n(x, y) = 0, of which the
+  left-hand side breaks up into factors, represents all the loci
+  separately represented by F1(x, y) = 0, F2(x, y) = 0, ... F_n(x, y) =
+  0. In particular an equation of degree n which is free from x
+  represents n straight lines parallel to the axis of x, and one of
+  degree n which is homogeneous in x and y, i.e. one which upon division
+  by x^n, becomes an equation in the ratio y/x, represents n straight
+  lines through the origin.
+
+  Curves represented by equations of the third degree are called cubic
+  curves. The general equation of this degree will be written (*)(x, y,
+  I)³ = 0.
+
+  11. _Descriptive Geometry._--A geometrical proposition is either
+  descriptive or metrical: in the former case the statement of it is
+  independent of the idea of magnitude (length, inclination, &c.), and
+  in the latter it has reference to this idea. The method of coordinates
+  seems to be by its inception essentially metrical. Yet in dealing by
+  this method with descriptive propositions we are eminently free from
+  metrical considerations, because of our power to use general
+  equations, and to avoid all assumption that measurements implied are
+  any particular measurements.
+
+  [Illustration: FIG. 51.]
+
+  12. It is worth while to illustrate this by the instance of the
+  well-known theorem of the radical centre of three circles. The theorem
+  is that, given any three circles A, B, C (fig. 51), the common chords
+  [alpha][alpha]', ß[beta]', [gamma][gamma]' of the three pairs of
+  circles meet in a point.
+
+  The geometrical proof is metrical throughout:--
+
+  Take O the point of intersection of [alpha][alpha]', ß[beta]', and
+  joining this with [gamma]', suppose that [gamma]'O does not pass
+  through [gamma], but that it meets the circles A, B in two distinct
+  points [gamma]2, [gamma]1 respectively. We have then the known
+  metrical property of intersecting chords of a circle; viz. in circle
+  C, where [alpha][alpha]', ß[beta]', are chords meeting at a point O,
+
+    O[alpha]·O[alpha]' = Oß·Oß',
+
+  where, as well as in what immediately follows, O[alpha], &c. denote,
+  of course, _lengths_ or _distances_.
+
+  Similarly in circle A,
+
+    Oß·Oß' = O[gamma]2·O[gamma]',
+
+  and in circle B,
+
+    O[alpha]·O[alpha]' = O[gamma]1·O[gamma]'.
+
+  Consequently O[gamma]1·O[gamma]' = O[gamma]2·O[gamma]', that is,
+  O[gamma]1 = O[gamma]2, or the points [gamma]1 and [gamma]2 coincide;
+  that is, they each coincide with [gamma].
+
+  We contrast this with the analytical method:--
+
+  Here it only requires to be known that an equation Ax + By + C = 0
+  represents a line, and an equation x² + y² + Ax + By + C = 0
+  represents a circle. A, B, C have, in the two cases respectively,
+  metrical significations; but these we are not concerned with. Using S
+  to denote the function x² + y² + Ax + By + C, the equation of a circle
+  is S = o. Let the equation of any other circle be S', = x² + y² + A'x
+  + B'y + C' = 0; the equation S - S' = 0 is a linear equation (S - S'
+  is in fact = (A - A')x + (B - B')y + C - C), and it thus represents a
+  line; this equation is satisfied by the coordinates of each of the
+  points of intersection of the two circles (for at each of these points
+  S = 0 and S' = 0, therefore also S - S' = 0); hence the equation S -
+  S' = 0 is that of the line joining the two points of intersection of
+  the two circles, or say it is the equation of the common chord of the
+  two circles. Considering then a third circle S", = x² + y² + A"x + B"y
+  + C" = 0, the equations of the common chords are S-S' = 0, S - S" = 0,
+  S' - S" = 0 (each of these a linear equation); at the intersection of
+  the first and second of these lines S = S' and S = S", therefore also
+  S' = S", or the equation of the third line is satisfied by the
+  coordinates of the point in question; that is, the three chords
+  intersect in a point O, the coordinates of which are determined by the
+  equations S = S' = S".
+
+  It further appears that if the two circles S = 0, S' = 0 do not
+  intersect in any real points, they must be regarded as intersecting in
+  two imaginary points, such that the line joining them is the real line
+  represented by the equation S - S' = 0; or that two circles, whether
+  their intersections be real or imaginary, have always a real common
+  chord (or radical axis), and that for _any_ three circles the common
+  chords intersect in a point (of course real) which is the radical
+  centre. And by this very theorem, given two circles with imaginary
+  intersections, we can, by drawing circles which meet each of them in
+  real points, construct the radical axis of the first-mentioned two
+  circles.
+
+  13. The principle employed in showing that the equation of the common
+  chord of two circles is S - S' = 0 is one of very extensive
+  application, and some more illustrations of it may be given.
+
+  Suppose S = 0, S' = 0 are lines (that is, let S, S' now denote linear
+  functions Ax + By + C, A'x + B'y + C'), then S - kS' = 0 (k an
+  arbitrary constant) is the equation of any line passing through the
+  point of intersection of the two given lines. Such a line may be made
+  to pass through any given point, say the point (x0, y0); if S0, S'0
+  are what S, S' respectively become on writing for (x, y) the values
+  (x0, y0), then the value of k is k = S0 ÷ S'0. The equation in fact is
+  SS'0 - S0S' = 0; and starting from this equation we at once verify it
+  _a posteriori_; the equation is a linear equation satisfied by the
+  values of (x, y) which make S = 0, S' = 0; and satisfied also by the
+  values (x0, y0); and it is thus the equation of the line in question.
+
+  If, as before, S = 0, S' = 0 represent circles, then (k being
+  arbitrary) S - kS' = 0 is the equation of any circle passing through
+  the two points of intersection of the two circles; and to make this
+  pass through a given point (x0, y0) we have again k = S0 ÷ S'0. In the
+  particular case k = 1, the circle becomes the common chord (more
+  accurately it becomes the common chord together with the line
+  infinity; see § 23 below).
+
+  If S denote the general quadric function,
+
+    S = ax² +2hxy + by² + 2fy + 2gx + c,
+
+  then the equation S = 0 represents a conic; assuming this, then, if S'
+  = 0 represents another conic, the equation S - kS' = 0 represents
+  _any_ conic through the four points of intersection of the two conics.
+
+  [Illustration: FIG. 52.]
+
+  14. The object still being to illustrate the mode of working with
+  coordinates for descriptive purposes, we consider the theorem of the
+  polar of a point in regard to a circle. Given a circle and a point O
+  (fig. 52), we draw through O any two lines meeting the circle in the
+  points A, A' and B, B' respectively, and then taking Q as the
+  intersection of the lines AB' and A'B, the theorem is that the locus
+  of the point Q is a right line depending only upon O and the circle,
+  but independent of the particular lines OAA' and OBB'.
+
+  Taking O as the origin, and for the axes any two lines through O at
+  right angles to each other, the equation of the circle will be
+
+    x² + y² + 2Ax + 2By + C = 0;
+
+  and if the equation of the line OAA' is taken to be y = mx, then the
+  points A, A' are found as the intersections of the straight line with
+  the circle; or to determine x we have
+
+    x²(1 + m²) + 2x(A + Bm) + C = 0.
+
+  If(x1, y1) are the coordinates of A, and (x2, y2) of A', then the
+  roots of this equation are x1, x2, whence easily
+
+    1    1        A + Bm
+    -- + --  = -2 ------.
+    x1   x2         C
+
+  And similarly, if the equation of the line OBB' is taken to be y =
+  m'x1 and the coordinates of B, B' to be (x3, y3) and (x4, y4)
+  respectively, then
+
+    1    1         A + Bm'
+    -- + --  =  -2 -------.
+    x3   x4           C'
+
+  We have then by § 8
+
+    x(y1 - y4) - y(x1 - x4) + x1y4 - x4y1 = 0,
+
+    x(y2 - y3) - y(x2 - x3) + x2y3 - x3y2 = 0,
+
+  as the equations of the lines AB' and A'B respectively. Reducing by
+  means of the relations y1 - mx1 = 0, y2 - mx2 = 0, y3 - m'x3 = 0, y4 -
+  m'x4 = 0, the two equations become
+
+    x(mx1 - m'x4) - y(x1 - x4) + (m'- m)x1x4 = 0,
+
+    x(mx2 - m'x3) - y(x2 - x3) + (m'- m)x2x3 = 0,
+
+  and if we divide the first of these equations by x1x4, and the second
+  by x2x3 and then add, we obtain
+      _                               _      _                       _
+     |   / 1    1  \       / 1    1 \  |    |  1    1     / 1    1 \  |
+    x| m(  -- + --  ) - m'(  -- + -- ) | - y|  -- + -- - (  -- + -- ) |
+     |_  \ x3   x4 /       \ x1   x2/ _|    |_ x3   x4    \ x1   x2/ _|
+
+      + 2m' - 2m = 0,
+
+  or, what is the same thing,
+
+     / 1    1  \              / 1    1  \
+    (  -- + --  )(y - m'x) - (  -- + --  )(y - mx) + 2m' - 2m = 0,
+     \ x1   x2 /              \ x3   x4 /
+
+  which by what precedes is the equation of a line through the point Q.
+  Substituting herein for 1/x1 + 1/x2, 1/x3 + 1/x4 their foregoing
+  values, the equation becomes
+
+    -(A + Bm)(y - m'x) + (A + Bm')(y - mx) + C(m' - m) = 0;
+
+  that is,
+
+    (m - m')(Ax + By + C) = 0;
+
+  or finally it is Ax + By + C = 0, showing that the point Q lies in a
+  line the position of which is independent of the particular lines
+  OAA', OBB' used in the construction. It is proper to notice that there
+  is no correspondence to each other of the points A, A' and B, B'; the
+  grouping might as well have been A, A' and B', B; and it thence
+  appears that the line Ax + By + C = 0 just obtained is in fact the
+  line joining the point Q with the point R which is the intersection of
+  AB and A'B'.
+
+  15. In § 8 it has been seen that two conditions determine the equation
+  of a straight line, because in Ax + By + C = 0 one of the coefficients
+  may be divided out, leaving only two parameters to be determined.
+  Similarly five conditions instead of six determine an equation of the
+  second degree (a, b, c, f, g, h)(x, y, 1)² = 0, and nine instead of
+  ten determine a cubic (*)(x, y, 1)³ = 0. It thus appears that a cubic
+  can be made to pass through 9 given points, and that the cubic so
+  passing through 9 given points is completely determined. There is,
+  however, a remarkable exception. Considering two given cubic curves S
+  = 0, S' = 0, these intersect in 9 points, and through these 9 points
+  we have the whole series of cubics S - kS' = 0, where k is an
+  arbitrary constant: k may be determined so that the cubic shall pass
+  through a given tenth point (k = S0 ÷ S'0, if the coordinates are (x0,
+  y0), and S0, S'0 denote the corresponding values of S, S'). The
+  resulting curve SS'0 - S'S0 = 0 may be regarded as the cubic
+  determined by the conditions of passing through 8 of the 9 points and
+  through the given point (x0, y0); and from the equation it thence
+  appears that the curve passes through the remaining one of the 9
+  points. In other words, we thus have the theorem, any cubic curve
+  which passes through 8 of the 9 intersections of two given cubic
+  curves passes through the 9th intersection.
+
+  The applications of this theorem are very numerous; for instance, we
+  derive from it Pascal's theorem of the inscribed hexagon. Consider a
+  hexagon inscribed in a conic. The three alternate sides constitute a
+  cubic, and the other three alternate sides another cubic. The cubics
+  intersect in 9 points, being the 6 vertices of the hexagon, and the 3
+  Pascalian points, or intersections of the pairs of opposite sides of
+  the hexagon. Drawing a line through two of the Pascalian points, the
+  conic and this line constitute a cubic passing through 8 of the 9
+  points of intersection, and it therefore passes through the remaining
+  point of intersection--that is, the third Pascalian point; and since
+  obviously this does not lie on the conic, it must lie on the
+  line--that is, we have the theorem that the three Pascalian points (or
+  points of intersection of the pairs of opposite sides) lie on a line.
+
+  16. _Metrical Theory resumed. Projections and Perpendiculars._--It
+  is a metrical fact of fundamental importance, already used in § 8,
+  that, if a finite line PQ be projected on any other line OO' by
+  perpendiculars PP', QQ' to OO', the length of the projection P'Q' is
+  equal to that of PQ multiplied by the cosine of the acute angle
+  between the two lines. Also the algebraical sum of the projections of
+  the sides of any closed polygon upon any line is zero, because as a
+  point goes round the polygon, from any vertex A to A again, the point
+  which is its projection on the line passes from A' the projection of A
+  to A' again, i.e. traverses equal distances along the line in positive
+  and negative senses. If we consider the polygon as consisting of two
+  broken lines, each extending from the same initial to the same
+  terminal point, the sum of the projections of the lines which compose
+  the one is equal, in sign and magnitude, to the sum of the projections
+  of the lines composing the other. Observe that the projection on a
+  line of a length perpendicular to the line is zero.
+
+  Let us hence find the equation of a straight line such that the
+  perpendicular OD on it from the origin is of length [rho] taken as
+  positive, and is inclined to the axis of x at an angle xOD = [alpha],
+  measured counter-clockwise from Ox. Take any point P (x, y) on the
+  line, and construct OM and MP as in fig. 48. The sum of the
+  projections of OM and MP on OD is OD itself; and this gives the
+  equation of the line
+
+    x cos [alpha] + y sin [alpha] = [rho].
+
+  Observe that cos [alpha] and sin [alpha] here are the sin [alpha] and
+  -cos [alpha], or the -sin [alpha] and cos [alpha] of § 8 according to
+  circumstances.
+
+  We can write down an expression for the perpendicular distance from
+  this line of any point (x', y') which does not lie upon it. If the
+  parallel through (x', y') to the line meet OD in E, we have x' cos
+  [alpha] + y' sin [alpha] = OE, and the perpendicular distance required
+  is OD - OE, i.e. [rho] - x' cos [alpha] - y' sin [alpha]; it is the
+  perpendicular distance taken positively or negatively according as
+  (x', y') lies on the same side of the line as the origin or not.
+
+  The general equation Ax + By + C = 0 may be given the form x cos
+  [alpha] + y sin [alpha] - [rho] = 0 by dividing it by [root](A² + B³).
+  Thus (Ax' + By' + C) ÷ [root](A² + B²) is in absolute value the
+  perpendicular distance of (x', y') from the line Ax + By + C = 0.
+  Remember, however, that there is an essential ambiguity of sign
+  attached to a square root. The expression found gives the distance
+  taken positively when (x', y') is on the origin side of the line, if
+  the sign of C is given to [root](A² + B²).
+
+  17. _Transformation of Coordinates._--We often need to adopt new axes
+  of reference in place of old ones; and the above principle of
+  projections readily expresses the old coordinates of any point in
+  terms of the new.
+
+  [Illustration: FIG. 53.]
+
+  Suppose, for instance, that we want to take for new origin the point
+  O' of old coordinates OA = h, AO' = k, and for new axes of X and Y
+  lines through O' obtained by rotating parallels to the old axes of x
+  and y through an angle [theta] counter-clockwise. Construct (fig. 53)
+  the old and new coordinates of any point P. Expressing that the
+  projections, first on the old axis of x and secondly on the old axis
+  of y, of OP are equal to the sums of the projections, on those axes
+  respectively, of the parts of the broken line OO'M'P, we obtain:
+
+    x = h + X cos [theta] + Y cos ([theta] + ½[pi]) = h + X cos [theta] -
+      Y sin [theta],
+
+  and
+
+    y = k + X cos (½[pi] - [theta]) + Y cos [theta] = k + X sin [theta] +
+      Y cos [theta].
+
+  Be careful to observe that these formulae do not apply to every
+  conceivable change of reference from one set of rectangular axes to
+  another. It might have been required to take O'X, O'Y' for the
+  positive directions of the new axes, so that the change of directions
+  of the axes could not be effected by rotation. We must then write -Y
+  for Y in the above.
+
+  Were the new axes oblique, making angles [alpha], ß respectively with
+  the old axis of x, and so inclined at the angle ß - [alpha], the same
+  method would give the formulae
+
+    x = h + X cos [alpha] + Y cos ß, y = k + X sin [alpha] + Y sin ß.
+
+  18. _The Conic Sections._--The conics, as they are now called, were at
+  first defined as curves of intersection of planes and a cone; but
+  Apollonius substituted a definition free from reference to space of
+  three dimensions. This, in effect, is that a conic is the locus of a
+  point the distance of which from a given point, called the focus, has
+  a given ratio to its distance from a given line, called the directrix
+  (see CONIC SECTION). If e : 1 is the ratio, e is called the
+  eccentricity. The distances are considered signless.
+
+  Take (h, k) for the focus, and x cos [alpha] + y sin [alpha] - p = 0
+  for the directrix. The absolute values of [root] {(x - h)² + (y - k)²}
+  and p - x cos [alpha] -y sin [alpha] are to have the ratio e : 1; and
+  this gives
+
+    (x - h)² + (y - k)² = e²(p - x cos [alpha] - y sin [alpha])²
+
+  as the general equation, in rectangular coordinates, of a conic.
+
+  It is of the second degree, and is the general equation of that
+  degree. If, in fact, we multiply it by an unknown [lambda], we can, by
+  solving six simultaneous equations in the six unknowns [lambda], h, k,
+  e, p, [alpha], so choose values for these as to make the coefficients
+  in the equation equal to those in any equation of the second degree
+  which may be given. There is no failure of this statement in the
+  special case when the given equation represents two straight lines, as
+  in § 10, but there is speciality: if the two lines intersect, the
+  intersection and either bisector of the angle between them are a focus
+  and directrix; if they are united in one line, any point on the line
+  and a perpendicular to it through the point are: if they are parallel,
+  the case is a limiting one in which e and h² + k² have become infinite
+  while e^(-2)(h² + k²) remains finite. In the case (§ 9) of an equation
+  such as represents a circle there is another instance of proceeding to
+  a limit: e has to become 0, while ep remains finite: moreover [alpha]
+  is indeterminate. The centre of a circle is its focus, and its
+  directrix has gone to infinity, having no special direction. This last
+  fact illustrates the necessity, which is also forced on plane geometry
+  by three-dimensional considerations, of treating all points at
+  infinity in a plane as lying on a single straight line.
+
+  Sometimes, in reducing an equation to the above focus and directrix
+  form, we find for h, k, e, p, tan [alpha], or some of them, only
+  imaginary values, as quadratic equations have to be solved; and we
+  have in fact to contemplate the existence of entirely imaginary
+  conics. For instance, no real values of x and y satisfy x² + 2y² + 3 =
+  0. Even when the locus represented is real, we obtain, as a rule, four
+  sets of values of h, k, e, p, of which two sets are imaginary; a real
+  conic has, besides two real foci and corresponding directrices, two
+  others that are imaginary.
+
+  In oblique as well as rectangular coordinates equations of the second
+  degree represent conics.
+
+  19. _The three Species of Conics._--A real conic, which does not
+  degenerate into straight lines, is called an ellipse, parabola or
+  hyperbola according as e <, = , or > 1. To trace the three forms it is best so to
+  choose the axes of reference as to simplify their equations.
+
+  In the case of a parabola, let 2c be the distance between the given
+  focus and directrix, and take axes referred to which these are the
+  point (c, 0) and the line x = - c. The equation becomes (x - c)² + y²
+  = (x + c)², i.e. y² = 4cx.
+
+  In the other cases, take a such that a(e ~ e^(-1)) is the distance of
+  focus from directrix, and so choose axes that these are (ae, 0) and x
+  = ae^(-1), thus getting the equation(x - ae)² + y² = e²(x - ae^(-1))²,
+  i.e. (1 - e²)x² + y² = a²(1 - e²). When e < 1, i.e. in the case of an
+  ellipse, this may be written x²/a² + y²/b² = 1, where b² = a²(1 - e²);
+  and when e > 1, i.e. in the case of an hyperbola, x²/a² - y²/b² = 1,
+  where b² = a²(e² - 1). The axes thus chosen for the ellipse and
+  hyperbola are called the principal axes.
+
+  In figs. 54, 55, 56 in order, conics of the three species, thus
+  referred, are depicted.
+
+  [Illustration: FIG. 54]
+
+  [Illustration: FIG. 55]
+
+  [Illustration: FIG. 56.]
+
+  The oblique straight lines in fig. 56 are the _asymptotes_ x/a = ±y/b
+  of the hyperbola, lines to which the curve tends with unlimited
+  closeness as it goes to infinity. The hyperbola would have an equation
+  of the form xy = c if referred to its asymptotes as axes, the
+  coordinates being then oblique, unless a = b, in which case the
+  hyperbola is called rectangular. An ellipse has two imaginary
+  asymptotes. In particular a circle x² + y² = a², a particular ellipse,
+  has for asymptotes the imaginary lines x = ±y [root]-1. These run from
+  the centre to the so-called circular points at infinity.
+
+  20. _Tangents and Curvature._--Let (x', y') and (x' + h, y' + k) be
+  two neighbouring points P, P' on a curve. The equation of the line on
+  which both lie is h(y - y') = k(x - x'). Now keep P fixed, and let P'
+  move towards coincidence with it along the curve. The connecting line
+  will tend towards a limiting position, to which it can never attain as
+  long as P and P' are distinct. The line which occupies this limiting
+  position is the tangent at P. Now if we subtract the equation of the
+  curve, with (x', y') for the coordinates in it, from the like equation
+  in (x' + h, y' + k), we obtain a relation in h and k, which will, as a
+  rule, be of the form 0 = Ah + Bk + terms of higher degrees in h and k,
+  where A, B and the other coefficients involve x' and y'. This gives
+  k/h = -A/B + terms which tend to vanish as h and k do, so that -A : B
+  is the limiting value tended to by k : h. Hence the equation of the
+  tangent is B(y - y') + A(x - x') = 0.
+
+  The _normal_ at (x', y') is the line through it at right angles to the
+  tangent, and its equation is A(y - y') - B(x - x') = 0.
+
+  In the case of the conic (a, b, c, f, g, h) (x, y, 1)² = 0 we find
+  that A/B = (ax' + hy' + g)/(hx' + by' + f).
+
+  We can obtain the coordinates of Q, the intersection of the normals
+  QP, QP' at (x', y') and (x' + h, y' + k), and then, using the limiting
+  value of k : h, deduce those of its limiting position as P' moves up
+  to P. This is the _centre of curvature_ of the curve at P (x', y'),
+  and is so called because it is the centre of the circle of closest
+  contact with the curve at that point. That it is so follows from the
+  facts that the closest circle is the limit tended to by the circle
+  which touches the curve at P and passes through P', and that the arc
+  from P to P' of this circle lies between the circles of centre Q and
+  radii QP, QP', which circles tend, not to different limits as P' moves
+  up to P, but to one. The distance from P to the centre of curvature is
+  the _radius of curvature_.
+
+  21. _Differential Plane Geometry._--The language and notation of the
+  differential calculus are very useful in the study of tangents and
+  curvature. Denoting by ([xi], [eta]) the current coordinates, we find,
+  as above, that the tangent at a point (x, y) of a curve is [eta] - y =
+  ([xi] - x)dy/dx, where dy/dx is found from the equation of the curve.
+  If this be f(x, y) = 0 the tangent is ([xi] - x) (dPf/dPx) + ([eta] - y)
+  (dPf/dPy) = 0. If [rho] and ([alpha], ß) are the radius and centre of
+  curvature at (x, y), we find that q([alpha] - x) = -p(1 + p²), q(ß -
+  y) = 1 + p², q²[rho]² = (1 + p²)³, where p, q denote dy/dx, d²y/dx²
+  respectively. (See INFINITESIMAL CALCULUS.)
+
+  In any given case we can, at all events in theory, eliminate x, y
+  between the above equations for [alpha] - x and ß - y, and the
+  equation of the curve. The resulting equation in ([alpha], ß)
+  represents the locus of the centre of curvature. This is the _evolute_
+  of the curve.
+
+  22. _Polar Coordinates._--In plane geometry the distance of any point
+  P from a fixed origin (or pole) O, and the inclination xOP of OP to a
+  fixed line Ox, determine the point: r, the numerical measure of OP,
+  the _radius vector_, and [theta], the circular measure of xOP, the
+  _inclination_, are called polar coordinates of P. The formulae x = r
+  cos [theta], y = r sin [theta] connect Cartesian and polar
+  coordinates, and make transition from either system to the other easy.
+  In polar coordinates the equations of a circle through O, and of a
+  conic with O as focus, take the simple forms r = 2a cos
+  ([theta]-[alpha]), r {1 - e cos ([theta]-[alpha])} = l. The use of
+  polar coordinates is very convenient in discussing curves which have
+  properties of symmetry akin to that of a regular polygon, such curves
+  for instance as r = a cos m [theta], with m integral, and also the
+  curves called spirals, which have equations giving r as functions of
+  [theta] itself, and not merely of sin [theta] and cos [theta]. In the
+  geometry of motion under central forces the advantage of working with
+  polar coordinates is great.
+
+  23. _Trilinear and Areal Coordinates._--Consider a fixed triangle ABC,
+  and regard its sides as produced without limit. Denote, as in
+  trigonometry, by a, b, c the positive numbers of units of a chosen
+  scale contained in the lengths BC, CA, AB, by A, B, C the angles, and
+  by [Delta] the area, of the triangle. We might, as in § 6, take CA, CB
+  as axes of x and y, inclined at an angle C. Any point P (x, y) in the
+  plane is at perpendicular distances y sin C and x sin C from CA and
+  CB. Call these ß and [alpha] respectively. The signs of ß and [alpha]
+  are those of y and x, i.e. ß is positive or negative according as P
+  lies on the same side of CA as B does or the opposite, and similarly
+  for [alpha]. An equation in (x, y) of any degree may, upon replacing
+  in it x and y by [alpha] cosec C and ß cosec C, be written as one of
+  the same degree in ([alpha], ß). Now let [gamma] be the perpendicular
+  distance of P from the third side AB, taken as positive or negative as
+  P is on the C side of AB or not. The geometry of the figure tells us
+  that a[alpha] + bß + c[gamma] = 2[Delta]. By means of this relation in
+  [alpha], ß, [gamma] we can give an equation considered countless other
+  forms, involving two or all of [alpha], ß, [gamma]. In particular we
+  may make it _homogeneous_ in [alpha], ß, [gamma]: to do this we have
+  only to multiply the terms of every degree less than the highest
+  present in the equation by a power of (a[alpha] + bß +
+  c[gamma])/2[Delta] just sufficient to raise them, in each case, to the
+  highest degree.
+
+  We call ([alpha], ß, [gamma]) _trilinear coordinates_, and an equation
+  in them the trilinear equation of the locus represented. Trilinear
+  equations are, as a rule, dealt with in their homogeneous forms. An
+  advantage thus gained is that we need not mean by ([alpha], ß,
+  [gamma]) the actual measures of the perpendicular distances, but any
+  properly signed numbers which have the same ratio two and two as these
+  distances.
+
+  In place of [alpha], ß, [gamma] it is lawful to use, as coordinates
+  specifying the position of a point in the plane of a triangle of
+  reference ABC, any given multiples of these. For instance, we may use
+  x = a[alpha]/2[Delta], y = bß/2[Delta], z = c[gamma]/2[Delta], the
+  properly signed ratios of the triangular areas PBC, PCA, PAB to the
+  triangular area ABC. These are called the _areal_ coordinates of P. In
+  areal coordinates the relation which enables us to make any equation
+  homogeneous takes the simple form x + y + z = 1; and, as before, we
+  need mean by x, y, z, in a homogeneous equation, only signed numbers
+  in the right ratios.
+
+  Straight lines and conics are represented in trilinear and in areal,
+  because in Cartesian, coordinates by equations of the first and second
+  degrees respectively, and these degrees are preserved when the
+  equations are made homogeneous. What must be said about points
+  infinitely far off in order to make universal the statement, to which
+  there is no exception as long as finite distances alone are
+  considered, that _every_ homogeneous equation of the first degree
+  represents a straight line? Let the point of areal coordinates (x',
+  y', z') move infinitely far off, and mean by x, y, z finite quantities
+  in the ratios which x', y', z' tend to assume as they become infinite.
+  The relation x' + y' + z' = 1 gives that the limiting state of things
+  tended to is expressed by x + y + z = 0. This particular equation of
+  the first degree is satisfied by no point at a finite distance; but we
+  see the propriety of saying that it has to be taken as satisfied by
+  all the points conceived of as actually at infinity. Accordingly the
+  special property of these points is expressed by saying that they lie
+  on a special straight line, of which the areal equation is x + y + z =
+  0. In trilinear coordinates this _line at infinity_ has for equation
+  a[alpha] + bß + c[gamma] = 0.
+
+  On the one special line at infinity parallel lines are treated as
+  meeting. There are on it two special (imaginary) points, the circular
+  points at infinity of § 19, through which all circles pass in the same
+  sense. In fact if S = O be one circle, in areal coordinates, S + (x +
+  y + z)(lx + my + nz) = 0 may, by proper choice of l, m, n, be made any
+  other; since the added terms are once lx + my + nz, and have the
+  generality of any expression like a'x + b'y + c' in Cartesian
+  coordinates. Now these two circles intersect in the two points where
+  either meets x + y + z = 0 as well as in two points on the radical
+  axis lx + my + nz = 0.
+
+  24. Let us consider the perpendicular distance of a point ([alpha]',
+  ß', [gamma]') from a line l[alpha] + mß + n[gamma]. We can take
+  rectangular axes of Cartesian coordinates (for clearness as to
+  equalities of angle it is best to choose an origin inside ABC), and
+  refer to them, by putting expressions p - x cos[theta] - y sin[theta],
+  &c., for [alpha] &c.; we can then apply § 16 to get the perpendicular
+  distance; and finally revert to the trilinear notation. The result is
+  to find that the required distance is
+
+    (l[alpha]' + mß' + n[gamma]')/{l, m, n},
+
+  where {l, m, n}² = l² + m² + n² - 2mn cos A - 2nl cos B - 2lm cos C.
+
+  In areal coordinates the perpendicular distance from (x', y', z') to
+  lx + my + nz = 0 is 2[Delta](lx' + my' + nz')/{al, bm, cn}. In both
+  cases the coordinates are of course actual values.
+
+  Now let [xi], [eta], [zeta] be the perpendiculars on the line from the
+  vertices A, B, C, i.e. the points (1, 0, 0), (0, 1, 0), (0, 0, 1),
+  with signs in accord with a convention that oppositeness of sign
+  implies distinction between one side of the line and the other. Three
+  applications of the result above give
+
+    [xi]/l = 2[Delta]/{al, bm, cn} = [eta]/m = [zeta]/n;
+
+  and we thus have the important fact that [xi]x' + [eta]y' + [zeta]z'
+  is the perpendicular distance between a point of areal coordinates
+  (x'y'z') and a line on which the perpendiculars from A, B, C are [xi],
+  [eta], [zeta] respectively. We have also that [xi]x + [eta]y + [zeta]z
+  = 0 is the areal equation of the line on which the perpendiculars are
+  [xi], [eta], [zeta]; and, by equating the two expressions for the
+  perpendiculars from (x', y', z') on the line, that in all cases
+  {a[xi], b[eta], c[zeta]}² = 4[Delta]².
+
+  25. _Line-coordinates. Duality._--A quite different order of ideas
+  may be followed in applying analysis to geometry. The notion of a
+  straight line specified may precede that of a point, and points may be
+  dealt with as the intersections of lines. The specification of a line
+  may be by means of coordinates, and that of a point by an equation,
+  satisfied by the coordinates of lines which pass through it. Systems
+  of _line-coordinates_ will here be only briefly considered. Every such
+  system is allied to some system of point-coordinates; and space will
+  be saved by giving prominence to this fact, and not recommencing _ab
+  initio_.
+
+  Suppose that any particular system of point-coordinates, in which lx +
+  my + nz = 0 may represent any straight line, is before us: notice that
+  not only are trilinear and areal coordinates such systems, but
+  Cartesian coordinates also, since we may write x/z, y/z for the
+  Cartesian x, y, and multiply through by z. The line is exactly
+  assigned if l, m, n, or their mutual ratios, are known. Call (l, m, n)
+  the _coordinates_ of the line. Now keep x, y, z constant, and let the
+  coordinates of the line vary, but always so as to satisfy the
+  equation. This equation, which we now write xl + ym + zn = 0, is
+  satisfied by the coordinates of every line through a certain fixed
+  point, and by those of no other line; it is the equation of that point
+  in the line-coordinates l, m, n.
+
+  Line-coordinates are also called _tangential_ coordinates. A curve is
+  the envelope of lines which touch it, as well as the locus of points
+  which lie on it. A homogeneous equation of degree above the first in
+  l, m, n is a relation connecting the coordinates of every line which
+  touches some curve, and represents that curve, regarded as an
+  envelope. For instance, the condition that the line of coordinates (l,
+  m, n), i.e. the line of which the allied point-coordinate equation is
+  lx + my + nz = 0, may touch a conic (a, b, c, f, g, h) (x, y, z)² = 0,
+  is readily found to be of the form (A, B, C, F, G, H) (l, m, n)² = 0,
+  i.e. to be of the second degree in the line-coordinates. It is not
+  hard to show that the _general_ equation of the second degree in l, m,
+  n thus represents a conic; but the degenerate conics of
+  line-coordinates are not line-pairs, as in point-coordinates, but
+  point-pairs.
+
+  The degree of the point-coordinate equation of a curve is the _order_
+  of the curve, the number of points in which it cuts a straight line.
+  That of the line-coordinate equation is its _class_, the number of
+  tangents to it from a point. The order and class of a curve are
+  generally different when either exceeds two.
+
+  26. The system of line-coordinates allied to the areal system of
+  point-coordinates has special interest.
+
+  The l, m, n of this system are the perpendiculars [xi], [eta], [zeta]
+  of § 24; and x'[xi] + y'[eta] + z'[zeta] = 0 is the equation of the
+  point of areal coordinates (x', y', z'), i.e. is a relation which the
+  perpendiculars from the vertices of the triangle of reference on every
+  line through the point, but no other line, satisfy. Notice that a
+  non-homogeneous equation of the first degree in [xi], [eta], [zeta]
+  does not, as a homogeneous one does, represent a point, but a circle.
+  In fact x'[xi] + y'[eta] + z'[zeta] = R expresses the constancy of the
+  perpendicular distance of the fixed point x'[xi] + y'[eta] + z'[zeta]
+  = 0 from the variable line ([xi], [eta], [zeta]), i.e. the fact that
+  ([xi], [eta], [zeta]) touches a circle with the fixed point for
+  centre. The relation in any [xi], [eta], [zeta] which enables us to
+  make an equation homogeneous is not linear, as in point-coordinates,
+  but quadratic, viz. it is the relation {a[xi], b[eta], c[zeta]}² =
+  4[Delta]² of § 24. Accordingly the homogeneous equation of the above
+  circle is
+
+    4[Delta]²(x'[xi] + y'[eta] + z'[zeta])² = R²{a[xi], b[eta], c[zeta]}².
+
+  Every circle has an equation of this form in the present system of
+  line-coordinates. Notice that the equation of any circle is satisfied
+  by those coordinates of lines which satisfy both x'[xi] + y'[eta] +
+  z'[zeta] = 0, the equation of its centre, and {a[xi], b[eta],
+  c[zeta]}² = 0. This last equation, of which the left-hand side
+  satisfies the condition for breaking up into two factors, represents
+  the two imaginary circular points at infinity, through which all
+  circles and their asymptotes pass.
+
+  There is strict duality in descriptive geometry between
+  point-line-locus and line-point-envelope theorems. But in metrical
+  geometry duality is encumbered by the fact that there is in a plane
+  one special line only, associated with distance, while of special
+  points, associated with direction, there are two: moreover the line is
+  real, and the points both imaginary.
+
+
+  II. _Solid Analytical Geometry._
+
+  27. Any point in space may be specified by three coordinates. We
+  consider three fixed planes of reference, and generally, as in all
+  that follows, three which are at right angles two and two. They
+  intersect, two and two, in lines x'Ox, y'Oy, z'Oz, called the axes of
+  x, y, z respectively, and divide all space into eight parts called
+  octants. If from any point P in space we draw PN parallel to zOz' to
+  meet the plane xOy in N, and then from N draw NM parallel to yOy' to
+  meet x'Ox in M, the coordinates (x, y, z) of P are the numerical
+  measures of OM, MN, NP; in the case of rectangular coordinates these
+  are the perpendicular distances of P from the three planes of
+  reference. The sign of each coordinate is positive or negative as P
+  lies on one side or the other of the corresponding plane. In the
+  octant delineated the signs are taken all positive.
+
+  [Illustration: FIG. 57.]
+
+  [Illustration: FIG. 58.]
+
+  In fig. 57 the delineation is on a plane of the paper taken parallel
+  to the plane zOx, the points of a solid figure being projected on that
+  plane by parallels to some chosen line through O in the positive
+  octant. Sometimes it is clearer to delineate, as in fig. 58, by
+  projection parallel to that line in the octant which is equally
+  inclined to Ox, Oy, Oz upon a plane of the paper perpendicular to it.
+  It is possible by parallel projection to delineate equal scales along
+  Ox, Oy, Oz by scales having any ratios we like along lines in a plane
+  having any mutual inclinations we like.
+
+  [Illustration: FIG. 59.]
+
+  For the delineation of a surface of simple form it frequently suffices
+  to delineate the sections by the coordinate planes; and, in
+  particular, when the surface has symmetry about each coordinate plane,
+  to delineate the quarter-sections belonging to a single octant. Thus
+  fig. 59 conveniently represents an octant of the wave surface, which
+  cuts each coordinate plane in a circle and an ellipse. Or we may
+  delineate a series of contour lines, i.e. sections by planes parallel
+  to xOy, or some other chosen plane; of course other sections may be
+  indicated too for greater clearness. For the delineation of a curve a
+  good method is to represent, as above, a series of points P thereof,
+  each accompanied by its ordinate PN, which serves to refer it to the
+  plane of xy. The employment of stereographic projection is also
+  interesting.
+
+  28. In plane geometry, reckoning the line as a curve of the first
+  order, we have only the point and the curve. In solid geometry,
+  reckoning a line as a curve of the first order, and the plane as a
+  surface of the first order, we have the point, the curve and the
+  surface; but the increase of complexity is far greater than would
+  hence at first sight appear. In plane geometry a curve is considered
+  in connexion with lines (its tangents); but in solid geometry the
+  curve is considered in connexion with lines and planes (its tangents
+  and osculating planes), and the surface also in connexion with lines
+  and planes (its tangent lines and tangent planes); there are surfaces
+  arising out of the line--cones, skew surfaces, developables, doubly
+  and triply infinite systems of lines, and whole classes of theories
+  which have nothing analogous to them in plane geometry: it is thus a
+  very small part indeed of the subject which can be even referred to in
+  the present article.
+
+  In the case of a surface we have between the coordinates (x, y, z) a
+  single, or say a onefold relation, which can be represented by a
+  single relation [f](x, y, z) = 0; or we may consider the coordinates
+  expressed each of them as a given function of two variable parameters
+  p, q; the form z = [f](x, y) is a particular case of each of these
+  modes of representation; in other words, we have in the first mode
+  [f](x, y, z) = z - [f](x, y), and in the second mode x = p, y = q for
+  the expression of two of the coordinates in terms of the parameters.
+
+  In the case of a curve we have between the coordinates (x, y, z) a
+  twofold relation: two equations [f](x, y, z) = 0, [phi](x, y, z) = 0
+  give such a relation; i.e. the curve is here considered as the
+  intersection of two surfaces (but the curve is not always the complete
+  intersection of two surfaces, and there are hence difficulties); or,
+  again, the coordinates may be given each of them as a function of a
+  single variable parameter. The form y = [phi](x), z = [psi](x), where
+  two of the coordinates are given in terms of the third, is a
+  particular case of each of these modes of representation.
+
+  29. The remarks under plane geometry as to descriptive and metrical
+  propositions, and as to the non-metrical character of the method of
+  coordinates when used for the proof of a descriptive proposition,
+  apply also to solid geometry; and they might be illustrated in like
+  manner by the instance of the theorem of the radical centre of four
+  spheres. The proof is obtained from the consideration that S and S'
+  being each of them a function of the form x² + y² + z² + ax + by + cz
+  + d, the difference S-S' is a mere linear function of the coordinates,
+  and consequently that S-S' = 0 is the equation of the plane containing
+  the circle of intersection of the two spheres S = 0 and S' = 0.
+
+  [Illustration: FIG. 60.]
+
+  30. _Metrical Theory._--The foundation in solid geometry of the
+  metrical theory is in fact the before-mentioned theorem that if a
+  finite right line PQ be projected upon any other line OO' by lines
+  perpendicular to OO', then the length of the projection P'Q' is equal
+  to the length of PQ into the cosine of its inclination to P'Q'--or (in
+  the form in which it is now convenient to state the theorem) the
+  perpendicular distance P'Q' of two parallel planes is equal to the
+  inclined distance PQ into the cosine of the inclination. The principle
+  of § 16, that the algebraical sum of the projections of the sides of
+  any closed polygon on any line is zero, or that the two sets of sides
+  of the polygon which connect a vertex A and a vertex B have the same
+  sum of projections on the line, in sign and magnitude, as we pass from
+  A to B, is applicable when the sides do not all lie in one plane.
+
+  31. Consider the skew quadrilateral QMNP, the sides QM, MN, NP being
+  respectively parallel to the three rectangular axes Ox, Oy, Oz; let
+  the lengths of these sides be [xi], [eta], [zeta], and that of the
+  side QP be = [rho]; and let the cosines of the inclinations (or say
+  the cosine-inclinations) of [rho] to the three axes be [alpha], ß,
+  [gamma]; then projecting successively on the three sides and on QP we
+  have
+
+    [xi], [eta], [zeta] = [rho][alpha], [rho]ß, [rho][gamma],
+
+  and
+
+    [rho] = [alpha][xi] + ß[eta] + [gamma][zeta],
+
+  whence [rho]² = [xi]² + [eta]² + [zeta]², which is the relation
+  between a distance [rho] and its projections [xi], [eta], [zeta] upon
+  three rectangular axes. And from the same equations we obtain [alpha]²
+  + ß² + [gamma]² = 1, which is a relation connecting the
+  cosine-inclinations of a line to three rectangular axes.
+
+  Suppose we have through Q any other line QT, and let the
+  cosine-inclinations of this to the axes be [alpha]', ß', [gamma]', and
+  [delta] be its cosine-inclination to QP; also let [rho] be the length
+  of the projection of QP upon QT; then projecting on QT we have
+
+    [rho] = [alpha]'[xi] + ß'[eta] + [gamma]'[zeta] = [rho][delta].
+
+  And in the last equation substituting for [xi], [eta], [zeta] their
+  values [rho][alpha], [rho]ß, [rho][gamma] we find
+
+    [delta] = [alpha][alpha]' + ß[beta]' + [gamma][gamma]',
+
+  which is an expression for the mutual cosine-inclination of two lines,
+  the cosine-inclinations of which to the axes are [alpha], ß, [gamma]
+  and [alpha]', ß', [gamma]' respectively. We have of course [alpha]² +
+  ß² + [gamma]² = 1 and [alpha]'² + ß'² + [gamma]'² = 1; and hence also
+
+    1 - [delta]² = ([alpha]² + ß² + [gamma]²)([alpha]'² + ß'² + [gamma]'²)
+      - ([alpha][alpha]' + ß[beta]' + [gamma][gamma]')²,
+
+    = (ß[gamma]' - ß'[gamma])² + ([gamma][alpha]' - [gamma]'[alpha])² +
+      ([alpha]ß' - [alpha]'ß)²;
+
+  so that the sine of the inclination can only be expressed as a square
+  root. These formulae are the foundation of spherical trigonometry.
+
+  32. _Straight Lines, Planes and Spheres._--The foregoing formulae give
+  at once the equations of these loci.
+
+  For first, taking Q to be a fixed point, coordinates (a, b, c), and
+  the cosine-inclinations ([alpha], ß, [gamma]) to be constant, then P
+  will be a point in the line through Q in the direction thus
+  determined; or, taking (x, y, z) for its coordinates, these will be
+  the current coordinates of a point in the line. The values of [xi],
+  [eta], [zeta] then are x - a, y - b, z - c, and we thus have
+
+     x - a    y - b    z - c
+    ------- = ----- = ------- (= [rho]),
+    [alpha]     ß     [gamma]
+
+  which (omitting the last equation, = [rho]) are the equations of the
+  line through the point (a, b, c), the cosine-inclinations to the axes
+  being [alpha], ß, [gamma], and these quantities being connected by the
+  relation [alpha]² + ß² + [gamma]² = 1. This equation may be omitted,
+  and then [alpha], ß, [gamma], instead of being equal, will only be
+  proportional, to the cosine-inclinations.
+
+  Using the last equation, and writing
+
+    x, y, z = a + [alpha][rho], b + ß[rho], c + [gamma][rho],
+
+  these are expressions for the current coordinates in terms of a
+  parameter [rho], which is in fact the distance from the fixed point
+  (a, b, c).
+
+  It is easy to see that, if the coordinates (x, y, z) are connected by
+  any two linear equations, these equations can always be brought into
+  the foregoing form, and hence that the two linear equations represent
+  a line.
+
+  Secondly, taking for greater simplicity the point Q to be coincident
+  with the origin, and [alpha]', ß', [gamma]', p to be constant, then p
+  is the perpendicular distance of a plane from the origin, and
+  [alpha]', ß', [gamma]' are the cosine-inclinations of this distance to
+  the axes ([alpha]'² + ß'² + [gamma]'² = 1). P is any point in this
+  plane, and taking its coordinates to be (x, y, z) then ([xi], [eta],
+  [zeta]) are = (x, y, z), and the foregoing equation p = [alpha]'[xi] +
+  ß'[eta] + [gamma]'[zeta] becomes
+
+    [alpha]'x + ß'y + [gamma]'z = p,
+
+  which is the equation of the plane in question.
+
+  If, more generally, Q is not coincident with the origin, then, taking
+  its coordinates to be (a, b, c), and writing p1 instead of p, the
+  equation is
+
+    [alpha]'(x - a) + ß'(y - b) + [gamma]'(z - c) = p1;
+
+  and we thence have p1 = p - (a[alpha]' + bß' + c[gamma]'), which is an
+  expression for the perpendicular distance of the point (a, b, c) from
+  the plane in question.
+
+  It is obvious that any linear equation Ax + By + Cz + D = O between
+  the coordinates can always be brought into the foregoing form, and
+  hence that such an equation represents a plane.
+
+  Thirdly, supposing Q to be a fixed point, coordinates (a, b, c), and
+  the distance QP = [rho], to be constant, say this is = d, then, as
+  before, the values of [xi], [eta], [zeta] are x - a, y - b, z - c, and
+  the equation [xi]² + [eta]² + [zeta]² = [rho]² becomes
+
+    (x - a)² + (y - b)² + (z - c)² = d²,
+
+  which is the equation of the sphere, coordinates of the centre = (a,
+  b, c), and radius = d.
+
+  A quadric equation wherein the terms of the second order are x² + y² +
+  z², viz. an equation
+
+    x² + y² + z² + Ax + By + Cz + D = 0,
+
+  can always, it is clear, be brought into the foregoing form; and it
+  thus appears that this is the equation of a sphere, coordinates of the
+  centre -½A, -½B, -½C, and squared radius = ¼(A² + B² + C²) - D.
+
+  33. _Cylinders, Cones, ruled Surfaces._--If the two equations of a
+  straight line involve a parameter to which any value may be given, we
+  have a singly infinite system of lines. They cover a surface, and the
+  equation of the surface is obtained by eliminating the parameter
+  between the two equations.
+
+  If the lines all pass through a given point, then the surface is a
+  cone; and, in particular, if the lines are all parallel to a given
+  line, then the surface is a cylinder.
+
+  Beginning with this last case, suppose the lines are parallel to the
+  line x = mz, y = nz, the equations of a line of the system are x = mz
+  + a, y = nz + b,--where a, b are supposed to be functions of the
+  variable parameter, or, what is the same thing, there is between them
+  a relation f(a, b) = 0: we have a = x - mz, b = y - nz, and the result
+  of the elimination of the parameter therefore is [f](x - mz, y - nz) =
+  0, which is thus the general equation of the cylinder the generating
+  lines whereof are parallel to the line x = mz, y = nz. The equation of
+  the section by the plane z = 0 is [f](x, y) = 0, and conversely if the
+  cylinder be determined by means of its curve of intersection with the
+  plane z = 0, then, taking the equation of this curve to be f(x, y) =
+  0, the equation of the cylinder is [f](x - mz, y - nz) = 0. Thus, if
+  the curve of intersection be the circle (x - [alpha])² + (y - ß)² =
+  [gamma]², we have (x - mz - [alpha])² + (y - nz - ß)² = [gamma]² as
+  the equation of an oblique cylinder on this base, and thus also (x -
+  [alpha])² + (y - ß)² = [gamma]² as the equation of the right cylinder.
+
+  If the lines all pass through a given point (a, b, c), then the
+  equations of a line are x - a = [alpha](z - c), y - b = ß(z - c),
+  where [alpha], ß are functions of the variable parameter, or, what is
+  the same thing, there exists between them an equation f([alpha], ß) =
+  0; the elimination of the parameter gives, therefore, f[(x - a)/(x -
+  c'), (y - b)/(z - c)] = 0; and this equation, or, what is the same
+  thing, any homogeneous equation f(x - a, y - b, z - c) = 0, or, taking
+  f to be a rational and integral function of the order n, say (*)(x -
+  a, y - b, z - c)^n = 0, is the general equation of the cone having the
+  point (a, b, c) for its vertex. Taking the vertex to be at the origin,
+  the equation is (*)(x, y, z)^n = 0; and, in particular, (*)(x, y, z)²
+  = 0 is the equation of a cone of the second order, or quadricone,
+  having the origin for its vertex.
+
+  34. In the general case of a singly infinite system of lines, the
+  locus is a ruled surface (or _regulus_). Now, when a line is changing
+  its position in space, it may be looked upon as in a state of turning
+  about some point in itself, while that point is, as a rule, in a state
+  of moving out of the plane in which the turning takes place. If
+  instantaneously it is only in a state of turning, it is usual, though
+  not strictly accurate, to say that it intersects its consecutive
+  position. A regulus such that consecutive lines on it do not
+  intersect, in this sense, is called a skew surface, or _scroll_; one
+  on which they do is called a developable surface or _torse_.
+
+  Suppose, for instance, that the equations of a line (depending on the
+  variable parameter [theta]) are x/a + y/c = [theta] (1 + y/b), x/a -
+  z/c = 1/[theta] (1 - y/b); then, eliminating [theta] we have x²/a² -
+  z²/c² = 1 - y²/b², or say, x²/a² + z²/b² - z²/c² = 1, the equation of
+  a quadric surface, afterwards called the hyperboloid of one sheet;
+  this surface is consequently a scroll. It is to be remarked that we
+  have upon the surface a second singly infinite series of lines; the
+  equations of a line of this second system (depending on the variable
+  parameter [phi]) are
+
+    x    z          /    y \   x    z      1    /    y \
+    -- + -- = [phi]( 1 - -- ), -- - -- = ----- ( 1 + -- ).
+    a    c          \    b /   a    c    [phi]  \    b /
+
+  It is easily shown that any line of the one system intersects every
+  line of the other system.
+
+  Considering any curve (of double curvature) whatever, the tangent
+  lines of the curve form a singly infinite system of lines, each line
+  intersecting the consecutive line of the system,--that is, they form a
+  developable, or torse; the curve and torse are thus inseparably
+  connected together, forming a single geometrical figure. An osculating
+  plane of the curve (see § 38 below) is a tangent plane of the torse
+  all along a generating line.
+
+  35. _Transformation of Coordinates._--There is no difficulty in
+  changing the origin, and it is for brevity assumed that the origin
+  remains unaltered. We have, then, two sets of rectangular axes, Ox,
+  Oy, Oz, and Ox1, Oy1, Ozx1, the mutual cosine-inclinations being shown
+  by the diagram--
+
+        |    x    |    y  |    z    |
+    ----+---------+-------+---------+
+    x1  | [alpha] |   ß   | [gamma] |
+    ----+---------+-------+---------+
+    y1  | [alpha] |   ß'  | [gamma]'|
+    ----+---------+-------+---------+
+    z1  | [alpha]"|   ß"  | [gamma]"|
+    ----+---------+-------+---------+
+
+  that is, [alpha], ß, [gamma] are the cosine-inclinations of Ox1 to Ox,
+  Oy, Oz; [alpha]', ß', [gamma]' those of Oy1, &c.
+
+  And this diagram gives also the linear expressions of the coordinates
+  (x1, y1, z1) or (x, y, z) of either set in terms of those of the other
+  set; we thus have
+
+    x1 = [alpha] x + ß y + [gamma] z,
+     x = [alpha]x1 + [alpha]'y1 + [alpha]"z1,
+
+    y1 = [alpha]'x + ß'y + [gamma]'z,
+     y = ßx1 + ß'y1 + ß"z1,
+
+    z1 = [alpha]"x + ß"y + [gamma]"z,
+     z = [gamma]x1 + [gamma]'y1 + [gamma]"z1,
+
+  which are obtained by projection, as above explained. Each of these
+  equations is, in fact, nothing else than the before-mentioned equation
+  p = [alpha]'[xi] + ß'[eta] + [gamma]'[zeta], adapted to the problem in
+  hand.
+
+  But we have to consider the relations between the nine coefficients.
+  By what precedes, or by the consideration that we must have
+  identically x² + y² + z² = x1² + y1² + z1², it appears that these
+  satisfy the relations--
+
+    a² + ß² + [gamma]² = 1,
+    [alpha]² + [alpha]'² + [alpha]"² = 1,
+
+    [alpha]'² + ß'² + [gamma]'² = 1,
+    ß² + ß'² + ß"² = 1,
+
+    [alpha]"² + ß"² + [gamma]"² = 1,
+    [gamma]² + [gamma]'² + [gamma]"² = 1,
+
+    a'a" + ß'ß" + [gamma]'[gamma]" = 0,
+    ß[gamma] +ß'[gamma]' + ß"[gamma]" = 0,
+
+    [alpha]"[alpha] + ß"ß + [gamma]"[gamma] = 0,
+    [gamma][alpha] + [gamma]'[alpha]' + [gamma]"[alpha]" = 0,
+
+    [alpha][alpha]' + ß[beta]' + [gamma][gamma]' = 0,
+    [alpha]ß +[alpha]'ß' + [alpha]"ß" = 0,
+
+  either set of six equations being implied in the other set.
+
+  It follows that the square of the determinant
+
+    |[alpha],  ß,  [gamma] |
+    |                      |
+    |[alpha]', ß', [gamma]'|
+    |                      |
+    |[alpha]", ß", [gamma]"|
+
+  is = 1; and hence that the determinant itself is = ± 1. The
+  distinction of the two cases is an important one: if the determinant
+  is = + 1, then the axes Ox1, Oy1, Oz1 are such that they can by a
+  rotation about O be brought to coincide with Ox, Oy, Oz respectively;
+  if it is = -1, then they cannot. But in the latter case, by measuring
+  x1, y1, z1 in the opposite directions we change the signs of all the
+  coefficients and so make the determinant to be = + 1; hence the former
+  case need alone be considered, and it is accordingly assumed that the
+  determinant is = + 1. This being so, it is found that we have the
+  equality [alpha] = ß'[gamma]" - ß"[gamma]', and eight like ones,
+  obtained from this by cyclical interchanges of the letters [alpha], ß,
+  [gamma], and of unaccented, singly and doubly accented letters.
+
+  36. The nine cosine-inclinations above are, as has been seen,
+  connected by six equations. It ought then to be possible to express
+  them all in terms of three parameters. An elegant means of doing this
+  has been given by Rodrigues, who has shown that the tabular expression
+  of the formulae of transformation may be written
+
+        |            x             |            y             |            z             |
+    ----+--------------------------+--------------------------+--------------------------+
+    x1  |1 + [lambda]² - µ² - [nu]²|    2([lambda]µ - [nu])   |   2([nu][lambda] + µ)    |
+    ----+--------------------------+--+-----------------------+-----+--------------------+
+    y1  |    2([lambda]µ + [nu])   |1 - [lambda]² + µ² - [nu]²|   2(µ[nu] + [lambda])    |
+    ----+--------------------------+--+-----------------------+-----+--------------------+
+    z1  |    2([nu][lambda] - µ)   |    2(µ[nu] + [lambda])   |1 - [lambda]² - µ² + [nu]²|
+    ----+--------------------------+--------------------------+--------------------------+
+       ÷(1 + [lambda]² + µ² + [nu]²),
+
+  the meaning being that the coefficients in the transformation are
+  fractions, with numerators expressed as in the table, and the common
+  denominator.
+
+  37. _The Species of Quadric Surfaces_.--Surfaces represented by
+  equations of the second degree are called _quadric_ surfaces. Quadric
+  surfaces are either _proper_ or _special_. The special ones arise when
+  the coefficients in the general equation are limited to satisfy
+  certain special equations; they comprise (1) plane-pairs, including in
+  particular one plane twice repeated, and (2) cones, including in
+  particular cylinders; there is but one form of cone, but cylinders may
+  be elliptic, parabolic or hyperbolic.
+
+  A discussion of the general equation of the second degree shows that
+  the _proper_ quadric surfaces are of five kinds, represented
+  respectively, when referred to the most convenient axes of reference,
+  by equations of the five types (a and b positive):
+
+             x²   y²
+    (1)  z = -- + --, elliptic paraboloid.
+             2a   2b
+
+             x²   y²
+    (2)  z = -- - --, hyperbolic paraboloid.
+             2a   2b
+
+         x²   y²   z²
+    (3)  -- + -- + -- = 1, ellipsoid.
+         a²   b²   c²
+
+         x²   y²   z²
+    (4)  -- + -- - -- = 1, hyperboloid of one sheet.
+         a²   b²   c²
+
+         x²   y²   z²
+    (5)  -- + -- - -- = -1, hyperboloid of two sheets.
+         a²   b²   c²
+
+  It is at once seen that these are distinct surfaces; and the equations
+  also show very readily the general form and mode of generation of the
+  several surfaces.
+
+  [Illustration: FIG. 61.]
+
+  In the elliptic paraboloid (fig. 61) the sections by the planes of zx
+  and zy are the parabolas
+
+        x²      y²
+    z = --, z = --
+        2a      2b
+
+  having the common axes Oz; and the section by any plane z = [gamma]
+  parallel to that of xy is the ellipse
+
+              x²   y²
+    [gamma] = -- + --;
+              2a   2b
+
+  so that the surface is generated by a variable ellipse moving parallel
+  to itself along the parabolas as directrices.
+
+  [Illustration: FIG. 62.]
+
+  [Illustration: FIG. 63.]
+
+  In the hyperbolic paraboloid (figs. 62 and 63) the sections by the
+  planes of zx, zy are the parabolas z = x²/2a, z = - y²/2b, having the
+  opposite axes Oz, Oz', and the section by a plane z = [gamma] parallel
+  to that of xy is the hyperbola [gamma] = x²/2a - y²/2b, which has its
+  transverse axis parallel to Ox or Oy according as [gamma] is positive
+  or negative. The surface is thus generated by a variable hyperbola
+  moving parallel to itself along the parabolas as directrices. The form
+  is best seen from fig. 63, which represents the sections by planes
+  parallel to the plane of xy, or say the contour lines; the continuous
+  lines are the sections above the plane of xy, and the dotted lines the
+  sections below this plane. The form is, in fact, that of a saddle.
+
+  [Illustration: FIG. 64.]
+
+  In the ellipsoid (fig. 64) the sections by the planes of zx, zy, and
+  xy are each of them an ellipse, and the section by any parallel plane
+  is also an ellipse. The surface may be considered as generated by an
+  ellipse moving parallel to itself along two ellipses as directrices.
+
+  In the hyperboloid of one sheet (fig. 65), the sections by the planes
+  of zx, zy are the hyperbolas
+
+    x²   z²      y²   z²
+    -- - -- = 1, -- - -- = 1,
+    c²   c²      b²   c²
+
+  having a common conjugate axis zOz'; the section by the plane of x, y,
+  and that by any parallel plane, is an ellipse; and the surface may be
+  considered as generated by a variable ellipse moving parallel to
+  itself along the two hyperbolas as directrices. If we imagine two
+  equal and parallel circular disks, their points connected by strings
+  of equal lengths, so that these are the generators of a right circular
+  cylinder, and if we turn one of the disks about its centre through an
+  angle in its plane, the strings in their new positions will be one
+  system of generators of a hyperboloid of one sheet, for which a = b;
+  and if we turn it through the same angle in the opposite direction, we
+  get in like manner the generators of the other system; there will be
+  the same general configuration when a = | b. The hyperbolic paraboloid
+  is also covered by two systems of rectilinear generators as a method
+  like that used in § 34 establishes without difficulty. The figures
+  should be studied to see how they can lie.
+
+  [Illustration: FIG. 65.]
+
+  [Illustration: FIG. 66.]
+
+  In the hyperboloid of two sheets (fig. 66) the sections by the planes
+  of zx and zy are the hyperbolas
+
+    z²   x²      z²   y²
+    -- - -- = 1, -- - -- = 1,
+    c²   a²      c²   b²
+
+  having a common transverse axis along z'Oz; the section by any plane z
+  = ±[gamma] parallel to that of xy is the ellipse
+
+    x²   y²   [gamma]²
+    -- + -- = -------- - 1,
+    a²   b²      c²
+
+  provided [gamma]² > c², and the surface, consisting of two distinct
+  portions or sheets, may be considered as generated by a variable
+  ellipse moving parallel to itself along the hyperbolas as directrices.
+
+  38. _Differential Geometry of Curves._--For convenience consider the
+  coordinates (x, y, z) of a point on a curve in space to be given as
+  functions of a variable parameter [theta], which may in particular be
+  one of themselves. Use the notation x', x" for dx/d[theta],
+  d²x/d[theta]², and similarly as to y and z. Only a few formulae will
+  be given. Call the current coordinates ([xi], [eta], [zeta]).
+
+  The _tangent_ at (x, y, z) is the line tended to as a limit by the
+  connector of (x, y, z) and a neighbouring point of the curve when the
+  latter moves up to the former: its equations are
+
+    ([xi] - x)/x' = ([eta] - y)/y' = ([zeta] - z)/z'.
+
+  The _osculating plane_ at (x, y, z) is the plane tended to as a limit
+  by that through (x, y, z) and two neighbouring points of the curve as
+  these, remaining distinct, both move up to (x, y, z): its one equation
+  is
+
+    ([xi] - x)(y'z" - y"z') + ([eta] - y)(z'x" - z"x') + ([zeta] - z)
+      (x'y" - x"y') = 0.
+
+  The _normal plane_ is the plane through (x, y, z) at right angles to
+  the tangent line, i.e. the plane
+
+    x'([xi] - x) + y'([eta] - y) + z'([zeta] - z) = 0.
+
+  It cuts the osculating plane in a line called the _principal normal_.
+  Every line through (x, y, z) in the normal plane is a normal. The
+  normal perpendicular to the osculating plane is called the _binormal_.
+  A tangent, principal normal, and binormal are a convenient set of
+  rectangular axes to use as those of reference, when the nature of a
+  curve near a point on it is to be discussed.
+
+  Through (x, y, z) and three neighbouring points, all on the curve,
+  passes a single sphere; and as the three points all move up to (x, y,
+  z) continuing distinct, the sphere tends to a limiting size and
+  position. The limit tended to is the sphere of closest contact with
+  the curve at (x, y, z); its centre and radius are called the centre
+  and radius of _spherical curvature_. It cuts the osculating plane in a
+  circle, called the _circle of absolute curvature_; and the centre and
+  radius of this circle are the centre and radius of absolute curvature.
+  The centre of absolute curvature is the limiting position of the point
+  where the principal normal at (x, y, z) is cut by the normal plane at
+  a neighbouring point, as that point moves up to (x, y, z).
+
+  39. _Differential Geometry of Surfaces._--Let (x, y, z) be any chosen
+  point on a surface [f](x, y, z) = 0. As a second point of the surface
+  moves up to (x, y, z), its connector with (x, y, z) tends to a
+  limiting position, a tangent line to the surface at (x, y, z). All
+  these tangent lines at (x, y, z), obtained by approaching (x, y, z)
+  from different directions on a surface, lie in one plane
+
+    dP[f]              dP[f]               dP[f]
+    ----- ([xi] - x) + ----- ([eta] - y) + ----- ([zeta] - z) = 0.
+     dPx                dPy                 dPz
+
+  This plane is called the _tangent plane_ at (x, y, z). One line
+  through (x, y, z) is at right angles to the tangent plane. This is the
+  normal
+
+                /dP[f]                /dP[f]                   /dP[f]
+    ([xi] - x) / ----- = ([eta] - y) / ----- = ([zeta] - z) = / -----.
+              /   dPx               /   dPy                  /   dPz
+
+  The tangent plane is cut by the surface in a curve, real or imaginary,
+  with a node or double point at (x, y, z). Two of the tangent lines
+  touch this curve at the node. They are called the "chief tangents"
+  (_Haupt-tangenten_) at (x, y, z); they have closer contact with the
+  surface than any other tangents.
+
+  In the case of a quadric surface the curve of intersection of a
+  tangent and the surface is of the second order and has a node, it must
+  therefore consist of two straight lines. Consequently a quadric
+  surface is covered by two sets of straight lines, a pair through every
+  point on it; these are imaginary for the ellipsoid, hyperboloid of two
+  sheets, and elliptic paraboloid.
+
+  A surface of any order is covered by two singly infinite systems of
+  curves, a pair through every point, the tangents to which are all
+  chief tangents at their respective points of contact. These are called
+  _chief-tangent curves_; on a quadric surface they are the above
+  straight lines.
+
+  40. The tangents at a point of a surface which bisect the angles
+  between the chief tangents are called the _principal tangents_ at the
+  point. They are at right angles, and together with the normal
+  constitute a convenient set of rectangular axes to which to refer the
+  surface when its properties near the point are under discussion. At a
+  special point which is such that the chief tangents there run to the
+  circular points at infinity in the tangent plane, the principal
+  tangents are indeterminate; such a special point is called an umbilic
+  of the surface.
+
+  There are two singly infinite systems of curves on a surface, a pair
+  cutting one another at right angles through every point upon it, all
+  tangents to which are principal tangents of the surface at their
+  respective points of contact. These are called _lines of curvature_,
+  because of a property next to be mentioned.
+
+  As a point Q moves in an arbitrary direction on a surface from
+  coincidence with a chosen point P, the normal at it, as a rule, at
+  once fails to meet the normal at P; but, if it takes the direction of
+  a line of curvature through P, this is instantaneously not the case.
+  We have thus on the normal two centres of curvature, and the distances
+  of these from the point on the surface are the two _principal radii of
+  curvature_ of the surface at that point; these are also the radii of
+  curvature of the sections of the surface by planes through the normal
+  and the two principal tangents respectively; or say they are the radii
+  of curvature of the normal sections through the two principal tangents
+  respectively. Take at the point the axis of z in the direction of the
+  normal, and those of x and y in the directions of the principal
+  tangents respectively, then, if the radii of curvature be a, b (the
+  signs being such that the coordinates of the two centres of curvature
+  are z = a and z = b respectively), the surface has in the
+  neighbourhood of the point the form of the paraboloid
+
+        x²   y²
+    z = -- + --,
+        2a   2b
+
+  and the chief-tangents are determined by the equation 0 = x²/2a +
+  y²/2b. The two centres of curvature may be on the same side of the
+  point or on opposite sides; in the former case a and b have the same
+  sign, the paraboloid is elliptic, and the chief-tangents are
+  imaginary; in the latter case a and b have opposite signs, the
+  paraboloid is hyperbolic, and the chief-tangents are real.
+
+  The normal sections of the surface and the paraboloid by the same
+  plane have the same radius of curvature; and it thence readily follows
+  that the radius of curvature of a normal section of the surface by a
+  plane inclined at an angle [theta] to that of zx is given by the
+  equation
+
+      1     cos² [theta]   sin² [theta]
+    ----- = ------------ + ------------.
+    [rho]        a               b
+
+  The section in question is that by a plane through the normal and a
+  line in the tangent plane inclined at an angle [theta] to the
+  principal tangent along the axis of x. To complete the theory,
+  consider the section by a plane having the same trace upon the tangent
+  plane, but inclined to the normal at an angle [phi]; then it is shown
+  without difficulty (Meunier's theorem) that the radius of curvature of
+  this inclined section of the surface is = [rho] cos [phi].
+
+  AUTHORITIES.--The above article is largely based on that by Arthur
+  Cayley in the 9th edition of this work. Of early and important recent
+  publications on analytical geometry, special mention is to be made of
+  R. Descartes, _Géométrie_ (Leyden, 1637); John Wallis, _Tractatus de
+  sectionibus conicis nova methodo expositis_ (1655, _Opera
+  mathematica_, i., Oxford, 1695); de l'Hospital, _Traité analytique des
+  sections coniques_ (Paris, 1720); Leonhard Euler, _Introductio in
+  analysin infinitorum_, ii. (Lausanne, 1748); Gaspard Monge,
+  "Application d'algčbre ą la géométrie" (_Journ. École Polytech._,
+  1801); Julius Plücker, _Analytisch-geometrische Entwickelungen_, 3
+  Bde. (Essen, 1828-1831); _System der analytischen Geometrie_ (Berlin,
+  1835); G. Salmon, _A Treatise on Conic Sections_ (Dublin, 1848; 6th
+  ed., London, 1879); Ch. Briot and J. Bouquet, _Leēons de géométrie
+  analytique_ (Paris, 1851; 16th ed., 1897); M. Chasles, _Traité de
+  géométrie supérieure_ (Paris, 1852); Wilhelm Fiedler, _Analytische
+  Geometrie der Kegelschnitte_ nach G. Salmon frei bearbeitet (Leipzig,
+  5te Aufl., 1887-1888); N.M. Ferrers, _An Elementary Treatise on
+  Trilinear Coordinates_ (London, 1861); Otto Hesse, _Vorlesungen aus
+  der analytischen Geometrie_ (Leipzig, 1865, 1881); W.A. Whitworth,
+  _Trilinear Coordinates and other Methods of Modern Analytical
+  Geometry_ (Cambridge, 1866); J. Booth, _A Treatise on Some New
+  Geometrical Methods_ (London, i., 1873; ii., 1877); A. Clebsch-F.
+  Lindemann, _Vorlesungen über Geometrie_, Bd. i. (Leipzig, 1876, 2te
+  Aufl., 1891); R. Baltser, _Analytische Geometrie_ (Leipzig, 1882);
+  Charlotte A. Scott, _Modern Methods of Analytical Geometry_ (London,
+  1894); G. Salmon, _A Treatise on the Analytical Geometry of three
+  Dimensions_ (Dublin, 1862; 4th ed., 1882); Salmon-Fiedler,
+  _Analytische Geometrie des Raumes_ (Leipzig, 1863; 4te Aufl., 1898);
+  P. Frost, _Solid Geometry_ (London, 3rd ed., 1886; 1st ed., Frost and
+  J. Wolstenholme). See also E. Pascal, _Repertorio di matematiche
+  superiori, II. Geometria_ (Milan, 1900), and articles now appearing in
+  the _Encyklopädie der mathematischen Wissenschaften_, Bd. iii. 1, 2.
+       (E. B. El.)
+
+
+V. LINE GEOMETRY
+
+Line geometry is the name applied to those geometrical investigations in
+which the straight line replaces the point as element. Just as ordinary
+geometry deals primarily with points and systems of points, this theory
+deals in the first instance with straight lines and systems of straight
+lines. In two dimensions there is no necessity for a special line
+geometry, inasmuch as the straight line and the point are
+interchangeable by the principle of duality; but in three dimensions the
+straight line is its own reciprocal, and for the better discussion of
+systems of lines we require some new apparatus, e.g., a system of
+coordinates applicable to straight lines rather than to points. The
+essential features of the subject are most easily elucidated by
+analytical methods: we shall therefore begin with the notion of line
+coordinates, and in order to emphasize the merits of the system of
+coordinates ultimately adopted, we first notice a system without these
+advantages, but often useful in special investigations.
+
+  In ordinary Cartesian coordinates the two equations of a straight line
+  may be reduced to the form y = rx + s, z = tx + u, and r, s, t, u may
+  be regarded as the four coordinates of the line. These coordinates
+  lack symmetry: moreover, in changing from one base of reference to
+  another the transformation is not linear, so that the degree of an
+  equation is deprived of real significance. For purposes of the general
+  theory we employ homogeneous coordinates; if x1y1z1w1 and x2y2z2w2 are
+  two points on the line, it is easily verified that the six
+  determinants of the array
+
+    |x1y1z1w1|
+    |x2y2z2w2|
+
+  are in the same ratios for all point-pairs on the line, and further,
+  that when the point coordinates undergo a linear transformation so
+  also do these six determinants. We therefore adopt these six
+  determinants for the coordinates of the line, and express them by the
+  symbols l, [lambda], m, µ, n, [nu] where l = x1w2 - x2w1, [lambda] =
+  y1z2 - y2z1, &c. There is the further advantage that if a1b1c1d1 and
+  a2b2c2d2 be two planes through the line, the six determinants
+
+    |a1b1c1d1|
+    |a2b2c2d2|
+
+  are in the same ratios as the foregoing, so that except as regards a
+  factor of proportionality we have [lambda] = b1c2 - b2c1, l = c1d2 -
+  c2d1, &c. The identical relation l[lambda] + mµ + n[nu] = o reduces
+  the number of independent constants in the six coordinates to four,
+  for we are only concerned with their mutual ratios; and the quadratic
+  character of this relation marks an essential difference between point
+  geometry and line geometry. The condition of intersection of two lines
+  is
+
+    l[lambda]' + l'[lambda] + mµ' + m'µ + n[nu]' + n'[nu] = 0
+
+  where the accented letters refer to the second line. If the
+  coordinates are Cartesian and l, m, n are direction cosines, the
+  quantity on the left is the mutual moment of the two lines.
+
+  Since a line depends on four constants, there are three distinct types
+  of configurations arising in line geometry--those containing a
+  triply-infinite, a doubly-infinite and a singly-infinite number of
+  lines; they are called Complexes, Congruences, and Ruled Surfaces or
+  Skews respectively. A _Complex_ is thus a system of lines satisfying
+  one condition--that is, the coordinates are connected by a single
+  relation; and the degree of the complex is the degree of this equation
+  supposing it to be algebraic. The lines of a complex of the nth degree
+  which pass through any point lie on a cone of the nth degree, those
+  which lie in any plane envelop a curve of the nth class and there are
+  n lines of the complex in any plane pencil; the last statement
+  combines the former two, for it shows that the cone is of the nth
+  degree and the curve is of the nth class. To find the lines common to
+  four complexes of degrees n1, n2, n3, n4, we have to solve five
+  equations, viz. the four complex equations together with the quadratic
+  equation connecting the line coordinates, therefore the number of
+  common lines is 2n1n2n3n4. As an example of complexes we have the
+  lines meeting a twisted curve of the nth degree, which form a complex
+  of the nth degree.
+
+  A _Congruence_ is the set of lines satisfying two conditions: thus a
+  finite number m of the lines pass through any point, and a finite
+  number n lie in any plane; these numbers are called the degree and
+  class respectively, and the congruence is symbolically written (m, n).
+
+  The simplest example of a congruence is the system of lines
+  constituted by all those that pass through m points and those that lie
+  in n planes; through any other point there pass m of these lines, and
+  in any other plane there lie n, therefore the congruence is of degree
+  m and class n. It has been shown by G.H. Halphen that the number of
+  lines common to two congruences is mm' + nn', which may be verified by
+  taking one of them to be of this simple type. The lines meeting two
+  fixed lines form the general (1, 1) congruence; and the chords of a
+  twisted cubic form the general type of a (1, 3) congruence; Halphen's
+  result shows that two twisted cubics have in general ten common
+  chords. As regards the analytical treatment, the difficulty is of the
+  same nature as that arising in the theory of curves in space, for a
+  congruence is not in general the complete intersection of two
+  complexes.
+
+  A _Ruled Surface_, _Regulus_ or _Skew_ is a configuration of lines
+  which satisfy three conditions, and therefore depend on only one
+  parameter. Such lines all lie on a surface, for we cannot draw one
+  through an arbitrary point; only one line passes through a point of
+  the surface; the simplest example, that of a quadric surface, is
+  really two skews on the same surface.
+
+  The degree of a ruled surface _qua_ line geometry is the number of its
+  generating lines contained in a linear complex. Now the number which
+  meets a given line is the degree of the surface _qua_ point geometry,
+  and as the lines meeting a given line form a particular case of linear
+  complex, it follows that the degree is the same from whichever point
+  of view we regard it. The lines common to three complexes of degrees,
+  n1n2n3, form a ruled surface of degree 2n1n2n3; but not every ruled
+  surface is the complete intersection of three complexes.
+
+
+    Linear complex.
+
+  In the case of a complex of the first degree (or linear complex) the
+  lines through a fixed point lie in a plane called the polar plane or
+  nul-plane of that point, and those lying in a fixed plane pass through
+  a point called the nul-point or pole of the plane. If the nul-plane of
+  A pass through B, then the nul-plane of B will pass through A; the
+  nul-planes of all points on one line l1 pass through another line l2.
+  The relation between l1 and l2 is reciprocal; any line of the complex
+  that meets one will also meet the other, and every line meeting both
+  belongs to the complex. They are called conjugate or polar lines with
+  respect to the complex. On these principles can be founded a theory of
+  reciprocation with respect to a linear complex.
+
+  This may be aptly illustrated by an elegant example due to A. Voss.
+  Since a twisted cubic can be made to satisfy twelve conditions, it
+  might be supposed that a finite number could be drawn to touch four
+  given lines, but this is not the case. For, suppose one such can be
+  drawn, then its reciprocal with respect to any linear complex
+  containing the four lines is a curve of the third class, i.e. another
+  twisted cubic, touching the same four lines, which are unaltered in
+  the process of reciprocation; as there is an infinite number of
+  complexes containing the four lines, there is an infinite number of
+  cubics touching the four lines, and the problem is poristic.
+
+  The following are some geometrical constructions relating to the
+  unique linear complex that can be drawn to contain five arbitrary
+  lines:
+
+  To construct the nul-plane of any point O, we observe that the two
+  lines which meet any four of the given five are conjugate lines of the
+  complex, and the line drawn through O to meet them is therefore a ray
+  of the complex; similarly, by choosing another four we can find
+  another ray through O: these rays lie in the nul-plane, and there is
+  clearly a result involved that the five lines so obtained all lie in
+  one plane. A reciprocal construction will enable us to find the
+  nul-point of any plane. Proceeding now to the metrical properties and
+  the statical and dynamical applications, we remark that there is just
+  one line such that the nul-plane of any point on it is perpendicular
+  to it. This is called the central axis; if d be the shortest distance,
+  [theta] the angle between it and a ray of the complex, then d tan
+  [theta] = p, where p is a constant called the pitch or parameter. Any
+  system of forces can be reduced to a force R along a certain line, and
+  a couple G perpendicular to that line; the lines of nul-moment for the
+  system form a linear complex of which the given line is the central
+  axis and the quotient G/R is the pitch. Any motion of a rigid body can
+  be reduced to a screw motion about a certain line, i.e. to an angular
+  velocity [omega] about that line combined with a linear velocity u
+  along the line. The plane drawn through any point perpendicular to the
+  direction of its motion is its nul-plane with respect to a linear
+  complex having this line for central axis, and the quotient u/[omega]
+  for pitch (cf. Sir R.S. Ball, _Theory of Screws_).
+
+  The following are some properties of a configuration of two linear
+  complexes:
+
+  The lines common to the two-complexes also belong to an infinite
+  number of linear complexes, of which two reduce to single straight
+  lines. These two lines are conjugate lines with respect to each of the
+  complexes, but they may coincide, and then some simple modifications
+  are required. The locus of the central axis of this system of
+  complexes is a surface of the third degree called the cylindroid,
+  which plays a leading part in the theory of screws as developed
+  synthetically by Ball. Since a linear complex has an invariant of the
+  second degree in its coefficients, it follows that two linear
+  complexes have a lineo-linear invariant. This invariant is
+  fundamental: if the complexes be both straight lines, its vanishing is
+  the condition of their intersection as given above; if only one of
+  them be a straight line, its vanishing is the condition that this line
+  should belong to the other complex. When it vanishes for any two
+  complexes they are said to be in _involution_ or _apolar_; the
+  nul-points P, Q of any plane then divide harmonically the points in
+  which the plane meets the common conjugate lines, and each complex is
+  its own reciprocal with respect to the other. As regards a
+  configuration of these linear complexes, the common lines from one
+  system of generators of a quadric, and the doubly infinite system of
+  complexes containing the common lines, include an infinite number of
+  straight lines which form the other system of generators of the same
+  quadric.
+
+
+    General line coordinates.
+
+  If the equation of a linear complex is Al + Bm + Cn + D[lambda] + Eµ +
+  F[nu] = 0, then for a line not belonging to the complex we may regard
+  the expression on the left-hand side as a multiple of the moment of
+  the line with respect to the complex, the word moment being used in
+  the statical sense; and we infer that when the coordinates are
+  replaced by linear functions of themselves the new coordinates are
+  multiples of the moments of the line with respect to six fixed
+  complexes. The essential features of this coordinate system are the
+  same as those of the original one, viz. there are six coordinates
+  connected by a quadratic equation, but this relation has in general a
+  different form. By suitable choice of the six fundamental complexes,
+  as they may be called, this connecting relation may be brought into
+  other simple forms of which we mention two: (i.) When the six are
+  mutually in involution it can be reduced to x1² + x2² + x3² + x4² +
+  x5² + x6² = 0; (ii.) When the first four are in involution and the
+  other two are the lines common to the first four it is x1² + x2² + x3²
+  + x4² - 2x5x6 = 0. These generalized coordinates might be explained
+  without reference to actual magnitude, just as homogeneous point
+  coordinates can be; the essential remark is that the equation of any
+  coordinate to zero represents a linear complex, a point of view which
+  includes our original system, for the equation of a coordinate to zero
+  represents all the lines meeting an edge of the fundamental
+  tetrahedron.
+
+  The system of coordinates referred to six complexes mutually in
+  involution was introduced by Felix Klein, and in many cases is more
+  useful than that derived directly from point coordinates; e.g. in the
+  discussion of quadratic complexes: by means of it Klein has developed
+  an analogy between line geometry and the geometry of spheres as
+  treated by G. Darboux and others. In fact, in that geometry a point is
+  represented by _five_ coordinates, connected by a relation of the same
+  type as the one just mentioned when the five fundamental spheres are
+  mutually at right angles and the equation of a sphere is of the first
+  degree. Extending this to four dimensions of space, we obtain an exact
+  analogue of line geometry, in which (i.) a point corresponds to a
+  line; (ii.) a linear complex to a hypersphere; (iii.) two linear
+  complexes in involution to two orthogonal hyperspheres; (iv.) a linear
+  complex and two conjugate lines to a hypersphere and two inverse
+  points. Many results may be obtained by this principle, and more still
+  are suggested by trying to extend the properties of circles to spheres
+  in three and four dimensions. Thus the elementary theorem, that, given
+  four lines, the circles circumscribed to the four triangles formed by
+  them are concurrent, may be extended to six hyperplanes in four
+  dimensions; and then we can derive a result in line geometry by
+  translating the inverse of this theorem. Again, just as there is an
+  infinite number of spheres touching a surface at a given point, two of
+  them having contact of a closer nature, so there is an infinite number
+  of linear complexes touching a non-linear complex at a given line, and
+  _three_ of these have contact of a closer nature (cf. Klein, _Math.
+  Ann._ v.).
+
+  Sophus Lie has pointed out a different analogy with sphere geometry.
+  Suppose, in fact, that the equation of a sphere of radius r is
+
+    x² + y² + z² + 2ax + 2by + 2cz + d = 0,
+
+  so that r² = a² + b² + c² - d; then introducing the quantity e to make
+  this equation homogeneous, we may regard the sphere as given by the
+  six coordinates a, b, c, d, e, r connected by the equation a² + b² +
+  c² - r² - de = 0, and it is easy to see that two spheres touch, if the
+  polar form 2aa1 + 2bb1 + 2cc1 - 2rr1 - de1 - d1e vanishes. Comparing
+  this with the equation x1² + x2² + x3² + x4² - 2x5x6 = 0 given above,
+  it appears that this sphere geometry and line geometry are identical,
+  for we may write a = x1, b = x2, c = x3, r = x4(/[delta] - 1), d = x5,
+  e = ½x6; but it is to be noticed that a sphere is really replaced by
+  two lines whose coordinates only differ in the sign of x4, so that
+  they are polar lines with respect to the complex x4 = 0. Two spheres
+  which touch correspond to two lines which intersect, or more
+  accurately to two pairs of lines (p, p') and (q, q'), of which the
+  pairs (p, q) and (p', q') both intersect. By this means the problem of
+  describing a sphere to touch four given spheres is reduced to that of
+  drawing a pair of lines (t, t') (of which t intersects one line of the
+  four pairs (pp'), (qq'), (rr'), (ss'), and t' intersects the remaining
+  four). We may, however, ignore the accented letters in translating
+  theorems, for a configuration of lines and its polar with respect to a
+  linear complex have the same projective properties. In Lie's
+  transformation a linear complex corresponds to the totality of spheres
+  cutting a given sphere at a given angle. A most remarkable result is
+  that lines of curvature in the sphere geometry become asymptotic lines
+  in the line geometry.
+
+  Some of the principles of line geometry may be brought into clearer
+  light by admitting the ideas of space of four and five dimensions.
+
+  Thus, regarding the coordinates of a line as homogeneous coordinates
+  in five dimensions, we may say that line geometry is equivalent to
+  geometry on a quadric surface in five dimensions. A linear complex is
+  represented by a hyperplane section; and if two such complexes are in
+  involution, the corresponding hyperplanes are conjugate with respect
+  to the fundamental quadric. By projecting this quadric
+  stereographically into space of four dimensions we obtain Klein's
+  analogy. In the same way geometry in a linear complex is equivalent to
+  geometry on a quadric in four dimensions; when two lines intersect the
+  representative points are on the same generator of this quadric.
+  Stereographic projection, therefore, converts a curve in a linear
+  complex, i.e. one whose tangents all belong to the complex, into one
+  whose tangents intersect a fixed conic: when this conic is the
+  imaginary circle at infinity the curve is what Lie calls a minimal
+  curve. Curves in a linear complex have been extensively studied. The
+  osculating plane at any point of such a curve is the nul-plane of the
+  point with respect to the complex, and points of superosculation
+  always coincide in pairs at the points of contact of stationary
+  tangents. When a point of such a curve is given, the osculating plane
+  is determined, hence all the curves through a given point with the
+  same tangent have the same torsion.
+
+
+    Non-linear complexes.
+
+  The lines through a given point that belong to a complex of the nth
+  degree lie on a cone of the nth degree: if this cone has a double line
+  the point is said to be a singular point. Similarly, a plane is said
+  to be singular when the envelope of the lines in it has a double
+  tangent. It is very remarkable that the same surface is the locus of
+  the singular points and the envelope of the singular planes: this
+  surface is called the singular surface, and both its degree and class
+  are in general 2n(n - 1)², which is equal to four for the quadratic
+  complex.
+
+  The singular lines of a complex F = 0 are the lines common to F and
+  the complex
+
+    [delta]F    [delta]F       [delta]F  [delta]F   [delta]F   [delta]F
+    -------- --------------- + --------  -------- + --------  ----------- = 0.
+    [delta]l [delta][lambda]   [delta]m  [delta]µ   [delta]n  [delta][nu]
+
+  As already mentioned, at each line l of a complex there is an infinite
+  number of tangent linear complexes, and they all contain the lines
+  adjacent to l. If now l be a singular line, these complexes all reduce
+  to straight lines which form a plane pencil containing the line l.
+  Suppose the vertex of the pencil is A, its plane a, and one of its
+  lines [xi], then l' being a complex line near l, meets [xi], or more
+  accurately the mutual moment of l', and is of the second order of
+  small quantities. If P be a point on l, a line through P quite near l
+  in the plane a will meet [xi] and is therefore a line of the complex;
+  hence the complex-cones of all points on l touch a and the
+  complex-curves of all planes through l touch l at A. It follows that l
+  is a double line of the complex-cone of A, and a double tangent of the
+  complex-curve of a. Conversely, a double line of a cone or curve is a
+  singular line, and a singular line clearly touches the curves of all
+  planes through it in the same point. Suppose now that the consecutive
+  line l' is also a singular line, A' being the allied singular point,
+  a' the singular plane and [xi]' any line of the pencil (A', a') so
+  that [xi]' is a tangent line at l' to the complex: the mutual moments
+  of the pairs l', [xi] and l, [xi] are each of the second order; hence
+  the plane a' meets the lines l and [xi]' in two points very near A.
+  This being true for all singular planes, near a the point of contact
+  of a with its envelope is in A, i.e. the locus of singular points is
+  the same as the envelope of singular planes. Further, when a line
+  touches a complex it touches the singular surface, for it belongs to a
+  plane pencil like (Aa), and thus in Klein's analogy the analogue of a
+  focus of a hyper-surface being a bitangent line of the complex is also
+  a bitangent line of the singular surface. The theory of cosingular
+  complexes is thus brought into line with that of confocal surfaces in
+  four dimensions, and guided by these principles the existence of
+  cosingular quadratic complexes can easily be established, the analysis
+  required being almost the same as that invented for confocal cyclides
+  by Darboux and others. Of cosingular complexes of higher degree
+  nothing is known.
+
+  Following J. Plücker, we give an account of the lines of a quadratic
+  complex that meet a given line.
+
+  The cones whose vertices are on the given line all pass through eight
+  fixed points and envelop a surface of the fourth degree; the conics
+  whose planes contain the given line all lie on a surface of the fourth
+  class and touch eight fixed planes. It is easy to see by elementary
+  geometry that these two surfaces are identical. Further, the given
+  line contains four singular points A1, A2, A3, A4, and the planes into
+  which their cones degenerate are the eight common tangent planes
+  mentioned above; similarly, there are four singular planes, a1, a2,
+  a3, a4, through the line, and the eight points into which their conics
+  degenerate are the eight common points above. The locus of the pole of
+  the line with respect to all the conics in planes through it is a
+  straight line called the _polar line_ of the given one; and through
+  this line passes the polar plane of the given line with respect to
+  each of the cones. The name polar is applied in the ordinary
+  analytical sense; any line has an infinite number of polar complexes
+  with respect to the given complex, for the equation of the latter can
+  be written in an infinite number of ways; one of these polars is a
+  straight line, and is the polar line already introduced. The surface
+  on which lie all the conics through a line l is called the Plücker
+  surface of that line: from the known properties of (2, 2)
+  correspondences it can be shown that the Plücker surface of l cuts l1
+  in a range of the same cross ratio as that of the range in which the
+  Plücker surface of l1 cuts l. Applying this to the case in which l1 is
+  the polar of l, we find that the cross ratios of (A1, A2, A3, A4) and
+  (a1, a2, a3, a4) are equal. The identity of the locus of the A's with
+  the envelope of the a's follows at once; moreover, a line meets the
+  singular surface in four points having the same cross ratio as that of
+  the four tangent planes drawn through the line to touch the surface.
+  The Plücker surface has eight nodes, eight singular tangent planes,
+  and is a double line. The relation between a line and its polar line
+  is not a reciprocal one with respect to the complex; but W. Stahl has
+  pointed out that the relation is reciprocal as far as the singular
+  surface is concerned.
+
+
+    Quadratic complexes.
+
+  To facilitate the discussion of the general quadratic complex we
+  introduce Klein's canonical form. We have, in fact, to deal with two
+  quadratic equations in six variables; and by suitable linear
+  transformations these can be reduced to the form
+
+    a1x1² + a2x2² + a3x3² + a4x4² + a5x5² +  a6x6² = 0
+     x1²  +  x2²  +  x3²  +  x4²  +  x5²  +   x6²  = 0
+
+  subject to certain exceptions, which will be mentioned later.
+
+  Taking the first equation to be that of the complex, we remark that
+  both equations are unaltered by changing the sign of any coordinate;
+  the geometrical meaning of this is, that the quadratic complex is its
+  own reciprocal with respect to each of the six fundamental complexes,
+  for changing the sign of a coordinate is equivalent to taking the
+  polar of a line with respect to the corresponding fundamental complex.
+  It is easy to establish the existence of six systems of bitangent
+  linear complexes, for the complex l1x1 + l2x2 + l3x3 + l4x4 + l5x5 +
+  l6x6 = 0 is a bitangent when
+
+                  l2²       l3²       l4²       l5²       l6²
+    l1 = 0, and ------- + ------- + ------- + ------- + ------- = 0
+                a2 - a1   a3 - a1   a4 - a1   a5 - a1   a6 - a1
+
+  and its lines of contact are conjugate lines with respect to the first
+  fundamental complex. We therefore infer the existence of six systems
+  of bitangent lines of the complex, of which the first is given by
+
+              x2²       x3²       x4²       x5²       x6²
+    x1 = 0, ------- + ------- + ------- + ------- + ------- = 0.
+            a2 - a1   a3 - a1   a4 - a1   a5 - a1   a6 - a1
+
+  Each of these lines is a bitangent of the singular surface, which is
+  therefore completely determined as being the focal surface of the (2,
+  2) congruence above. It is thence easy to verify that the two
+  complexes [Sigma]ax² = 0 and [Sigma]bx² = 0 are cosingular if b_r =
+  a_r[lambda] + µ/a_r[nu] + [rho].
+
+  The singular surface of the general quadratic complex is the famous
+  quartic, with sixteen nodes and sixteen singular tangent planes, first
+  discovered by E.E. Kümmer.
+
+  We cannot give a full account of its properties here, but we deduce at
+  once from the above that its bitangents break up into six (2, 2)
+  congruences, and the six linear complexes containing these are
+  mutually in involution. The nodes of the singular surface are points
+  whose complex cones are coincident planes, and the complex conic in a
+  singular tangent plane consists of two coincident points. This
+  configuration of sixteen points and planes has many interesting
+  properties; thus each plane contains six points which lie on a conic,
+  while through each point there pass six planes which touch a quadric
+  cone. In many respects the Kümmer quartic plays a part in three
+  dimensions analogous to the general quartic curve in two; it further
+  gives a natural representation of certain relations between
+  hyperelliptic functions (cf. R.W.H.T. Hudson, _Kümmer's Quartic_,
+  1905).
+
+
+    Classification of quadratic complexes.
+
+  As might be expected from the magnitude of a form in six variables,
+  the number of projectivally distinct varieties of quadratic complexes
+  is very great; and in fact Adolf Weiler, by whom the question was
+  first systematically studied on lines indicated by Klein, enumerated
+  no fewer than forty-nine different types. But the principle of the
+  classification is so important, and withal so simple, that we give a
+  brief sketch which indicates its essential features.
+
+  We have practically to study the intersection of two quadrics F and F'
+  in six variables, and to classify the different cases arising we make
+  use of the results of Karl Weierstrass on the equivalence conditions
+  of two pairs of quadratics. As far as at present required, they are as
+  follows: Suppose that the factorized form of the determinantal
+  equation Disct (F + [lambda]F') = 0 is
+
+    ([lambda] - [alpha])^(s1 + s2 + s3 ...)
+      ([lambda] - ß)^(t1 + t2 + t3 + ...) ...
+
+  where the root [alpha] occurs s1 + s2 + s3 ... times in the
+  determinant, s2 + s3 ... times in every first minor, s3 + ... times in
+  every second minor, and so on; the meaning of each exponent is then
+  perfectly definite. Every factor of the type ([lambda] - [alpha])^s is
+  called an _elementartheil_ (elementary divisor) of the determinant,
+  and the condition of equivalence of two pairs of quadratics is simply
+  that their determinants have the same elementary divisors. We write
+  the pair of forms symbolically thus [(s1s2 ...), (t1t2 ...), ...],
+  letters in the inner brackets referring to the same factor. Returning
+  now to the two quadratics representing the complex, the sum of the
+  exponents will be six, and two complexes are put in the same class if
+  they have the same symbolical expression; i.e. the actual values of
+  the roots of the determinantal equation need not be the same for both,
+  but their manner of occurrence, as far as here indicated, must be
+  identical in the two. The enumeration of all possible cases is thus
+  reduced to a simple question in combinatorial analysis, and the actual
+  study of any particular case is much facilitated by a useful rule of
+  Klein's for writing down in a simple form two quadratics belonging to
+  a given class--one of which, of course, represents the equation
+  connecting line coordinates, and the other the equation of the
+  complex. The general complex is naturally [111111]; the complex of
+  tangents to a quadric is [(111), (111)] and that of lines meeting a
+  conic is [(222)]. Full information will be found in Weiler's memoir,
+  _Math. Ann._ vol. vii.
+
+  The detailed study of each variety of complex opens up a vast subject;
+  we only mention two special cases, the harmonic complex and the
+  tetrahedral complex.
+
+  The harmonic complex, first studied by Battaglini, is generated in an
+  infinite number of ways by the lines cutting two quadrics
+  harmonically. Taking the most general case, and referring the quadrics
+  to their common self-conjugate tetrahedron, we can find its equation
+  in a simple form, and verify that this complex really depends only on
+  seventeen constants, so that it is not the most general quadratic
+  complex. It belongs to the general type in so far as it is discussed
+  above, but the roots of the determinant are in involution. The
+  singular surface is the "tetrahedroid" discussed by Cayley. As a
+  particular case, from a metrical point of view, we have L.F. Painvin's
+  complex generated by the lines of intersection of perpendicular
+  tangent planes of a quadric, the singular surface now being Fresnel's
+  wave surface. The tetrahedral or Reye complex is the simplest and best
+  known of proper quadratic complexes. It is generated by the lines
+  which cut the faces of a tetrahedron in a constant cross ratio, and
+  therefore by those subtending the same cross ratio at the four
+  vertices. The singular surface is made up of the faces or the vertices
+  of the fundamental tetrahedron, and each edge of this tetrahedron is a
+  double line of the complex. The complex was first discussed by K.T.
+  Reye as the assemblage of lines joining corresponding points in a
+  homographic transformation of space, and this point of view leads to
+  many important and elegant properties. A (metrically) particular case
+  of great interest is the complex generated by the normals to a family
+  of confocal quadrics, and for many investigations it is convenient to
+  deal with this complex referred to the principal axes. For example,
+  Lie has developed the theory of curves in a Reye complex (i.e. curves
+  whose tangents belong to the complex) as solutions of a differential
+  equation of the form (b - c)xdydz + (c - a)ydzdx + (a - b)zdxdy = 0,
+  and we can simplify this equation by a logarithmic transformation.
+  Many theorems connecting complexes with differential equations have
+  been given by Lie and his school. A line complex, in fact, corresponds
+  to a Mongian equation having [oo]^3 line integrals.
+
+
+    Congruences.
+
+  As the coordinates of a line belonging to a congruence are functions
+  of two independent parameters, the theory of congruences is analogous
+  to that of surfaces, and we may regard it as a fundamental inquiry to
+  find the simplest form of surface into which a given congruence can be
+  transformed. Most of those whose properties have been extensively
+  discussed can be represented on a plane by a birational
+  transformation. But in addition to the difficulties of the theory of
+  algebraic surfaces, a subject still in its infancy, the theory of
+  congruences has other difficulties in that a congruence is seldom
+  completely represented, even by two equations.
+
+  A fundamental theorem is that the lines of a congruence are in general
+  bitangents of a surface; in fact, since the condition of intersection
+  of two consecutive straight lines is ld[lambda] + dmdµ + dnd[nu] = 0,
+  a line l of the congruence meets two adjacent lines, say l1 and l2.
+  Suppose l, l1 lie in the plane pencil (A1a1) and l, l2 in the plane
+  pencil (A2a2), then the locus of the A's is the same as the envelope
+  of the a's, but a2 is the tangent plane at A1 and a1 at A2. This
+  surface is called the focal surface of the congruence, and to it all
+  the lines l are bitangent. The distinctive property of the points A is
+  that two of the congruence lines through them coincide, and in like
+  manner the planes a each contain two coincident lines. The focal
+  surface consists of two sheets, but one or both may degenerate into
+  curves; thus, for example, the normals to a surface are bitangents of
+  the surface of centres, and in the case of Dupin's cyclide this
+  surface degenerates into two conics.
+
+  In the discussion of congruences it soon becomes necessary to
+  introduce another number r, called the rank, which expresses the
+  number of plane pencils each of which contains an arbitrary line and
+  two lines of the congruence. The order of the focal surface is 2m(n -
+  1) - 2r, and its class is m(m - 1) - 2r. Our knowledge of congruences
+  is almost exclusively confined to those in which either m or n does
+  not exceed two. We give a brief account of those of the second order
+  without singular lines, those of order unity not being especially
+  interesting. A congruence generally has singular points through which
+  an infinite number of lines pass; a singular point is said to be of
+  order r when the lines through it lie on a cone of the rth degree. By
+  means of formulae connecting the number of singular points and their
+  orders with the class m of quadratic congruence Kümmer proved that the
+  class cannot exceed seven. The focal surface is of degree four and
+  class 2m; this kind of quartic surface has been extensively studied by
+  Kümmer, Cayley, Rohn and others. The varieties (2, 2), (2, 3), (2, 4),
+  (2, 5) all belong to at least one Reye complex; and so also does the
+  most important class of (2, 6) congruences which includes all the
+  above as special cases. The congruence (2, 2) belongs to a linear
+  complex and forty different Reye complexes; as above remarked, the
+  singular surface is Kümmer's sixteen-nodal quartic, and the same
+  surface is focal for six different congruences of this variety. The
+  theory of (2, 2) congruences is completely analogous to that of the
+  surfaces called cyclides in three dimensions. Further particulars
+  regarding quadratic congruences will be found in Kümmer's memoir of
+  1866, and the second volume of Sturm's treatise. The properties of
+  quadratic congruences having singular lines, i.e. degenerate focal
+  surfaces, are not so interesting as those of the above class; they
+  have been discussed by Kümmer, Sturm and others.
+
+
+    Ruled surfaces.
+
+  Since a ruled surface contains only [infinity]¹ elements, this theory
+  is practically the same as that of curves. If a linear complex
+  contains more than n generators of a ruled surface of the nth degree,
+  it contains all the generators, hence for n = 2 there are three
+  linearly independent complexes, containing all the generators, and
+  this is a well-known property of quadric surfaces. In ruled cubics the
+  generators all meet two lines which may or may not coincide; these two
+  cases correspond to the two main classes of cubics discussed by Cayley
+  and Cremona. As regards ruled quartics, the generators must lie in one
+  and may lie in two linear complexes. The first class is equivalent to
+  a quartic in four dimensions and is always rational, but the latter
+  class has to be subdivided into the elliptic and the rational, just
+  like twisted quartic curves. A quintic skew may not lie in a linear
+  complex, and then it is unicursal, while of sextics we have two
+  classes not in a linear complex, viz. the elliptic variety, having
+  thirty-six places where a linear complex contains six consecutive
+  generators, and the rational, having six such places.
+
+  The general theory of skews in two linear complexes is identical with
+  that of curves on a quadric in three dimensions and is known. But for
+  skews lying in only one linear complex there are difficulties; the
+  curve now lies in four dimensions, and we represent it in three by
+  stereographic projection as a curve meeting a given plane in n points
+  on a conic. To find the maximum deficiency for a given degree would
+  probably be difficult, but as far as degree eight the space-curve
+  theory of Halphen and Nöther can be translated into line geometry at
+  once. When the skew does not lie in a linear complex at all the theory
+  is more difficult still, and the general theory clearly cannot advance
+  until further progress is made in the study of twisted curves.
+
+  REFERENCES.--The earliest works of a general nature are Plücker, _Neue
+  Geometrie des Raumes_ (Leipzig, 1868); and Kümmer, "Über die
+  algebraischen Strahlensysteme," _Berlin Academy_ (1866). Systematic
+  development on purely synthetic lines will be found in the three
+  volumes of Sturm, _Liniengeometrie_ (Leipzig, 1892, 1893, 1896); vol.
+  i. deals with the linear and Reye complexes, vols. ii. and iii. with
+  quadratic congruences and complexes respectively. For a highly
+  suggestive review by Gino Loria see _Bulletin des sciences
+  mathématiques_ (1893, 1897). A shorter treatise, giving a very
+  interesting account of Klein's coordinates, is the work of Koenigs,
+  _La Géométrie réglée et ses applications_ (Paris, 1898). English
+  treatises are C.M. Jessop, _Treatise on the Line Complex_ (1903);
+  R.W.H.T. Hudson, _Kümmer's Quartic_ (1905). Many references to memoirs
+  on line geometry will be found in Hagen, _Synopsis der höheren
+  Mathematik_, ii. (Berlin, 1894); Loria, _Il passato ed il presente
+  delle principali teorie geometriche_ (Milan, 1897); a clear résumé of
+  the principal results is contained in the very elegant volume of
+  Pascal, _Repertorio di mathematiche superiori_, ii. (Milan, 1900).
+  Another treatise dealing extensively with line geometry is Lie,
+  _Geometrie der Berührungstransformationen_ (Leipzig, 1896). Many
+  memoirs on the subject have appeared in the _Mathematische Annalen_; a
+  full list of these will be found in the index to the first fifty
+  volumes, p. 115. Perhaps the two memoirs which have left most
+  impression on the subsequent development of the subject are Klein,
+  "Zur Theorie der Liniencomplexe des ersten und zweiten Grades," _Math.
+  Ann._ ii.; and Lie, "Über Complexe, insbesondere Linien- und
+  Kugelcomplexe," _Math. Ann._ v.     (J. H. Gr.)
+
+
+VI. NON-EUCLIDEAN GEOMETRY
+
+The various metrical geometries are concerned with the properties of the
+various types of congruence-groups, which are defined in the study of
+the _axioms_ of _geometry_ and of their immediate consequences. But this
+point of view of the subject is the outcome of recent research, and
+historically the subject has a different origin. Non-Euclidean geometry
+arose from the discussion, extending from the Greek period to the
+present day, of the various assumptions which are implicit in the
+traditional Euclidean system of geometry. In the course of these
+investigations it became evident that metrical geometries, each
+internally consistent but inconsistent in many respects with each other
+and with the Euclidean system, could be developed. A short historical
+sketch will explain this origin of the subject, and describe the famous
+and interesting progress of thought on the subject. But previously a
+description of the chief characteristic properties of elliptic and of
+hyperbolic geometries will be given, assuming the standpoint arrived at
+below under VII. _Axioms of Geometry_.
+
+First assume the equation to the absolute (cf. _loc. cit._) to be w² -
+x² - y² - z² = 0. The absolute is then real, and the geometry is
+hyberbolic.
+
+  The distance (d12) between the two points (x1, y1, z1, w1) and (x2,
+  y2, z2, w2) is given by
+
+    cosh (d12/[gamma]) = (w1w2 - x1x2 - y1y2 - z1z2)/[(w1² - x1² - y1² - Z1²)
+      (w2² - x2² - y2² - z2²)]½     (1)
+
+  The only points to which the metrical geometry applies are those
+  within the region enclosed by the quadric; the other points are
+  "improper ideal points." The angle ([theta]12) between two planes, l1x
+  + m1y + n1z + r1w = 0 and l2x + m2y + n2z + r2w = 0, is given by
+
+  cos [theta]12 = (l1l2 + m1m2 + n1n2 - r1r2)/{(l1² + m1² + n1² - r1²)
+    (l2² + m2² + n2² - r2²)}^½       (2)
+
+  These planes only have a real angle of inclination if they possess a
+  line of intersection within the actual space, i.e. if they intersect.
+  Planes which do not intersect possess a shortest distance along a line
+  which is perpendicular to both of them. If this shortest distance is
+  [delta]12, we have
+
+    cosh ([delta]12/[gamma]) = (l1l2 + m1m2 + n1n2 - r1r2)/(l1² + m1² + n1² - r1²)
+      (l2² + m2² + n2² - r2²)½     (3)
+
+  [Illustration: FIG. 67.]
+
+  Thus in the case of the two planes one and only one of the two,
+  [theta]12 and [delta]12, is real. The same considerations hold for
+  coplanar straight lines (see VII. _Axioms of Geometry_). Let O (fig.
+  67) be the point (0, 0, 0, 1), OX the line y = 0, z = 0, OY the line z
+  = 0, x = 0, and OZ the line x = 0, y = 0. These are the coordinate
+  axes and are at right angles to each other. Let P be any point, and
+  let [rho] be the distance OP, [theta] the angle POZ, and [phi] the
+  angle between the planes ZOX and ZOP. Then the coordinates of P can be
+  taken to be
+
+    sinh ([rho]/[gamma]) sin [theta] cos [phi], sinh ([rho]/[gamma]) sin [theta]
+      sin [phi], sinh ([rho]/[gamma]) cos[theta], cosh ([rho]/[gamma]).
+
+  [Illustration: FIG. 68.]
+
+  If ABC is a triangle, and the sides and angles are named according to
+  the usual convention, we have
+
+    sinh (a/[gamma])/sin A = sinh (b/[gamma])/sin B = sinh (c/[gamma])/sin C, (4)
+
+  and also
+
+    cosh (a/[gamma]) = cosh (b/[gamma]) cosh (c/[gamma]) -
+      sinh (b/[gamma]) sinh (c/[gamma]) cos A,   (5)
+
+  with two similar equations. The sum of the three angles of a triangle
+  is always less than two right angles. The area of the triangle ABC is
+  [lambda]²([pi] - A - B - C). If the base BC of a triangle is kept
+  fixed and the vertex A moves in the fixed plane ABC so that the area
+  ABC is constant, then the locus of A is a line of equal distance from
+  BC. This locus is not a straight line. The whole theory of similarity
+  is inapplicable; two triangles are either congruent, or their angles
+  are not equal two by two. Thus the elements of a triangle are
+  determined when its three angles are given. By keeping A and B and the
+  line BC fixed, but by making C move off to infinity along BC, the
+  lines BC and AC become parallel, and the sides a and b become
+  infinite. Hence from equation (5) above, it follows that two parallel
+  lines (cf. Section VII. _Axioms of Geometry_) must be considered as
+  making a zero angle with each other. Also if B be a right angle, from
+  the equation (5), remembering that, in the limit,
+
+    cosh (a/[gamma])/cosh (b/[gamma]) = cosh (a/[gamma])/sinh (b/[gamma]) = 1,
+
+  we have cos A = tanh (c/2[gamma]) .... (6).
+
+  The angle A is called by N.I. Lobatchewsky the "angle of parallelism."
+
+  The whole theory of lines and planes at right angles to each other is
+  simply the theory of conjugate elements with respect to the absolute,
+  where ideal lines and planes are introduced.
+
+  Thus if l and l' be any two conjugate lines with respect to the
+  absolute (of which one of the two must be improper, say l'), then any
+  plane through l' and containing proper points is perpendicular to l.
+  Also if p is any plane containing proper points, and P is its pole,
+  which is necessarily improper, then the lines through P are the
+  normals to P. The equation of the sphere, centre (x1, y1, z1, w1) and
+  radius [rho], is
+
+    (w1²- x1²- y1²- z1²)(w² - x² - y² - z²) cosh²([rho]/[gamma]) = (w1w -
+      x1x - y1y -z1z)²   (7).
+
+  The equation of the surface of equal distance ([sigma]) from the plane
+  lx + my + nz + rw = 0 is
+
+    (l² + m² + n² - r²)(w² - x² - y² - z²) sinh²([sigma]/[gamma]) = (rw +
+      lx + my + nz)²    (8).
+
+  A surface of equal distance is a sphere whose centre is improper; and
+  both types of surface are included in the family
+
+    k²(w² - x² - y² - z²) = (ax + by + cz + dw)²   (9).
+
+  But this family also includes a third type of surfaces, which can be
+  looked on either as the limits of spheres whose centres have
+  approached the absolute, or as the limits of surfaces of equal
+  distance whose central planes have approached a position tangential to
+  the absolute. These surfaces are called limit-surfaces. Thus (9)
+  denotes a limit-surface, if d² - a² - b² - c² = 0. Two limit-surfaces
+  only differ in position. Thus the two limit-surfaces which touch the
+  plane YOZ at O, but have their concavities turned in opposite
+  directions, have as their equations
+
+    w² - x² - y² - z² = (w ± x)².
+
+  The geodesic geometry of a sphere is elliptic, that of a surface of
+  equal distance is hyperbolic, and that of a limit-surface is parabolic
+  (i.e. _Euclidean_). The equation of the surface (cylinder) of equal
+  distance ([delta]) from the line OX is
+
+    (w² - x²) tanh²([delta]/[gamma]) - y² - z² = 0.
+
+  This is not a ruled surface. Hence in this geometry it is not possible
+  for two straight lines to be at a constant distance from each other.
+
+  Secondly, let the equation of the absolute be x² + y² + z² + w² = 0.
+  The absolute is now imaginary and the geometry is elliptic.
+
+  The distance (d12) between the two points (x1, y1, z1, w1) and (x2,
+  y2, z2, w2) is given by
+
+    cos (d12/[gamma]) = ±(x1x2 + y1y2 + z1z2 + w1w2)
+      / {(x1² + y1² + z1² + w1²) {(x2² + y2² + z2² + w2²)}^½   (10).
+
+  Thus there are two distances between the points, and if one is d12,
+  the other is [pi][gamma]-d12. Every straight line returns into itself,
+  forming a closed series. Thus there are two segments between any two
+  points, together forming the whole line which contains them; one
+  distance is associated with one segment, and the other distance with
+  the other segment. The complete length of every straight line is
+  [pi][gamma].
+
+  The angle between the two planes l1x + m1y + n1z + r + 1w = 0 and l2x
+  + m2y + n2z + r2w = 0 is
+
+    cos [theta]12 = (l1l2 + m1m2 + n1n2 + r1r2)/ {(l1² + m1² + n1² +r1²)
+      (l2² + m2² + n2² + r2²)}^½ (11).
+
+  The polar plane with respect to the absolute of the point (x1, y1, z1,
+  w1) is the real plane x1x + y1y + z1z + w1w = 0, and the pole of the
+  plane l1x + m1y + n1z + r1w = 0 is the point (l1, m1, n1, r1). Thus
+  (from equations 10 and 11) it follows that the angle between the polar
+  planes of the points (x1, ...) and (x2, ...) is d12/[gamma], and that
+  the distance between the poles of the planes (l1, ...) and (l2, ...)
+  is [gamma][theta]12. Thus there is complete reciprocity between points
+  and planes in respect to all properties. This complete reign of the
+  principle of duality is one of the great beauties of this geometry.
+  The theory of lines and planes at right angles is simply the theory of
+  conjugate elements with respect to the absolute. A tetrahedron
+  self-conjugate with respect to the absolute has all its intersecting
+  elements (edges and planes) at right angles. If l and l' are two
+  conjugate lines, the planes through one are the planes perpendicular
+  to the other. If P is the pole of the plane p, the lines through P are
+  the normals to the plane p. The distance from P to p is ½[pi][gamma].
+  Thus every sphere is also a surface of equal distance from the polar
+  of its centre, and conversely. A plane does not divide space; for the
+  line joining any two points P and Q only cuts the plane once, in L
+  say, then it is always possible to go from P to Q by the segment of
+  the line PQ which does not contain L. But P and Q may be said to be
+  separated by a plane p, if the point in which PQ cuts p lies on the
+  shortest segment between P and Q. With this sense of "separation," it
+  is possible[2] to find three points P, Q, R such that P and Q are
+  separated by the plane p, but P and R are not separated by p, nor are
+  Q and R.
+
+  Let A, B, C be any three non-collinear points, then four triangles are
+  defined by these points. Thus if a, b, c and A, B, C are the elements
+  of any one triangle, then the four triangles have as their elements:
+
+    (1)      a,                  b,                  c,             A,           B,           C.
+
+    (2)      a,            [pi][gamma] - b,   [pi][gamma] - c,      A,       [pi] - B,    [pi] - C.
+
+    (3) [pi][gamma] - a,          b,          [pi][gamma] - c,   [pi] - A,       B,       [pi] - C.
+
+    (4) [pi][gamma] - a,   [pi][gamma] - b,          c,          [pi] - A,   [pi] - B,        C.
+
+  The formulae connecting the elements are
+
+    sin A/sin (a/[gamma]) = sin B/sin (b/[gamma]) = sin C/sin (c/[gamma]),
+         (12)
+
+  and
+
+    cos (a/[gamma]) = cos (b/[gamma]) cos (c/[gamma]) + sin (b/[gamma])
+      sin (c/[gamma]) cos A,   (13)
+
+  with two similar equations.
+
+  Two cases arise, namely (I.) according as one of the four triangles
+  has as its sides the shortest segments between the angular points, or
+  (II.) according as this is not the case. When case I. holds there is
+  said to be a "principal triangle."[3] If all the figures considered
+  lie within a sphere of radius ¼[pi][gamma] only case I. can hold, and
+  the principal triangle is the triangle wholly within this sphere, also
+  the peculiarities in respect to the separation of points by a plane
+  cannot then arise. The sum of the three angles of a triangle ABC is
+  always greater than two right angles, and the area of the triangle is
+  [gamma]²(A + B + C--[pi]). Thus as in hyperbolic geometry the theory
+  of similarity does not hold, and the elements of a triangle are
+  determined when its three angles are given. The coordinates of a point
+  can be written in the form
+
+    sin ([rho]/[gamma]) sin [Phi] cos [phi], sin ([rho]/[gamma]) sin [Phi]
+      sin [phi], sin ([rho]/[gamma]) cos [Phi], cos ([rho]/[gamma]),
+
+  where [rho], [Phi] and [phi] have the same meanings as in the
+  corresponding formulae in hyperbolic geometry. Again, suppose a watch
+  is laid on the plane OXY, face upwards with its centre at O, and the
+  line 12 to 6 (as marked on dial) along the line YOY. Let the watch be
+  continually pushed along the plane along the line OX, that is, in the
+  direction 9 to 3. Then the line XOX being of finite length, the watch
+  will return to O, but at its first return it will be found to be face
+  downwards on the other side of the plane, with the line 12 to 6
+  reversed in direction along the line YOY. This peculiarity was first
+  pointed out by Felix Klein. The theory of parallels as it exists in
+  hyperbolic space has no application in elliptic geometry. But another
+  property of Euclidean parallel lines holds in elliptic geometry, and
+  by the use of it parallel lines are defined. For the equation of the
+  surface (cylinder) of equal distance ([delta]) from the line XOX is
+
+    (x² + w²) tan²([delta]/[gamma]) - (y² + z²) = 0.
+
+  This is also the surface of equal distance, ½[pi][gamma]-[delta], from
+  the line conjugate to XOX. Now from the form of the above equation
+  this is a ruled surface, and through every point of it two generators
+  pass. But these generators are lines of equal distance from XOX. Thus
+  throughout every point of space two lines can be drawn which are lines
+  of equal distance from a given line l. This property was discovered by
+  W.K. Clifford. The two lines are called Clifford's right and left
+  parallels to l through the point. This property of parallelism is
+  reciprocal, so that if m is a left parallel to l, then l is a left
+  parallel to m. Note also that two parallel lines l and m are not
+  coplanar. Many of those properties of Euclidean parallels, which do
+  not hold for Lobatchewsky's parallels in hyperbolic geometry, do hold
+  for Clifford's parallels in elliptic geometry. The geodesic geometry
+  of spheres is elliptic, the geodesic geometry of surfaces of equal
+  distance from lines (cylinders) is Euclidean, and surfaces of
+  revolution can be found[4] of which the geodesic geometry is
+  hyperbolic. But it is to be noticed that the connectivity of these
+  surfaces is different to that of a Euclidean plane. For instance there
+  are only [&infin]² congruence transformations of the cylindrical
+  surfaces of equal distance into themselves, instead of the [&infin]³
+  for the ordinary plane. It would obviously be possible to state
+  "axioms" which these geodesics satisfy, and thus to define
+  independently, and not as loci, quasi-spaces of these peculiar types.
+  The existence of such Euclidean quasi-geometries was first pointed out
+  by Clifford.[5]
+
+In both elliptic and hyperbolic geometry the spherical geometry, i.e.
+the relations between the angles formed by lines and planes passing
+through the same point, is the same as the "spherical trigonometry" in
+Euclidean geometry. The constant [gamma], which appears in the formulae
+both of hyperbolic and elliptic geometry, does not by its variation
+produce different types of geometry. There is only one type of elliptic
+geometry and one type of hyperbolic geometry; and the magnitude of the
+constant [gamma] in each case simply depends upon the magnitude of the
+arbitrary unit of length in comparison with the natural unit of length
+which each particular instance of either geometry presents. The
+existence of a natural unit of length is a peculiarity common both to
+hyperbolic and elliptic geometries, and differentiates them from
+Euclidean geometry. It is the reason for the failure of the theory of
+similarity in them. If [gamma] is very large, that is, if the natural
+unit is very large compared to the arbitrary unit, and if the lengths
+involved in the figures considered are not large compared to the
+arbitrary unit, then both the elliptic and hyperbolic geometries
+approximate to the Euclidean. For from formulae (4) and (5) and also
+from (12) and (13) we find, after retaining only the lowest powers of
+small quantities, as the formulae for any triangle ABC,
+
+  a/ sin A = b/ sin B = c/ sin C,
+
+and
+
+  a² = b² + c² - 2bc cos A,
+
+with two similar equations. Thus the geometries of small figures are in
+both types Euclidean.
+
+
+  Theory of parallels before Gauss.
+
+_History._--"In pulcherrimo Geometriae corpore," wrote Sir Henry Savile
+in 1621, "duo sunt naevi, duae labes nec quod sciam plures, in quibus
+eluendis et emaculendis cum veterum tum recentiorum ... vigilavit
+industria." These two blemishes are the theory of parallels and the
+theory of proportion. The "industry of the moderns," in both respects,
+has given rise to important branches of mathematics, while at the same
+time showing that Euclid is in these respects more free from blemish
+than had been previously credible. It was from endeavours to improve the
+theory of parallels that non-Euclidean geometry arose; and though it has
+now acquired a far wider scope, its historical origin remains
+instructive and interesting. Euclid's "axiom of parallels" appears as
+Postulate V. to the first book of his _Elements_, and is stated thus,
+"And that, if a straight line falling on two straight lines make the
+angles, internal and on the same side, less than two right angles, the
+two straight lines, being produced indefinitely, meet on the side on
+which are the angles less than two right angles." The original Greek is
+[Greek: kai ean eis duo eutheias eutheia empiptousa tas entos kai epi ta
+auta merź gōnias duo orthōn elassonas poiź, ekballomenas tas duo
+eutheias ep' apeiron sympiptein, eph' ha merź eisin hai tōn duo orthōn
+elassones].
+
+To Euclid's successors this axiom had signally failed to appear
+self-evident, and had failed equally to appear indemonstrable. Without
+the use of the postulate its converse is proved in Euclid's 28th
+proposition, and it was hoped that by further efforts the postulate
+itself could be also proved. The first step consisted in the discovery
+of equivalent axioms. Christoph Clavius in 1574 deduced the axiom from
+the assumption that a line whose points are all equidistant from a
+straight line is itself straight. John Wallis in 1663 showed that the
+postulate follows from the possibility of similar triangles on different
+scales. Girolamo Saccheri (1733) showed that it is sufficient to have a
+single triangle, the sum of whose angles is two right angles. Other
+equivalent forms may be obtained, but none shows any essential
+superiority to Euclid's. Indeed plausibility, which is chiefly aimed at,
+becomes a positive demerit where it conceals a real assumption.
+
+
+  Saccheri.
+
+A new method, which, though it failed to lead to the desired goal,
+proved in the end immensely fruitful, was invented by Saccheri, in a
+work entitled _Euclides ab omni naevo vindicatus_ (Milan, 1733). If the
+postulate of parallels is involved in Euclid's other assumptions,
+contradictions must emerge when it is denied while the others are
+maintained. This led Saccheri to attempt a _reductio ad absurdum_, in
+which he mistakenly believed himself to have succeeded. What is
+interesting, however, is not his fallacious conclusion, but the
+non-Euclidean results which he obtains in the process. Saccheri
+distinguishes three hypotheses (corresponding to what are now known as
+Euclidean or parabolic, elliptic and hyperbolic geometry), and proves
+that some one of the three must be universally true. His three
+hypotheses are thus obtained: equal perpendiculars AC, BD are drawn from
+a straight line AB, and CD are joined. It is shown that the angles ACD,
+BDC are equal. The first hypothesis is that these are both right angles;
+the second, that they are both obtuse; and the third, that they are both
+acute. Many of the results afterwards obtained by Lobatchewsky and
+Bolyai are here developed. Saccheri fails to be the founder of
+non-Euclidean geometry only because he does not perceive the possible
+truth of his non-Euclidean hypotheses.
+
+
+  Lambert.
+
+Some advance is made by Johann Heinrich Lambert in his _Theorie der
+Parallellinien_ (written 1766; posthumously published 1786). Though he
+still believed in the necessary truth of Euclidean geometry, he
+confessed that, in all his attempted proofs, something remained
+undemonstrated. He deals with the same three hypotheses as Saccheri,
+showing that the second holds on a sphere, while the third would hold on
+a sphere of purely imaginary radius. The second hypothesis he succeeds
+in condemning, since, like all who preceded Bernhard Riemann, he is
+unable to conceive of the straight line as finite and closed. But the
+third hypothesis, which is the same as Lobatchewsky's, is not even
+professedly refuted.[6]
+
+
+  Three periods of non-Euclidean geometry.
+
+Non-Euclidean geometry proper begins with Karl Friedrich Gauss. The
+advance which he made was rather philosophical than mathematical: it was
+he (probably) who first recognized that the postulate of parallels is
+possibly false, and should be empirically tested by measuring the angles
+of large triangles. The history of non-Euclidean geometry has been aptly
+divided by Felix Klein into three very distinct periods. The
+first--which contains only Gauss, Lobatchewsky and Bolyai--is
+characterized by its synthetic method and by its close relation to
+Euclid. The attempt at indirect proof of the disputed postulate would
+seem to have been the source of these three men's discoveries; but when
+the postulate had been denied, they found that the results, so far from
+showing contradictions, were just as self-consistent as Euclid. They
+inferred that the postulate, if true at all, can only be proved by
+observations and measurements. Only one kind of non-Euclidean space is
+known to them, namely, that which is now called hyperbolic. The second
+period is analytical, and is characterized by a close relation to the
+theory of surfaces. It begins with Riemann's inaugural dissertation,
+which regards space as a particular case of a _manifold_; but the
+characteristic standpoint of the period is chiefly emphasized by Eugenio
+Beltrami. The conception of measure of curvature is extended by Riemann
+from surfaces to spaces, and a new kind of space, finite but unbounded
+(corresponding to the second hypothesis of Saccheri and Lambert), is
+shown to be possible. As opposed to the second period, which is purely
+metrical, the third period is essentially projective in its method. It
+begins with Arthur Cayley, who showed that metrical properties are
+projective properties relative to a certain fundamental quadric, and
+that different geometries arise according as this quadric is real,
+imaginary or degenerate. Klein, to whom the development of Cayley's work
+is due, showed further that there are two forms of Riemann's space,
+called by him the elliptic and the spherical. Finally, it has been shown
+by Sophus Lie, that if figures are to be freely movable throughout all
+space in [oo]^6 ways, no other three-dimensional spaces than the above
+four are possible.
+
+
+  Gauss.
+
+Gauss published nothing on the theory of parallels, and it was not
+generally known until after his death that he had interested himself in
+that theory from a very early date. In 1799 he announces that Euclidean
+geometry would follow from the assumption that a triangle can be drawn
+greater than any given triangle. Though unwilling to assume this, we
+find him in 1804 still hoping to prove the postulate of parallels. In
+1830 he announces his conviction that geometry is not an a priori
+science; in the following year he explains that non-Euclidean geometry
+is free from contradictions, and that, in this system, the angles of a
+triangle diminish without limit when all the sides are increased. He
+also gives for the circumference of a circle of radius r the formula
+[pi]k(e^(r/k) - e^(r-/k)), where k is a constant depending upon the
+nature of the space. In 1832, in reply to the receipt of Bolyai's
+_Appendix_, he gives an elegant proof that the amount by which the sum
+of the angles of a triangle falls short of two right angles is
+proportional to the area of the triangle. From these and a few other
+remarks it appears that Gauss possessed the foundations of hyperbolic
+geometry, which he was probably the first to regard as perhaps true. It
+is not known with certainty whether he influenced Lobatchewsky and
+Bolyai, but the evidence we possess is against such a view.[7]
+
+
+  Lobatchewsky.
+
+The first to publish a non-Euclidean geometry was Nicholas Lobatchewsky,
+professor of mathematics in the new university of Kazań.[8] In the place
+of the disputed postulate he puts the following: "All straight lines
+which, in a plane, radiate from a given point, can, with respect to any
+other straight line in the same plane, be divided into two classes, the
+_intersecting_ and the _non-intersecting_. The _boundary line_ of the
+one and the other class is called _parallel to the given line_." It
+follows that there are two parallels to the given line through any
+point, each meeting the line at infinity, like a Euclidean parallel.
+(Hence a line has two distinct points at infinity, and not one only as
+in ordinary geometry.) The two parallels to a line through a point make
+equal acute angles with the perpendicular to the line through the point.
+If p be the length of the perpendicular, either of these angles is
+denoted by [Pi](p). The determination of [Pi](p) is the chief problem
+(cf. equation (6) above); it appears finally that, with a suitable
+choice of the unit of length,
+
+  tan ½ [Pi](p) = e^(-p).
+
+Before obtaining this result it is shown that spherical trigonometry is
+unchanged, and that the normals to a circle or a sphere still pass
+through its centre. When the radius of the circle or sphere becomes
+infinite all these normals become parallel, but the circle or sphere
+does not become a straight line or plane. It becomes what Lobatchewsky
+calls a limit-line or limit-surface. The geometry on such a surface is
+shown to be Euclidean, limit-lines replacing Euclidean straight lines.
+(It is, in fact, a surface of zero measure of curvature.) By the help of
+these propositions Lobatchewsky obtains the above value of [Pi](p), and
+thence the solution of triangles. He points out that his formulae result
+from those of spherical trigonometry by substituting ia, ib, ic, for the
+sides a, b, c.
+
+
+  Bolyai.
+
+John Bolyai, a Hungarian, obtained results closely corresponding to
+those of Lobatchewsky. These he published in an appendix to a work by
+his father, entitled _Appendix Scientiam spatii absolute veram exhibens:
+a veritate aut falsitate Axiomatis XI. Euclidei (a priori haud unquam
+decidenda) independentem: adjecta ad casum falsitatis, quadratura
+circuli geometrica_.[9] This work was published in 1831, but its
+conception dates from 1823. It reveals a profounder appreciation of the
+importance of the new ideas, but otherwise differs little from
+Lobatchewsky's. Both men point out that Euclidean geometry as a limiting
+case of their own more general system, that the geometry of very small
+spaces is always approximately Euclidean, that no a priori grounds exist
+for a decision, and that observation can only give an approximate
+answer. Bolyai gives also, as his title indicates, a geometrical
+construction, in hyperbolic space, for the quadrature of the circle, and
+shows that the area of the greatest possible triangle, which has all its
+sides parallel and all its angles zero, is [pi][iota]², where i is what
+we should now call the space-constant.
+
+
+  Riemann.
+
+The works of Lobatchewsky and Bolyai, though known and valued by Gauss,
+remained obscure and ineffective until, in 1866, they were translated
+into French by J. Hoüel. But at this time Riemann's dissertation, _Über
+die Hypothesen, welche der Geometrie zu Grunde liegen_,[10] was already
+about to be published. In this work Riemann, without any knowledge of
+his predecessors in the same field, inaugurated a far more profound
+discussion, based on a far more general standpoint; and by its
+publication in 1867 the attention of mathematicians and philosophers was
+at last secured. (The dissertation dates from 1854, but owing to changes
+which Riemann wished to make in it, it remained unpublished until after
+his death.)
+
+
+  Definition of a manifold.
+
+Riemann's work contains two fundamental conceptions, that of a manifold
+and that of the _measure of curvature_ of a continuous manifold
+possessed of what he calls flatness in the smallest parts. By means of
+these conceptions space is made to appear at the end of a gradual series
+of more and more specialized conceptions. Conceptions of magnitude, he
+explains, are only possible where we have a general conception capable
+of determination in various ways. The manifold consists of all these
+various determinations, each of which is an element of the manifold. The
+passage from one element to another may be discrete or continuous; the
+manifold is called discrete or continuous accordingly. Where it is
+discrete two portions of it can be compared, as to magnitude, by
+counting; where continuous, by measurement. But measurement demands
+superposition, and consequently some magnitude independent of its place
+in the manifold. In passing, in a continuous manifold, from one element
+to another in a determinate way, we pass through a series of
+intermediate terms, which form a one-dimensional manifold. If this whole
+manifold be similarly caused to pass over into another, each of its
+elements passes through a one-dimensional manifold, and thus on the
+whole a two-dimensional manifold is generated. In this way we can
+proceed to n dimensions. Conversely, a manifold of n dimensions can be
+analysed into one of one dimension and one of (n - 1) dimensions. By
+repetitions of this process the position of an element may be at last
+determined by n magnitudes. We may here stop to observe that the above
+conception of a manifold is akin to that due to Hermann Grassmann in the
+first edition (1847) of his _Ausdehnungslehre_.[11]
+
+
+  Measure of curvature.
+
+Both concepts have been elaborated and superseded by the modern
+procedure in respect to the axioms of geometry, and by the conception of
+abstract geometry involved therein. Riemann proceeds to specialize the
+manifold by considerations as to measurement. If measurement is to be
+possible, some magnitude, we saw, must be independent of position; let
+us consider manifolds in which lengths of lines are such magnitudes, so
+that every line is measurable by every other. The coordinates of a point
+being x1, x2, ... x_n, let us confine ourselves to lines along which the
+ratios dx1 : dx2 : ... : dx_n alter continuously. Let us also assume that
+the element of length, ds, is unchanged (to the first order) when all
+its points undergo the same infinitesimal motion. Then if all the
+increments dx be altered in the same ratio, ds is also altered in this
+ratio. Hence ds is a homogeneous function of the first degree of the
+increments dx. Moreover, ds must be unchanged when all the dx change
+sign. The simplest possible case is, therefore, that in which ds is the
+square root of a quadratic function of the dx. This case includes space,
+and is alone considered in what follows. It is called the case of
+flatness in the smallest parts. Its further discussion depends upon the
+measure of curvature, the second of Riemann's fundamental conceptions.
+This conception, derived from the theory of surfaces, is applied as
+follows. Any one of the shortest lines which issue from a given point
+(say the origin) is completely determined by the initial ratios of the
+dx. Two such lines, defined by dx and [delta]x say, determine a pencil,
+or one-dimensional series, of shortest lines, any one of which is
+defined by [lambda]dx + µ[delta]x, where the parameter [lambda] : µ may
+have any value. This pencil generates a two-dimensional series of
+points, which may be regarded as a surface, and for which we may apply
+Gauss's formula for the measure of curvature at any point. Thus at every
+point of our manifold there is a measure of curvature corresponding to
+every such pencil; but all these can be found when n.[/(n-1)]/2 of them
+are known. If figures are to be freely movable, it is necessary and
+sufficient that the measure of curvature should be the same for all
+points and all directions at each point. Where this is the case, if
+[alpha] be the measure of curvature, the linear element can be put into
+the form
+
+  ds = [root]([Sigma]dx²)/(1 + ¼[alpha][Sigma]x²).
+
+If [alpha] be positive, space is finite, though still unbounded, and
+every straight line is closed--a possibility first recognized by
+Riemann. It is pointed out that, since the possible values of a form a
+continuous series, observations cannot prove that our space is strictly
+Euclidean. It is also regarded as possible that, in the infinitesimal,
+the measure of curvature of our space should be variable.
+
+There are four points in which this profound and epoch-making work is
+open to criticism or development--(1) the idea of a manifold requires
+more precise determination; (2) the introduction of coordinates is
+entirely unexplained and the requisite presuppositions are unanalysed;
+(3) the assumption that ds is the square root of a quadratic function of
+dx1, dx2, ... is arbitrary; (4) the idea of superposition, or
+congruence, is not adequately analysed. The modern solution of these
+difficulties is properly considered in connexion with the general
+subject of the axioms of geometry.
+
+
+  Helmholtz.
+
+The publication of Riemann's dissertation was closely followed by two
+works of Hermann von Helmholtz,[12] again undertaken in ignorance of the
+work of predecessors. In these a proof is attempted that ds must be a
+rational integral quadratic function of the increments of the
+coordinates. This proof has since been shown by Lie to stand in need of
+correction (see VII. _Axioms of Geometry_). Helmholtz's remaining works
+on the subject[13] are of almost exclusively philosophical interest. We
+shall return to them later.
+
+
+  Beltrami.
+
+The only other writer of importance in the second period is Eugenio
+Beltrami, by whom Riemann's work was brought into connexion with that of
+Lobatchewsky and Bolyai. As he gave, by an elegant method, a convenient
+Euclidean interpretation of hyperbolic plane geometry, his results will
+be stated at some length[14]. The _Saggio_ shows that Lobatchewsky's
+plane geometry holds in Euclidean geometry on surfaces of constant
+negative curvature, straight lines being replaced by geodesics. Such
+surfaces are capable of a conformal representation on a plane, by which
+geodesics are represented by straight lines. Hence if we take, as
+coordinates on the surface, the Cartesian coordinates of corresponding
+points on the plane, the geodesics must have linear equations.
+
+  Hence it follows that
+
+    ds² = R²w^(-4){([alpha]² - v²)du² + 2uvdudv + ([alpha]² - u²)dv²}
+
+  where w² = [alpha]² - u² - v², and (-1)/R² is the measure of curvature
+  of our surface (note that k = [gamma] as used above). The angle
+  between two geodesics u = const., v = const. is [theta], where
+
+    cos [theta] = uv/[root]{([alpha]² - u²)([alpha]² - v²)}, sin [theta] =
+      aw/[root]{(a² - u²)(a² - v²)}.
+
+  Thus u = 0 is orthogonal to all geodesies v = const., and vice versa.
+  In order that sin [theta] may be real, w² must be positive; thus
+  geodesics have no real intersection when the corresponding straight
+  lines intersect outside the circle u² + v² = [alpha]². When they
+  intersect on this circle, [theta] = 0. Thus Lobatchewsky's parallels
+  are represented by straight lines intersecting on the circle. Again,
+  transforming to polar coordinates u = r cos µ, v = r sin µ, and
+  calling [rho] the geodesic distance of u, v from the origin, we have,
+  for a geodesic through the origin,
+
+    d[rho] = Radr/(a² - r²), [rho] = ½R log(a + r)/(a - r), r = a tan h
+      ([rho]/R).
+
+  Thus points on the surface corresponding to points in the plane on the
+  limiting circle r = a, are all at an infinite distance from the
+  origin. Again, considering r constant, the arc of a geodesic circle
+  subtending an angle µ at the origin is
+
+    [sigma] = Rrµ/[root](a² - r²) = µR sin h ([rho]/R),
+
+  whence the circumference of a circle of radius [rho] is 2[pi]R sin h
+  ([rho]/R). Again, if [alpha] be the angle between any two geodesics
+
+    V - v = m(U - u), V - v = n(U - u),
+
+  then tan [alpha] = a(n - m)w/{(1 + mn)a² - (v - mu) (v - nu)}.
+
+  Thus [alpha] is imaginary when u, v is outside the limiting circle,
+  and is zero when, and only when, u, v is on the limiting circle. All
+  these results agree with those of Lobatchewsky and Bolyai. The maximum
+  triangle, whose angles are all zero, is represented in the auxiliary
+  plane by a triangle inscribed in the limiting circle. The angle of
+  parallelism is also easily obtained. The perpendicular to v = 0 at a
+  distance [delta] from the origin is u = a tan h ([delta]/R), and the
+  parallel to this through the origin is u = v sin h ([delta]/R). Hence
+  [Pi] ([delta]), the angle which this parallel makes with v = 0, is
+  given by
+
+    tan [Pi]([delta]) . sin h ([delta]/R) = 1, or tan ½[Pi]([delta]) =
+      e^(-[delta]/R)
+
+  which is Lobatchewsky's formula. We also obtain easily for the area of
+  a triangle the formula R²([pi] - A - B - C).
+
+  Beltrami's treatment connects two curves which, in the earlier
+  treatment, had no connexion. These are limit-lines and curves of
+  constant distance from a straight line. Both may be regarded as
+  circles, the first having an infinite, the second an imaginary radius.
+  The equation to a circle of radius [rho] and centre u0v0 is
+
+    (a² - uu0 - vv0)² = cos h² ([rho]/R)w0²w² = C²w²     (say).
+
+  This equation remains real when [rho] is a pure imaginary, and remains
+  finite when w0 = 0, provided [rho] becomes infinite in such a way that
+  w0 cos h ([rho]/R) remains finite. In the latter case the equation
+  represents a limit-line. In the former case, by giving different
+  values to C, we obtain concentric circles with the imaginary centre
+  u0v0. One of these, obtained by putting C = 0, is the straight line a²
+  - uu0 - vv0 = 0. Hence the others are each throughout at a constant
+  distance from this line. (It may be shown that all motions in a
+  hyperbolic plane consist, in a general sense, of rotations; but three
+  types must be distinguished according as the centre is real, imaginary
+  or at infinity. All points describe, accordingly, one of the three
+  types of circles.)
+
+  The above Euclidean interpretation fails for three or more dimensions.
+  In the _Teoria fondamentale_, accordingly, where n dimensions are
+  considered, Beltrami treats hyperbolic space in a purely analytical
+  spirit. The paper shows that Lobatchewsky's space of any number of
+  dimensions has, in Riemann's sense, a constant negative measure of
+  curvature. Beltrami starts with the formula (analogous to that of the
+  _Saggio_)
+
+    ds² = R²x^(-2)(dx² + dx1² + dx2² + ... + dx_n²)
+
+  where x² + x1² + x2² + ... + x_n² = a².
+
+  He shows that geodesics are represented by linear equations between
+  x1, x2, ..., x_n, and that the geodesic distance [rho] between two
+  points x and x' is given by
+
+          [rho]                a² - x1x'1 - x2x'2 - ... - x_n x'_n
+     cosh ----- = -----------------------------------------------------------------
+            R     {(a² - x1² - x2² - ... - x_n²)(a² - x'1² - x'2² - ... - x'_n²)}^½
+
+  (a formula practically identical with Cayley's, though obtained by a
+  very different method). In order to show that the measure of curvature
+  is constant, we make the substitutions
+
+    x1 = r[lambda]1, x2 = r[lambda]2 ... x_n = r[lambda]_n, where
+      [Sigma][lambda]² = 1.
+
+  Hence
+                 _______
+    ds² = (Radr/(a² - r²)])² + R²r²d[Delta]²/(a² - r²).
+
+  where
+
+    d[Delta]² = [Sigma]d[lambda]².
+
+  Also calling [rho] the geodesic distance from the origin, we have
+
+         [rho]         a               [rho]          r
+    cosh ----- = ---------------, sinh ----- = ---------------.
+           R     [root](a² - r²)         R     [root](a² - r²)
+
+  Hence
+
+    ds² = d[rho]² + (R sin h ([rho]/R))²d[Delta]².
+
+  Putting
+
+    z1 = [rho][lambda]1, z2 = [rho][lambda]2, ... z_n = [rho][lambda]_n,
+
+  we obtain
+                               _                       _
+                          1   |  /  R        [rho]\²    |
+    ds² = [Sigma]dz² + ------ | ( ----- sinh ----- ) - 1| [Sigma](z_i dz_k - z_k dz_i)².
+                       [rho]² |_ \[rho]        R  /    _|
+
+  Hence when [rho] is small, we have approximately
+
+                        1
+    ds² = [Sigma]dz² + ---[Sigma](z_i dz_k - z_k dz_i)²    (1).
+                       3R²
+
+  Considering a surface element through the origin, we may choose our
+  axes so that, for this element,
+
+    z3 = Z4 = ... = z_n = 0.
+
+  Thus
+
+                         1
+    ds² = dz1² + dz2² + ---(z1dz2 - z2dz1)²    (2).
+                        3R²
+
+  Now the area of the triangle whose vertices are (0, 0), (z1, z2),
+  (dz1, dz2) is ½(z1, dz2 - z2dz1). Hence the quotient when the terms of
+  the fourth order in (2) are divided by the square of this triangle is
+  4/3R²; hence, returning to general axes, the same is the quotient when
+  the terms of the fourth order in (1) are divided by the square of the
+  triangle whose vertices are (0, 0, ... 0), (z1, z2, z3, ... z_n),
+  (dz1, dz2, dz3 ... dz_n). But -¾ of this quotient is defined by
+  Riemann as the measure of curvature.[15] Hence the measure of
+  curvature is -1/R², i.e. is constant and negative. The properties of
+  parallels, triangles, &c., are as in the _Saggio_. It is also shown
+  that the analogues of limit surfaces have zero curvature; and that
+  spheres of radius [rho] have constant positive curvature 1/R² sinh²
+  ([rho]/R), so that spherical geometry may be regarded as contained in
+  the pseudo-spherical (as Beltrami calls Lobatchewsky's system).
+
+
+  Transition to the projective method.
+
+The _Saggio_, as we saw, gives a Euclidean interpretation confined to
+two dimensions. But a consideration of the auxiliary plane suggests a
+different interpretation, which may be extended to any number of
+dimensions. If, instead of referring to the pseudosphere, we merely
+_define_ distance and angle, in the Euclidean plane, as those functions
+of the coordinates which gave us distance and angle on the pseudosphere,
+we find that the geometry of our plane has become Lobatchewsky's. All
+the points of the limiting circle are now at infinity, and points beyond
+it are imaginary. If we give our circle an imaginary radius the geometry
+on the plane becomes elliptic. Replacing the circle by a sphere, we
+obtain an analogous representation for three dimensions. Instead of a
+circle or sphere we may take any conic or quadric. With this definition,
+if the fundamental quadric be [Sigma]_(xx) = 0, and if [Sigma]_(xx)' be
+the polar form of [Sigma]_(xx), the distance [rho] between x and x' is
+given by the projective formula
+
+  cos([rho]/k) = [Sigma]_xx'/{[Sigma]_(xx)·[Sigma]_x'x'}^½.
+
+That this formula is projective is rendered evident by observing that
+e^(-2i[rho]/k) is the anharmonic ratio of the range consisting of the
+two points and the intersections of the line joining them with the
+fundamental quadric. With this we are brought to the third or projective
+period. The method of this period is due to Cayley; its application to
+previous non-Euclidean geometry is due to Klein. The projective method
+contains a generalization of discoveries already made by Laguerre[16] in
+1853 as regards Euclidean geometry. The arbitrariness of this procedure
+of deriving metrical geometry from the properties of conics is removed
+by Lie's theory of congruence. We then arrive at the stage of thought
+which finds its expression in the modern treatment of the axioms of
+geometry.
+
+
+  The two kinds of elliptic space.
+
+The projective method leads to a discrimination, first made by
+Klein,[17] of two varieties of Riemann's space; Klein calls these
+elliptic and spherical. They are also called the polar and antipodal
+forms of elliptic space. The latter names will here be used. The
+difference is strictly analogous to that between the diameters and the
+points of a sphere. In the polar form two straight lines in a plane
+always intersect in one and only one point; in the antipodal form they
+intersect always in two points, which are antipodes. According to the
+definition of geometry adopted in section VII. (_Axioms of Geometry_),
+the antipodal form is not to be termed "geometry," since any pair of
+coplanar straight lines intersect each other in two points. It may be
+called a "quasi-geometry." Similarly in the antipodal form two diameters
+always determine a plane, but two points on a sphere do not determine a
+great circle when they are antipodes, and two great circles always
+intersect in two points. Again, a plane does not form a boundary among
+lines through a point: we can pass from any one such line to any other
+without passing through the plane. But a great circle does divide the
+surface of a sphere. So, in the polar form, a complete straight line
+does not divide a plane, and a plane does not divide space, and does
+not, like a Euclidean plane, have two sides.[18] But, in the antipodal
+form, a plane is, in these respects, like a Euclidean plane.
+
+It is explained in section VII. in what sense the metrical geometry of
+the material world can be considered to be determinate and not a matter
+of arbitrary choice. The scientific question as to the best available
+evidence concerning the nature of this geometry is one beset with
+difficulties of a peculiar kind. We are obstructed by the fact that all
+existing physical science assumes the Euclidean hypothesis. This
+hypothesis has been involved in all actual measurements of large
+distances, and in all the laws of astronomy and physics. The principle
+of simplicity would therefore lead us, in general, where an observation
+conflicted with one or more of those laws, to ascribe this anomaly, not
+to the falsity of Euclidean geometry, but to the falsity of the laws in
+question. This applies especially to astronomy. On the earth our means
+of measurement are many and direct, and so long as no great accuracy is
+sought they involve few scientific laws. Thus we acquire, from such
+direct measurements, a very high degree of probability that the
+space-constant, if not infinite, is yet large as compared with
+terrestrial distances. But astronomical distances and triangles can only
+be measured by means of the received laws of astronomy and optics, all
+of which have been established by assuming the truth of the Euclidean
+hypothesis. It therefore remains possible (until a detailed proof of the
+contrary is forthcoming) that a large but finite space-constant, with
+different laws of astronomy and optics, would have equally explained the
+phenomena. We cannot, therefore, accept the measurements of stellar
+parallaxes, &c., as conclusive evidence that the space-constant is large
+as compared with stellar distances. For the present, on grounds of
+simplicity, we may rightly adopt this view; but it must remain possible
+that, in view of some hitherto undiscovered discrepancy, a slight
+correction of the sort suggested might prove the simplest alternative.
+But conversely, a finite parallax for very distant stars, or a negative
+parallax for any star, could not be accepted as conclusive evidence that
+our geometry is non-Euclidean, unless it were shown--and this seems
+scarcely possible--that no modification of astronomy or optics could
+account for the phenomenon. Thus although we may admit a probability
+that the space-constant is large in comparison with stellar distances, a
+conclusive proof or disproof seems scarcely possible.
+
+Finally, it is of interest to note that, though it is theoretically
+possible to prove, by scientific methods, that our geometry is
+non-Euclidean, it is wholly impossible to prove by such methods that it
+is accurately Euclidean. For the unavoidable errors of observation must
+always leave a slight margin in our measurements. A triangle might be
+found whose angles were certainly greater, or certainly less, than two
+right angles; but to prove them _exactly_ equal to two right angles must
+always be beyond our powers. If, therefore, any man cherishes a hope of
+proving the exact truth of Euclid, such a hope must be based, not upon
+scientific, but upon philosophical considerations.
+
+  BIBLIOGRAPHY.--The bibliography appended to section VII. should be
+  consulted in this connexion. Also, in addition to the citations
+  already made, the following works may be mentioned.
+
+  For Lobatchewsky's writings, cf. _Urkunden zur Geschichte der
+  nichteuklidischen Geometrie_, i., _Nikolaj Iwanowitsch Lobatschefsky_,
+  by F. Engel and P. Stäckel (Leipzig, 1898). For John Bolyai's
+  _Appendix_, cf. _Absolute Geometrie nach Johann Bolyai_, by J.
+  Frischauf (Leipzig, 1872), and also the new edition of his father's
+  large work, _Tentamen_ ..., published by the Mathematical Society of
+  Budapest; the second volume contains the appendix. Cf. also J.
+  Frischauf, _Elemente der absoluten Geometrie_ (Leipzig, 1876); M.L.
+  Gérard, _Sur la géométrie non-Euclidienne_ (thesis for doctorate)
+  (Paris, 1892); de Tilly, _Essai sur les principes fondamentales de la
+  géométrie et de la mécanique_ (Bordeaux, 1879); Sir R.S. Ball, "On the
+  Theory of Content," _Trans. Roy. Irish Acad._ vol. xxix. (1889); F.
+  Lindemann, "Mechanik bei projectiver Maasbestimmung," _Math. Annal._
+  vol. vii.; W.K. Clifford, "Preliminary Sketch of Biquaternions,"
+  _Proc. of Lond. Math. Soc._ (1873), and _Coll. Works_; A. Buchheim,
+  "On the Theory of Screws in Elliptic Space," _Proc. Lond. Math. Soc._
+  vols. xv., xvi., xvii.; H. Cox, "On the Application of Quaternions and
+  Grassmann's Algebra to different Kinds of Uniform Space," _Trans.
+  Camb. Phil. Soc._ (1882); M. Dehn, "Die Legendarischen Sätze über die
+  Winkelsumme im Dreieck," Math. Ann. vol. 53 (1900), and "Über den
+  Rauminhalt," _Math. Annal._ vol. 55 (1902).
+
+  For expositions of the whole subject, cf. F. Klein, _Nicht-Euklidische
+  Geometrie_ (Göttingen, 1893); R. Bonola, _La Geometria non-Euclidea_
+  (Bologna, 1906); P. Barbarin, _La Géométrie non-Euclidienne_ (Paris,
+  1902); W. Killing, _Die nicht-Euklidischen Raumformen in analytischer
+  Behandlung_ (Leipzig, 1885). The last-named work also deals with
+  geometry of more than three dimensions; in this connexion cf. also G.
+  Veronese, _Fondamenti di geometria a pił dimensioni ed a pił specie_
+  _di unitą rettilinee_ ... (Padua, 1891, German translation, Leipzig,
+  1894); G. Fontené, _L'Hyperespace ą (n-1) dimensions_ (Paris, 1892);
+  and A.N. Whitehead, _loc. cit._ Cf. also E. Study, "Über
+  nicht-Euklidische und Liniengeometrie," _Jahr. d. Deutsch. Math. Ver._
+  vol. xv. (1906); W. Burnside, "On the Kinematics of non-Euclidean
+  Space," _Proc. Lond. Math. Soc._ vol. xxvi. (1894). A bibliography on
+  the subject up to 1878 has been published by G.B. Halsted, _Amer.
+  Journ. of Math._ vols. i. and ii.; and one up to 1900 by R. Bonola,
+  _Index operum ad geometriam absolutam spectantium_ ... (1902, and
+  Leipzig, 1903).     (B. A. W. R.; A. N. W.)
+
+
+VII. AXIOMS OF GEOMETRY
+
+  Theories of space.
+
+Until the discovery of the non-Euclidean geometries (Lobatchewsky, 1826
+and 1829; J. Bolyai, 1832; B. Riemann, 1854), geometry was universally
+considered as being exclusively the science of existent space. (See
+section VI. _Non-Euclidean Geometry_.) In respect to the science, as
+thus conceived, two controversies may be noticed. First, there is the
+controversy respecting the absolute and relational theories of space.
+According to the absolute theory, which is the traditional view (held
+explicitly by Newton), space has an existence, in some sense whatever it
+may be, independent of the bodies which it contains. The bodies occupy
+space, and it is not intrinsically unmeaning to say that any definite
+body occupies _this_ part of space, and not _that_ part of space,
+without reference to other bodies occupying space. According to the
+relational theory of space, of which the chief exponent was
+Leibnitz,[19] space is nothing but a certain assemblage of the relations
+between the various particular bodies in space. The idea of space with
+no bodies in it is absurd. Accordingly there can be no meaning in saying
+that a body is _here_ and not _there_, apart from a reference to the
+other bodies in the universe. Thus, on this theory, absolute motion is
+intrinsically unmeaning. It is admitted on all hands that in practice
+only relative motion is directly measurable. Newton, however, maintains
+in the _Principia_ (scholium to the 8th definition) that it is
+indirectly measurable by means of the effects of "centrifugal force" as
+it occurs in the phenomena of rotation. This irrelevance of absolute
+motion (if there be such a thing) to science has led to the general
+adoption of the relational theory by modern men of science. But no
+decisive argument for either view has at present been elaborated.[20]
+Kant's view of space as being a form of perception at first sight
+appears to cut across this controversy. But he, saturated as he was with
+the spirit of the Newtonian physics, must (at least in both editions of
+the _Critique_) be classed with the upholders of the absolute theory.
+The form of perception has a type of existence proper to itself
+independently of the particular bodies which it contains. For example he
+writes:[21] "Space does not represent any quality of objects by
+themselves, or objects in their relation to one another, i.e. space does
+not represent any determination which is inherent in the objects
+themselves, and would remain, even if all subjective conditions of
+intuition were removed."
+
+
+  Axioms.
+
+The second controversy is that between the view that the axioms
+applicable to space are known only from experience, and the view that in
+some sense these axioms are given _a priori_. Both these views, thus
+broadly stated, are capable of various subtle modifications, and a
+discussion of them would merge into a general treatise on epistemology.
+The cruder forms of the _a priori_ view have been made quite untenable
+by the modern mathematical discoveries. Geometers now profess ignorance
+in many respects of the exact axioms which apply to existent space, and
+it seems unlikely that a profound study of the question should thus
+obliterate _a priori_ intuitions.
+
+Another question irrelevant to this article, but with some relevance to
+the above controversy, is that of the derivation of our perception of
+existent space from our various types of sensation. This is a question
+for psychology.[22]
+
+_Definition of Abstract Geometry._--Existent space is the subject matter
+of only one of the applications of the modern science of abstract
+geometry, viewed as a branch of pure mathematics. Geometry has been
+defined[23] as "the study of series of two or more dimensions." It has
+also been defined[24] as "the science of cross classification." These
+definitions are founded upon the actual practice of mathematicians in
+respect to their use of the term "Geometry." Either of them brings out
+the fact that geometry is not a science with a determinate subject
+matter. It is concerned with any subject matter to which the formal
+axioms may apply. Geometry is not peculiar in this respect. All branches
+of pure mathematics deal merely with types of relations. Thus the
+fundamental ideas of geometry (e.g. those of _points_ and of _straight
+lines_) are not ideas of determinate entities, but of any entities for
+which the axioms are true. And a set of formal geometrical axioms cannot
+in themselves be true or false, since they are not determinate
+propositions, in that they do not refer to a determinate subject matter.
+The axioms are propositional functions.[25] When a set of axioms is
+given, we can ask (1) whether they are consistent, (2) whether their
+"existence theorem" is proved, (3) whether they are independent. Axioms
+are consistent when the contradictory of any axiom cannot be deduced
+from the remaining axioms. Their existence theorem is the proof that
+they are true when the fundamental ideas are considered as denoting some
+determinate subject matter, so that the axioms are developed into
+determinate propositions. It follows from the logical law of
+contradiction that the proof of the existence theorem proves also the
+consistency of the axioms. This is the only method of proof of
+consistency. The axioms of a set are independent of each other when no
+axiom can be deduced from the remaining axioms of the set. The
+independence of a given axiom is proved by establishing the consistency
+of the remaining axioms of the set, together with the contradictory of
+the given axiom. The enumeration of the axioms is simply the enumeration
+of the hypotheses[26] (with respect to the undetermined subject matter)
+of which some at least occur in each of the subsequent propositions.
+
+Any science is called a "geometry" if it investigates the theory of the
+classification of a set of entities (the points) into classes (the
+straight lines), such that (1) there is one and only one class which
+contains any given pair of the entities, and (2) every such class
+contains more than two members. In the two geometries, important from
+their relevance to existent space, axioms which secure an order of the
+points on any line also occur. These geometries will be called
+"Projective Geometry" and "Descriptive Geometry." In projective geometry
+any two straight lines in a plane intersect, and the straight lines are
+closed series which return into themselves, like the circumference of a
+circle. In descriptive geometry two straight lines in a plane do not
+necessarily intersect, and a straight line is an open series without
+beginning or end. Ordinary Euclidean geometry is a descriptive geometry;
+it becomes a projective geometry when the so-called "points at infinity"
+are added.
+
+
+_Projective Geometry._
+
+Projective geometry may be developed from two undefined fundamental
+ideas, namely, that of a "point" and that of a "straight line." These
+undetermined ideas take different specific meanings for the various
+specific subject matters to which projective geometry can be applied.
+The number of the axioms is always to some extent arbitrary, being
+dependent upon the verbal forms of statement which are adopted. They
+will be presented[27] here as twelve in number, eight being "axioms of
+classification," and four being "axioms of order."
+
+_Axioms of Classification._--The eight axioms of classification are as
+follows:
+
+1. Points form a class of entities with at least two members.
+
+2. Any straight line is a class of points containing at least three
+members.
+
+3. Any two distinct points lie in one and only one straight line.
+
+4. There is at least one straight line which does not contain all the
+points.
+
+5. If A, B, C are non-collinear points, and A' is on the straight line
+BC, and B' is on the straight line CA, then the straight lines AA' and
+BB' possess a point in common.
+
+  _Definition._--If A, B, C are any three non-collinear points, the
+  _plane_ ABC is the class of points lying on the straight lines joining
+  A with the various points on the straight line BC.
+
+6. There is at least one plane which does not contain all the points.
+
+7. There exists a plane [alpha], and a point A not incident in [alpha],
+such that any point lies in some straight line which contains both A and
+a point in [alpha].
+
+  _Definition._--Harm. (ABCD) symbolizes the following conjoint
+  statements: (1) that the points A, B, C, D are collinear, and (2) that
+  a quadrilateral can be found with one pair of opposite sides
+  intersecting at A, with the other pair intersecting at C, and with its
+  diagonals passing through B and D respectively. Then B and D are said
+  to be "harmonic conjugates" with respect to A and C.
+
+8. Harm. (ABCD) implies that B and D are distinct points.
+
+In the above axioms 4 secures at least two dimensions, axiom 5 is the
+fundamental axiom of the plane, axiom 6 secures at least three
+dimensions, and axiom 7 secures at most three dimensions. From axioms
+1-5 it can be proved that any two distinct points in a straight line
+determine that line, that any three non-collinear points in a plane
+determine that plane, that the straight line containing any two points
+in a plane lies wholly in that plane, and that any two straight lines in
+a plane intersect. From axioms 1-6 Desargue's well-known theorem on
+triangles in perspective can be proved.
+
+  The enunciation of this theorem is as follows: If ABC and A'B'C' are
+  two coplanar triangles such that the lines AA', BB', CC' are
+  concurrent, then the three points of intersection of BC and B'C' of CA
+  and C'A', and of AB and A'B' are collinear; and conversely if the
+  three points of intersection are collinear, the three lines are
+  concurrent. The proof which can be applied is the usual projective
+  proof by which a third triangle A"B"C" is constructed not coplanar
+  with the other two, but in perspective with each of them.
+
+  It has been proved[28] that Desargues's theorem cannot be deduced from
+  axioms 1-5, that is, if the geometry be confined to two dimensions.
+  All the proofs proceed by the method of producing a specification of
+  "points" and "straight lines" which satisfies axioms 1-5, and such
+  that Desargues's theorem does not hold.
+
+  It follows from axioms 1-5 that Harm. (ABCD) implies Harm. (ADCB) and
+  Harm. (CBAD), and that, if A, B, C be any three distinct collinear
+  points, there exists at least one point D such that Harm. (ABCD). But
+  it requires Desargues's theorem, and hence axiom 6, to prove that
+  Harm. (ABCD) and Harm. (ABCD') imply the identity of D and D'.
+
+The necessity for axiom 8 has been proved by G. Fano,[29] who has
+produced a three dimensional geometry of fifteen points, i.e. a method
+of cross classification of fifteen entities, in which each straight line
+contains three points, and each plane contains seven straight lines. In
+this geometry axiom 8 does not hold. Also from axioms 1-6 and 8 it
+follows that Harm. (ABCD) implies Harm. (BCDA).
+
+  _Definitions._--When two plane figures can be derived from one another
+  by a single projection, they are said to be in _perspective_. When two
+  plane figures can be derived one from the other by a finite series of
+  perspective relations between intermediate figures, they are said to
+  be _projectively_ related. Any property of a plane figure which
+  necessarily also belongs to any projectively related figure, is called
+  a _projective_ property.
+
+  The following theorem, known from its importance as "the fundamental
+  theorem of projective geometry," cannot be proved[30] from axioms 1-8.
+  The enunciation is: "A projective correspondence between the points on
+  two straight lines is completely determined when the correspondents of
+  three distinct points on one line are determined on the other." This
+  theorem is equivalent[31] (assuming axioms 1-8) to another theorem,
+  known as Pappus's Theorem, namely: "If l and l' are two distinct
+  coplanar lines, and A, B, C are three distinct points on l, and A',
+  B', C' are three distinct points on l', then the three points of
+  intersection of AA' and B'C, of A'B and CC', of BB' and C'A, are
+  collinear." This theorem is obviously Pascal's well-known theorem
+  respecting a hexagon inscribed in a conic, for the special case when
+  the conic has degenerated into the two lines l and l'. Another theorem
+  also equivalent (assuming axioms 1-8) to the fundamental theorem is
+  the following:[32] If the three collinear pairs of points, A and A', B
+  and B', C and C', are such that the three pairs of opposite sides of a
+  complete quadrangle pass respectively through them, i.e. one pair
+  through A and A' respectively, and so on, and if also the three sides
+  of the quadrangle which pass through A, B, and C, are concurrent in
+  one of the corners of the quadrangle, then another quadrangle can be
+  found with the same relation to the three pairs of points, except that
+  its three sides which pass through A, B, and C, are not concurrent.
+
+  Thus, if we choose to take any one of these three theorems as an
+  axiom, all the theorems of projective geometry which do not require
+  ordinal or metrical ideas for their enunciation can be proved. Also a
+  conic can be defined as the locus of the points found by the usual
+  construction, based upon Pascal's theorem, for points on the conic
+  through five given points. But it is unnecessary to assume here any
+  one of the suggested axioms; for the fundamental theorem can be
+  deduced from the axioms of order together with axioms 1-8.
+
+_Axioms of Order._--It is possible to define (cf. Pieri, _loc. cit._)
+the property upon which the order of points on a straight line depends.
+But to secure that this property does in fact range the points in a
+serial order, some axioms are required. A straight line is to be a
+closed series; thus, when the points are in order, it requires two
+points on the line to divide it into two distinct complementary
+segments, which do not overlap, and together form the whole line.
+Accordingly the problem of the definition of order reduces itself to the
+definition of these two segments formed by any two points on the line;
+and the axioms are stated relatively to these segments.
+
+  _Definition._--If A, B, C are three collinear points, the points on
+  the _segment_ ABC are defined to be those points such as X, for which
+  there exist two points Y and Y' with the property that Harm. (AYCY')
+  and Harm. (BYXY') both hold. The _supplementary segment_ ABC is
+  defined to be the rest of the points on the line. This definition is
+  elucidated by noticing that with our ordinary geometrical ideas, if B
+  and X are any two points between A and C, then the two pairs of
+  points, A and C, B and X, define an involution with real double
+  points, namely, the Y and Y' of the above definition. The property of
+  belonging to a segment ABC is projective, since the harmonic relation
+  is projective.
+
+The first three axioms of order (cf. Pieri, _loc. cit._) are:
+
+9. If A, B, C are three distinct collinear points, the supplementary
+segment ABC is contained within the segment BCA.
+
+10. If A, B, C are three distinct collinear points, the common part of
+the segments BCA and CAB is contained in the supplementary segment ABC.
+
+11. If A, B, C are three distinct collinear points, and D lies In the
+segment ABC, then the segment ADC is contained within the segment ABC.
+
+From these axioms all the usual properties of a closed order follow. It
+will be noticed that, if A, B, C are any three collinear points, C is
+necessarily traversed in passing from A to B by one route along the
+line, and is not traversed in passing from A to B along the other route.
+Thus there is no meaning, as referred to closed straight lines, in the
+simple statement that C lies between A and B. But there may be a
+relation of separation between two pairs of collinear points, such as A
+and C, and B and D. The couple B and D is said to separate A and C, if
+the four points are collinear and D lies in the segment complementary to
+the segment ABC. The property of the separation of pairs of points by
+pairs of points is projective. Also it can be proved that Harm. (ABCD)
+implies that B and D separate A and C.
+
+  _Definitions._--A series of entities arranged in a serial order, open
+  or closed, is said to be _compact_, if the series contains no
+  immediately consecutive entities, so that in traversing the series
+  from any one entity to any other entity it is necessary to pass
+  through entities distinct from either. It was the merit of R. Dedekind
+  and of G. Cantor explicitly to formulate another fundamental property
+  of series. The Dedekind property[33] as applied to an open series can
+  be defined thus: An open series possesses the Dedekind property, if,
+  however, it be divided into two mutually exclusive classes u and v,
+  which (1) contain between them the whole series, and (2) are such that
+  every member of u precedes in the serial order every member of v,
+  there is always a member of the series, belonging to one of the two, u
+  or v, which precedes every member of v (other than itself if it belong
+  to v), and also succeeds every member of u (other than itself if it
+  belong to u). Accordingly in an open series with the Dedekind property
+  there is always a member of the series marking the junction of two
+  classes such as u and v. An open series is _continuous_ if it is
+  compact and possesses the Dedekind property. A closed series can
+  always be transformed into an open series by taking any arbitrary
+  member as the first term and by taking one of the two ways round as
+  the ascending order of the series. Thus the definitions of compactness
+  and of the Dedekind property can be at once transferred to a closed
+  series.
+
+12. The last axiom of order is that there exists at least one straight
+line for which the point order possesses the Dedekind property.
+
+It follows from axioms 1-12 by projection that the Dedekind property is
+true for all lines. Again the _harmonic system_ ABC, where A, B, C are
+collinear points, is defined[34] thus: take the harmonic conjugates A',
+B', C' of each point with respect to the other two, again take the
+harmonic conjugates of each of the six points A, B, C, A', B', C' with
+respect to each pair of the remaining five, and proceed in this way by
+an unending series of steps. The set of points thus obtained is called
+the harmonic system ABC. It can be proved that a harmonic system is
+compact, and that every segment of the line containing it possesses
+members of it. Furthermore, it is easy to prove that the fundamental
+theorem holds for harmonic systems, in the sense that, if A, B, C are
+three points on a line l, and A', B', C' are three points on a line l',
+and if by any two distinct series of projections A, B, C are projected
+into A', B', C', then any point of the harmonic system ABC corresponds
+to the same point of the harmonic system A'B'C' according to both the
+projective relations which are thus established between l and l'. It now
+follows immediately that the fundamental theorem must hold for all the
+points on the lines l and l', since (as has been pointed out) harmonic
+systems are "everywhere dense" on their containing lines. Thus the
+fundamental theorem follows from the axioms of order.
+
+A system of numerical coordinates can now be introduced, possessing the
+property that linear equations represent planes and straight lines. The
+outline of the argument by which this remarkable problem (in that
+"distance" is as yet undefined) is solved, will now be given. It is
+first proved that the points on any line can in a certain way be
+definitely associated with all the positive and negative real numbers,
+so as to form with them a one-one correspondence. The arbitrary elements
+in the establishment of this relation are the points on the line
+associated with 0, 1 and [oo].
+
+This association[35] is most easily effected by considering a class of
+projective relations of the line with itself, called by F. Schur (_loc.
+cit._) _prospectivities_.
+
+  Let l (fig. 69) be the given line, m and n any two lines intersecting
+  at U on l, S and S' two points on n. Then a projective relation
+  between l and itself is formed by projecting l from S on to m, and
+  then by projecting m from S' back on to l. All such projective
+  relations, however m, n, S and S' be varied, are called
+  "prospectivities," and U is the double point of the prospectivity. If
+  a point O on l is related to A by a prospectivity, then all
+  prospectivities, which (1) have the same double point U, and (2)
+  relate O to A, give the same correspondent (Q, in figure) to any point
+  P on the line l; in fact they are all the same prospectivity, however
+  m, n, S, and S' may have been varied subject to these conditions. Such
+  a prospectivity will be denoted by (OAU²).
+
+  [Illustration: FIG. 69.]
+
+  The sum of two prospectivities, written (OAU²) + (OBU²), is defined to
+  be that transformation of the line l into itself which is obtained by
+  first applying the prospectivity (OAU²) and then applying the
+  prospectivity (OBU²). Such a transformation, when the two summands
+  have the same double point, is itself a prospectivity with that double
+  point.
+
+  [Illustration: FIG. 70]
+
+  With this definition of addition it can be proved that prospectivities
+  with the same double point satisfy all the axioms of magnitude.
+  Accordingly they can be associated in a one-one correspondence with
+  the positive and negative real numbers. Let E (fig. 70) be any point
+  on l, distinct from O and U. Then the prospectivity (OEU²) is
+  associated with unity, the prospectivity (OOU²) is associated with
+  zero, and (OUU²) with [infinity]. The prospectivities of the type
+  (OPU²), where P is any point on the segment OEU, correspond to the
+  positive numbers; also if P' is the harmonic conjugate of P with
+  respect to O and U, the prospectivity (OP'U²) is associated with the
+  corresponding negative number. (The subjoined figure explains this
+  relation of the positive and negative prospectivities.) Then any point
+  P on l is associated with the same number as is the prospectivity
+  (OPU²).
+
+  [Illustration: FIG. 71.]
+
+  It can be proved that the order of the numbers in algebraic order of
+  magnitude agrees with the order on the line of the associated points.
+  Let the numbers, assigned according to the preceding specification, be
+  said to be associated with the points according to the
+  "numeration-system (OEU)." The introduction of a coordinate system for
+  a plane is now managed as follows: Take any triangle OUV in the plane,
+  and on the lines OU and OV establish the numeration systems (OE1U) and
+  (OE2V), where E1 and E2 are arbitrarily chosen. Then (cf. fig. 71) if
+  M and N are associated with the numbers x and y according to these
+  systems, the coordinates of P are x and y. It then follows that the
+  equation of a straight line is of the form ax + by + c = 0. Both
+  coordinates of any point on the line UV are infinite. This can be
+  avoided by introducing homogeneous coordinates X, Y, Z, where x = X/Z,
+  and y = Y/Z, and Z = 0 is the equation of UV.
+
+  [Illustration: FIG. 72.]
+
+  The procedure for three dimensions is similar. Let OUVW (fig. 72) be
+  any tetrahedron, and associate points on OU, OV, OW with numbers
+  according to the numeration systems (OE1U), (OE2V), and (OE3W). Let
+  the planes VWP, WUP, UVP cut OU, OV, OW in L, M, N respectively; and
+  let x, y, z be the numbers associated with L, M, N respectively. Then
+  P is the point (x, y, z). Also homogeneous coordinates can be
+  introduced as before, thus avoiding the infinities on the plane UVW.
+
+  The cross ratio of a range of four collinear points can now be defined
+  as a number characteristic of that range. Let the coordinates of any
+  point P_r of the range P1 P2 P3 P4 be
+
+    [lambda]_r a + µ_r + a'   [lambda]_r b + µ_r b'
+    -----------------------,  ---------------------,
+       [lambda]_r + µ_r         [lambda]_r + µ_r
+
+      [lambda]_r c + µ_r c'
+      ---------------------, (r = 1, 2, 3, 4)
+        [lambda]_r + µ_r
+
+  and let ([lambda]_r µ_s) be written for [lambda]_r µ_s -[lambda]_s
+  µ_r. Then the cross ratio {P1 P2 P3 P4} is defined to be the number
+  ([lambda]1µ2)([lambda]3µ4)/([lambda]1µ4)([lambda]3µ2). The equality of
+  the cross ratios of the ranges (P1 P2 P3 P4) and (Q1 Q2 Q3 Q4) is
+  proved to be the necessary and sufficient condition for their mutual
+  projectivity. The cross ratios of all harmonic ranges are then easily
+  seen to be all equal to -1, by comparing with the range (OE1UE'1) on
+  the axis of x.
+
+  Thus all the ordinary propositions of geometry in which distance and
+  angular measure do not enter otherwise than in cross ratios can now be
+  enunciated and proved. Accordingly the greater part of the analytical
+  theory of conics and quadrics belongs to geometry at this stage The
+  theory of distance will be considered after the principles of
+  descriptive geometry have been developed.
+
+
+_Descriptive Geometry._
+
+Descriptive geometry is essentially the science of multiple order for
+open series. The first satisfactory system of axioms was given by M.
+Pasch.[36] An improved version is due to G. Peano.[37] Both these
+authors treat the idea of the class of points constituting the segment
+lying _between_ two points as an undefined fundamental idea. Thus in
+fact there are in this system two fundamental ideas, namely, of points
+and of segments. It is then easy enough to define the prolongations of
+the segments, so as to form the complete straight lines. D.
+Hilbert's[38] formulation of the axioms is in this respect practically
+based on the same fundamental ideas. His work is justly famous for some
+of the mathematical investigations contained in it, but his exposition
+of the axioms is distinctly inferior to that of Peano. Descriptive
+geometry can also be considered[39] as the science of a class of
+relations, each relation being a two-termed serial relation, as
+considered in the logic of relations, ranging the points between which
+it holds into a linear open order. Thus the relations are the straight
+lines, and the terms between which they hold are the points. But a
+combination of these two points of view yields[40] the simplest
+statement of all. Descriptive geometry is then conceived as the
+investigation of an undefined fundamental relation between three terms
+(points); and when the relation holds between three points A, B, C, the
+points are said to be "in the [linear] order ABC."
+
+O. Veblen's axioms and definitions, slightly modified, are as follows:--
+
+1. If the points A, B, C are in the order ABC, they are in the order
+CBA.
+
+2. If the points A, B, C are in the order ABC, they are not in the order
+BCA.
+
+3. If the points A, B, C are in the order ABC, A is distinct from C.
+
+4. If A and B are any two distinct points, there exists a point C such
+that A, B, C are in the order ABC.
+
+  _Definition._--The _line_ AB (A =| B) consists of A and B, and of all
+  points X in one of the possible orders, ABX, AXB, XAB. The points X in
+  the order AXB constitute the _segment_ AB.
+
+5. If points C and D (C =| D) lie on the line AB, then A lies on the
+line CD.
+
+6. There exist three distinct points A, B, C not in any of the orders
+ABC, BCA, CAB.
+
+7. If three distinct points A, B, C (fig. 73) do not lie on the same
+line, and D and E are two distinct points in the orders BCD and CEA,
+then a point F exists in the order AFB, and such that D, E, F are
+collinear.
+
+[Illustration: FIG. 73.]
+
+  _Definition._--If A, B, C are three non-collinear points, the _plane_
+  ABC is the class of points which lie on any one of the lines joining
+  any two of the points belonging to the _boundary_ of the triangle ABC,
+  the boundary being formed by the segments BC, CA and AB. The
+  _interior_ of the triangle ABC is formed by the points in segments
+  such as PQ, where P and Q are points respectively on two of the
+  segments BC, CA, AB.
+
+8. There exists a plane ABC, which does not contain all the points.
+
+  _Definition._--If A, B, C, D are four non-coplanar points, the space
+  ABCD is the class of points which lie on any of the lines containing
+  two points on the surface of the tetrahedron ABCD, the _surface_ being
+  formed by the interiors of the triangles ABC, BCD, DCA, DAB.
+
+9. There exists a space ABCD which contains all the points.
+
+10. The Dedekind property holds for the order of the points on any
+straight line.
+
+It follows from axioms 1-9 that the points on any straight line are
+arranged in an open serial order. Also all the ordinary theorems
+respecting a point dividing a straight line into two parts, a straight
+line dividing a plane into two parts, and a plane dividing space into
+two parts, follow.
+
+  Again, in any plane [alpha] consider a line l and a point A (fig. 74).
+
+  [Illustration: FIG. 74.]
+
+  Let any point B divide l into two half-lines l1 and l2. Then it can be
+  proved that the set of half-lines, emanating from A and intersecting
+  l1 (such as m), are bounded by two half-lines, of which ABC is one.
+  Let r be the other. Then it can be proved that r does not intersect
+  l1. Similarly for the half-line, such as n, intersecting l2. Let s be
+  its bounding half-line. Then two cases are possible. (1) The
+  half-lines r and s are collinear, and together form one complete line.
+  In this case, there is one and only one line (viz. r + s) through A
+  and lying in [alpha] which does not intersect l. This is the Euclidean
+  case, and the assumption that this case holds is the _Euclidean
+  parallel axiom_. But (2) the half-lines r and s may not be collinear.
+  In this case there will be an infinite number of lines, such as k for
+  instance, containing A and lying in [alpha], which do not intersect l.
+  Then the lines through A in [alpha] are divided into two classes by
+  reference to l, namely, the _secant_ lines which intersect l, and the
+  _non-secant_ lines which do not intersect l. The two boundary
+  non-secant lines, of which r and s are respectively halves, may be
+  called the two parallels to l through A.
+
+  The perception of the possibility of case 2 constituted the
+  starting-point from which Lobatchewsky constructed the first explicit
+  coherent theory of non-Euclidean geometry, and thus created a
+  revolution in the philosophy of the subject. For many centuries the
+  speculations of mathematicians on the foundations of geometry were
+  almost confined to hopeless attempts to prove the "parallel axiom"
+  without the introduction of some equivalent axiom.[41]
+
+_Associated Projective and Descriptive Spaces._--A region of a
+projective space, such that one, and only one, of the two supplementary
+segments between any pair of points within it lies entirely within it,
+satisfies the above axioms (1-10) of descriptive geometry, where the
+points of the region are the descriptive points, and the portions of
+straight lines within the region are the descriptive lines. If the
+excluded part of the original projective space is a single plane, the
+Euclidean parallel axiom also holds, otherwise it does not hold for the
+descriptive space of the limited region. Again, conversely, starting
+from an original descriptive space an associated projective space can be
+constructed by means of the concept of _ideal points_.[42] These are
+also called _projective points_, where it is understood that the simple
+points are the points of the original descriptive space. An _ideal
+point_ is the class of straight lines which is composed of two coplanar
+lines a and b, together with the lines of intersection of all pairs of
+intersecting planes which respectively contain a and b, together with
+the lines of intersection with the plane ab of all planes containing any
+one of the lines (other than a or b) already specified as belonging to
+the ideal point. It is evident that, if the two original lines a and b
+intersect, the corresponding ideal point is nothing else than the whole
+class of lines which are concurrent at the point ab. But the essence of
+the definition is that an ideal point has an existence when the lines a
+and b do not intersect, so long as they are coplanar. An ideal point is
+termed _proper_, if the lines composing it intersect; otherwise it is
+_improper_.
+
+A theorem essential to the whole theory is the following: if any two of
+the three lines a, b, c are coplanar, but the three lines are not all
+coplanar, and similarly for the lines a, b, d, then c and d are
+coplanar. It follows that any two lines belonging to an ideal point can
+be used as the pair of guiding lines in the definition. An ideal point
+is said to be _coherent_ with a plane, if any of the lines composing it
+lie in the plane. An _ideal line_ is the class of ideal points each of
+which is coherent with two given planes. If the planes intersect, the
+ideal line is termed _proper_, otherwise it is _improper_. It can be
+proved that any two planes, with which any two of the ideal points are
+both coherent, will serve as the guiding planes used in the definition.
+The ideal planes are defined as in projective geometry, and all the
+other definitions (for segments, order, &c.) of projective geometry are
+applied to the ideal elements. If an ideal plane contains some proper
+ideal points, it is called _proper_, otherwise it is _improper_. Every
+ideal plane contains some improper ideal points.
+
+It can now be proved that all the axioms of projective geometry hold of
+the ideal elements as thus obtained; and also that the order of the
+ideal points as obtained by the projective method agrees with the order
+of the proper ideal points as obtained from that of the associated
+points of the descriptive geometry. Thus a projective space has been
+constructed out of the ideal elements, and the proper ideal elements
+correspond element by element with the associated descriptive elements.
+Thus the proper ideal elements form a region in the projective space
+within which the descriptive axioms hold. Accordingly, by substituting
+ideal elements, a descriptive space can always be considered as a region
+within a projective space. This is the justification for the ordinary
+use of the "points at infinity" in the ordinary Euclidean geometry; the
+reasoning has been transferred from the original descriptive space to
+the associated projective space of ideal elements; and with the
+Euclidean parallel axiom the improper ideal elements reduce to the ideal
+points on a single improper ideal plane, namely, the plane at
+infinity.[43]
+
+_Congruence and Measurement._--The property of physical space which is
+expressed by the term "measurability" has now to be considered. This
+property has often been considered as essential to the very idea of
+space. For example, Kant writes,[44] "Space is represented as an
+infinite given _quantity_." This quantitative aspect of space arises
+from the measurability of distances, of angles, of surfaces and of
+volumes. These four types of quantity depend upon the two first among
+them as fundamental. The measurability of space is essentially connected
+with the idea of _congruence_, of which the simplest examples are to be
+found in the proofs of equality by the method of superposition, as used
+in elementary plane geometry. The mere concepts of "part" and of "whole"
+must of necessity be inadequate as the foundation of measurement, since
+we require the comparison as to quantity of regions of space which have
+no portions in common. The idea of congruence, as exemplified by the
+method of superposition in geometrical reasoning, appears to be founded
+upon that of the "rigid body," which moves from one position to another
+with its internal spatial relations unchanged. But unless there is a
+previous concept of the metrical relations between the parts of the
+body, there can be no basis from which to deduce that they are
+unchanged.
+
+It would therefore appear as if the idea of the congruence, or metrical
+equality, of two portions of space (as empirically suggested by the
+motion of rigid bodies) must be considered as a fundamental idea
+incapable of definition in terms of those geometrical concepts which
+have already been enumerated. This was in effect the point of view of
+Pasch.[45] It has, however, been proved by Sophus Lie[46] that
+congruence is capable of definition without recourse to a new
+fundamental idea. This he does by means of his theory of finite
+continuous groups (see GROUPS, THEORY OF), of which the definition is
+possible in terms of our established geometrical ideas, remembering that
+coordinates have already been introduced. The displacement of a rigid
+body is simply a mode of defining to the senses a one-one transformation
+of all space into itself. For at any point of space a particle may be
+conceived to be placed, and to be rigidly connected with the rigid body;
+and thus there is a definite correspondence of any point of space with
+the new point occupied by the associated particle after displacement.
+Again two successive displacements of a rigid body from position A to
+position B, and from position B to position C, are the same in effect as
+one displacement from A to C. But this is the characteristic "group"
+property. Thus the transformations of space into itself defined by
+displacements of rigid bodies form a group.
+
+Call this group of transformations a congruence-group. Now according to
+Lie a congruence-group is defined by the following characteristics:--
+
+1. A congruence-group is a finite continuous group of one-one
+transformations, containing the identical transformation.
+
+2. It is a sub-group of the general projective group, i.e. of the group
+of which any transformation converts planes into planes, and straight
+lines into straight lines.
+
+3. An infinitesimal transformation can always be found satisfying the
+condition that, at least throughout a certain enclosed region, any
+definite line and any definite point on the line are latent, i.e.
+correspond to themselves.
+
+4. No infinitesimal transformation of the group exists, such that, at
+least in the region for which (3) holds, a straight line, a point on it,
+and a plane through it, shall all be latent.
+
+The property enunciated by conditions (3) and (4), taken together, is
+named by Lie "Free mobility in the infinitesimal." Lie proves the
+following theorems for a projective space:--
+
+  1. If the above four conditions are only satisfied by a group
+  throughout part of projective space, this part either ([alpha]) must
+  be the region enclosed by a real closed quadric, or (ß) must be the
+  whole of the projective space with the exception of a single plane. In
+  case ([alpha]) the corresponding congruence group is the continuous
+  group for which the enclosing quadric is latent; and in case (ß) an
+  imaginary conic (with a real equation) lying in the latent plane is
+  also latent, and the congruence group is the continuous group for
+  which the plane and conic are latent.
+
+  2. If the above four conditions are satisfied by a group throughout
+  the whole of projective space, the congruence group is the continuous
+  group for which some imaginary quadric (with a real equation) is
+  latent.
+
+  By a proper choice of non-homogeneous co-ordinates the equation of any
+  quadrics of the types considered, either in theorem 1 ([alpha]), or in
+  theorem 2, can be written in the form 1 +c(x² + y² + z²) = 0, where c
+  is negative for a real closed quadric, and positive for an imaginary
+  quadric. Then the general infinitesimal transformation is defined by
+  the three equations:
+
+    dx/dt = u - [omega]3y + [omega]2z + cx(ux + vy + wz), \
+    dy/dt = v - [omega]1z + [omega]3x + cy(ux + vy + wz),  > (A)
+    dz/dt = w - [omega]2x + [omega]1y + cz(ux + vy + wz). /
+
+  In the ease considered in theorem 1 (ß), with the proper choice of
+  co-ordinates the three equations defining the general infinitesimal
+  transformation are:
+
+    dx/dt = u - [omega]3y + [omega]2z, \
+    dy/dt = v - [omega]1z + [omega]3x,  > (B)
+    dz/dt = w - [omega]2x + [omega]1y. /
+
+  In this case the latent plane is the plane for which at least one of
+  x, y, z are infinite, that is, the plane 0.x + 0.y + 0.z + a = 0; and
+  the latent conic is the conic in which the cone x² + y² + z² = 0
+  intersects the latent plane.
+
+It follows from theorems 1 and 2 that there is not one unique
+congruence-group, but an indefinite number of them. There is one
+congruence-group corresponding to each closed real quadric, one to each
+imaginary quadric with a real equation, and one to each imaginary conic
+in a real plane and with a real equation. The quadric thus associated
+with each congruence-group is called the _absolute_ for that group, and
+in the degenerate case of 1 (ß) the absolute is the latent plane
+together with the latent imaginary conic. If the absolute is real, the
+congruence-group is _hyperbolic_; if imaginary, it is _elliptic_; if the
+absolute is a plane and imaginary conic, the group is parabolic.
+Metrical geometry is simply the theory of the properties of some
+particular congruence-group selected for study.
+
+  The definition of distance is connected with the corresponding
+  congruence-group by two considerations in respect to a range of five
+  points (A1, A2, P1, P2, P3), of which A1 and A2 are on the absolute.
+
+  Let {A1P1A2P2} stand for the cross ratio (as defined above) of the
+  range (A1P1A2P2), with a similar notation for the other ranges. Then
+
+  (1)   log{A1P1A2P2} + log{A1P2A2P3} = log{A1P1A2P3},
+
+  and
+
+  (2), if the points A1, A2, P1, P2 are transformed into A'1, A'2, P'1,
+  P'2 by any transformation of the congruence-group, ([alpha])
+  {A1P(1}A2P2 = {A'1P'1A'2P'2}, since the transformation is projective,
+  and (ß) A'1, A'2 are on the absolute since A1 and A2 are on it. Thus
+  if we define the distance P1P2 to be ½k log {A1P1A2P2}, where A1 and
+  A2 are the points in which the line P1P2 cuts the absolute, and k is
+  some constant, the two characteristic properties of distance, namely,
+  (1) the addition of consecutive lengths on a straight line, and (2)
+  the invariability of distances during a transformation of the
+  congruence-group, are satisfied. This is the well-known Cayley-Klein
+  projective definition[47] of distance, which was elaborated in view of
+  the addition property alone, previously to Lie's discovery of the
+  theory of congruence-groups. For a hyperbolic group when P1 and P2 are
+  in the region enclosed by the absolute, log {A1P1A2P2} is real, and
+  therefore k must be real. For an elliptic group A1 and A2 are
+  conjugate imaginaries, and log {A1P1A2P2} is a pure imaginary, and k
+  is chosen to be [kappa]/[iota], where [kappa] is real and [iota] =
+  [root]-.
+
+  Similarly the angle between two planes, p1 and p2, is defined to be
+  (1/2[iota]) log (t1p1t2p2), where t1 and t2 are tangent planes to the
+  absolute through the line p1p2. The planes t1 and t2 are imaginary for
+  an elliptic group, and also for an hyperbolic group when the planes p1
+  and p2 intersect at points within the region enclosed by the absolute.
+  The development of the consequences of these metrical definitions is
+  the subject of non-Euclidean geometry.
+
+  The definitions for the parabolic case can be arrived at as limits of
+  those obtained in either of the other two cases by making k ultimately
+  to vanish. It is also obvious that, if P1 and P2 be the points (x1,
+  y1, z1) and (x2, y2, z2), it follows from equations (B) above that
+  {(x1 - x2)² + (y1 - y2)² + (z1 - z2)²}^½ is unaltered by a congruence
+  transformation and also satisfies the addition property for collinear
+  distances. Also the previous definition of an angle can be adapted to
+  this case, by making t1 and t2 to be the tangent planes through the
+  line p1p2 to the imaginary conic. Similarly if p1 and p2 are
+  intersecting lines, the same definition of an angle holds, where t1
+  and t2 are now the lines from the point p1p2 to the two points where
+  the plane p1p2 cuts the imaginary conic. These points are in fact the
+  "circular points at infinity" on the plane. The development of the
+  consequences of these definitions for the parabolic case gives the
+  ordinary Euclidean metrical geometry.
+
+Thus the only metrical geometry for the whole of projective space is of
+the elliptic type. But the actual measure-relations (though not their
+general properties) differ according to the elliptic congruence-group
+selected for study. In a descriptive space a congruence-group should
+possess the four characteristics of such a group throughout the whole of
+the space. Then form the associated ideal projective space. The
+associated congruence-group for this ideal space must satisfy the four
+conditions throughout the region of the proper ideal points. Thus the
+boundary of this region is the absolute. Accordingly there can be no
+metrical geometry for the whole of a descriptive space unless its
+boundary (in the associated ideal space) is a closed quadric or a plane.
+If the boundary is a closed quadric, there is one possible
+congruence-group of the hyperbolic type. If the boundary is a plane (the
+plane at infinity), the possible congruence-groups are parabolic; and
+there is a congruence-group corresponding to each imaginary conic in
+this plane, together with a Euclidean metrical geometry corresponding to
+each such group. Owing to these alternative possibilities, it would
+appear to be more accurate to say that systems of quantities can be
+found in a space, rather than that space is a quantity.
+
+Lie has also deduced[48] the same results with respect to
+congruence-groups from another set of defining properties, which
+explicitly assume the existence of a quantitative relation (the
+distance) between any two points, which is invariant for any
+transformation of the congruence-group.[49]
+
+The above results, in respect to congruence and metrical geometry,
+considered in relation to existent space, have led to the doctrine[50]
+that it is intrinsically unmeaning to ask which system of metrical
+geometry is true of the physical world. Any one of these systems can be
+applied, and in an indefinite number of ways. The only question before
+us is one of convenience in respect to simplicity of statement of the
+physical laws. This point of view seems to neglect the consideration
+that science is to be relevant to the definite perceiving minds of men;
+and that (neglecting the ambiguity introduced by the invariable slight
+inexactness of observation which is not relevant to this special
+doctrine) we have, in fact, presented to our senses a definite set of
+transformations forming a congruence-group, resulting in a set of
+measure relations which are in no respect arbitrary. Accordingly our
+scientific laws are to be stated relevantly to that particular
+congruence-group. Thus the investigation of the type (elliptic,
+hyperbolic or parabolic) of this special congruence-group is a perfectly
+definite problem, to be decided by experiment. The consideration of
+experiments adapted to this object requires some development of
+non-Euclidean geometry (see section VI., _Non-Euclidean Geometry_). But
+if the doctrine means that, assuming some sort of objective reality for
+the material universe, beings can be imagined, to whom _either_ all
+congruence-groups are equally important, _or_ some other
+congruence-group is specially important, the doctrine appears to be an
+immediate deduction from the mathematical facts. Assuming a definite
+congruence-group, the investigation of surfaces (or three-dimensional
+loci in space of four dimensions) with geodesic geometries of the form
+of metrical geometries of other types of congruence-groups forms an
+important chapter of non-Euclidean geometry. Arising from this
+investigation there is a widely-spread fallacy, which has found its way
+into many philosophic writings, namely, that the possibility of the
+geometry of existent three-dimensional space being other than Euclidean
+depends on the physical existence of Euclidean space of four or more
+dimensions. The foregoing exposition shows the baselessness of this
+idea.
+
+  BIBLIOGRAPHY.--For an account of the investigations on the axioms of
+  geometry during the Greek period, see M. Cantor, _Vorlesungen über die
+  Geschichte der Mathematik_, Bd. i. and iii.; T.L. Heath, _The Thirteen
+  Books of Euclid's Elements, a New Translation from the Greek, with
+  Introductory Essays and Commentary, Historical, Critical, and
+  Explanatory_ (Cambridge, 1908)--this work is the standard source of
+  information; W.B. Frankland, _Euclid, Book I., with a Commentary_
+  (Cambridge, 1905)--the commentary contains copious extracts from the
+  ancient commentators. The next period of really substantive importance
+  is that of the 18th century. The leading authors are: G. Saccheri,
+  S.J., _Euclides ab omni naevo vindicatus_ (Milan, 1733). Saccheri was
+  an Italian Jesuit who unconsciously discovered non-Euclidean geometry
+  in the course of his efforts to prove its impossibility. J.H. Lambert,
+  _Theorie der Parallellinien_ (1766); A.M. Legendre, _Éléments de
+  géométrie_ (1794). An adequate account of the above authors is given
+  by P. Stäckel and F. Engel, _Die Theorie der Parallellinien von Euklid
+  bis auf Gauss_ (Leipzig, 1895). The next period of time (roughly from
+  1800 to 1870) contains two streams of thought, both of which are
+  essential to the modern analysis of the subject. The first stream is
+  that which produced the discovery and investigation of non-Euclidean
+  geometries, the second stream is that which has produced the geometry
+  of position, comprising both projective and descriptive geometry not
+  very accurately discriminated. The leading authors on non-Euclidean
+  geometry are K.F. Gauss, in private letters to Schumacher, cf. Stäckel
+  and Engel, _loc. cit._; N. Lobatchewsky, rector of the university of
+  Kazan, to whom the honour of the effective discovery of non-Euclidean
+  geometry must be assigned. His first publication was at Kazan in 1826.
+  His various memoirs have been re-edited by Engel; cf. _Urkunden zur
+  Geschichte der nichteuklidischen Geometrie_ by Stäckel and Engel, vol.
+  i. "Lobatchewsky." J. Bolyai discovered non-Euclidean geometry
+  apparently in independence of Lobatchewsky. His memoir was published
+  in 1831 as an appendix to a work by his father W. Bolyai, _Tentamen
+  juventutem...._ This memoir has been separately edited by J.
+  Frischauf, _Absolute Geometrie nach J. Bolyai_ (Leipzig, 1872); B.
+  Riemann, _Über die Hypothesen, welche der Geometrie zu Grunde liegen_
+  (1854); cf. _Gesamte Werke_, a translation in The Collected Papers of
+  W.K. Clifford. This is a fundamental memoir on the subject and must
+  rank with the work of Lobatchewsky. Riemann discovered elliptic
+  metrical geometry, and Lobatchewsky hyperbolic geometry. A full
+  account of Riemann's ideas, with the subsequent developments due to
+  Clifford, F. Klein and W. Killing, will be found in _The Boston
+  Colloquium for 1903_ (New York, 1905), article "Forms of Non-Euclidean
+  Space," by F.S. Woods. A. Cayley, _loc. cit._ (1859), and F. Klein,
+  "Über die sogenannte nichteuklidische Geometrie," _Math. Annal._ vols.
+  iv. and vi. (1871 and 1872), between them elaborated the projective
+  theory of distance; H. Helmholtz, "Über die thatsächlichen Grundlagen
+  der Geometrie" (1866), and "Über die Thatsachen, die der Geometrie zu
+  Grunde liegen" (1868), both in his _Wissenschaftliche Abhandlungen_,
+  vol. ii., and S. Lie, _loc. cit._ (1890 and 1893), between them
+  elaborated the group theory of congruence.
+
+  The numberless works which have been written to suggest equivalent
+  alternatives to Euclid's parallel axioms may be neglected as being of
+  trivial importance, though many of them are marvels of geometric
+  ingenuity.
+
+  The second stream of thought confined itself within the circle of
+  ideas of Euclidean geometry. Its origin was mainly due to a succession
+  of great French mathematicians, for example, G. Monge, _Géométrie
+  descriptive_ (1800); J.V. Poncelet, _Traité des proprietés projectives
+  des figures_ (1822); M. Chasles, _Aperēu historique sur l'origine et
+  le développement des méthodes en géométrie_ (Bruxelles, 1837), and
+  _Traité de géométrie supérieure_ (Paris, 1852); and many others. But
+  the works which have been, and are still, of decisive influence on
+  thought as a store-house of ideas relevant to the foundations of
+  geometry are K.G.C. von Staudt's two works, _Geometrie der Lage_
+  (Nürnberg, 1847); and _Beiträge zur Geometrie der Lage_ (Nürnberg,
+  1856, 3rd ed. 1860).
+
+  The final period is characterized by the successful production of
+  exact systems of axioms, and by the final solution of problems which
+  have occupied mathematicians for two thousand years. The successful
+  analysis of the ideas involved in serial continuity is due to R.
+  Dedekind, _Stetigkeit und irrationale Zahlen_ (1872), and to G.
+  Cantor, _Grundlagen einer allgemeinen Mannigfaltigkeitslehre_
+  (Leipzig, 1883), and _Acta math._ vol. 2.
+
+  Complete systems of axioms have been stated by M. Pasch, _loc. cit._;
+  G. Peano, _loc. cit._; M. Pieri, _loc. cit._; B. Russell, _Principles
+  of Mathematics_; O. Veblen, _loc. cit._; and by G. Veronese in his
+  treatise, _Fondamenti di geometria_ (Padua, 1891; German transl. by A.
+  Schepp, _Grundzüge der Geometrie_, Leipzig, 1894). Most of the leading
+  memoirs on special questions involved have been cited in the text; in
+  addition there may be mentioned M. Pieri, "Nuovi principii di
+  geometria projettiva complessa," _Trans. Accad. R. d. Sci._ (Turin,
+  1905); E.H. Moore, "On the Projective Axioms of Geometry," _Trans.
+  Amer. Math. Soc._, 1902; O. Veblen and W.H. Bussey, "Finite Projective
+  Geometries," _Trans. Amer. Math. Soc._, 1905; A.B. Kempe, "On the
+  Relation between the Logical Theory of Classes and the Geometrical
+  Theory of Points," _Proc. Lond. Math. Soc._, 1890; J. Royce, "The
+  Relation of the Principles of Logic to the Foundations of Geometry,"
+  _Trans. of Amer. Math. Soc._, 1905; A. Schoenflies, "Über die
+  Möglichkeit einer projectiven Geometrie bei transfiniter
+  (nichtarchimedischer) Massbestimmung," _Deutsch. M.-V. Jahresb._,
+  1906.
+
+  For general expositions of the bearings of the above investigations,
+  cf. Hon. Bertrand Russell, _loc. cit._; L. Couturat, _Les Principes
+  des mathématiques_ (Paris, 1905); H. Poincaré, _loc. cit._; Russell
+  and Whitehead, _Principia mathematica_ (Cambridge, Univ. Press). The
+  philosophers whose views on space and geometric truth deserve especial
+  study are Descartes, Leibnitz, Hume, Kant and J.S. Mill.     (A. N. W.)
+
+
+FOOTNOTES:
+
+  [1] For Egyptian geometry see EGYPT, § _Science and Mathematics_.
+
+  [2] Cf. A.N. Whitehead, _Universal Algebra_, Bk. vi. (Cambridge,
+    1898).
+
+  [3] Cf. A.N. Whitehead, _loc. cit._
+
+  [4] Cf. A.N. Whitehead, "The Geodesic Geometry of Surfaces in
+    non-Euclidean Space," _Proc. Lond. Math. Soc._ vol. xxix.
+
+  [5] Cf. Klein, "Zur nicht-Euklidischen Geometrie," _Math. Annal._
+    vol. xxxvii.
+
+  [6] On the theory of parallels before Lobatchewsky, see Stäckel und
+    Engel, _Theorie der Parallellinien von Euklid bis auf Gauss_
+    (Leipzig, 1895). The foregoing remarks are based upon the materials
+    collected in this work.
+
+  [7] See Stäckel und Engel, _op. cit._, and "Gauss, die beiden Bolyai,
+    und die nicht-Euklidische Geometrie," _Math. Annalen_, Bd. xlix.;
+    also Engel's translation of Lobatchewsky (Leipzig, 1898), pp. 378 ff.
+
+  [8] Lobatchewsky's works on the subject are the following:--"On the
+    Foundations of Geometry," _Kazań Messenger_, 1829-1830; "New
+    Foundations of Geometry, with a complete Theory of Parallels,"
+    _Proceedings of the University of Kazań_, 1835 (both in Russian, but
+    translated into German by Engel, Leipzig, 1898); "Géométrie
+    imaginaire," Crelle's Journal, 1837; _Theorie der Parallellinien_
+    (Berlin, 1840; 2nd ed., 1887; translated by Halsted, Austin, Texas,
+    1891). His results appear to have been set forth in a paper (now
+    lost) which he read at Kazań in 1826.
+
+  [9] Translated by Halsted (Austin, Texas, 4th ed., 1896.)
+
+  [10] _Abhandlungen d. Königl. Ges. d. Wiss. zu Göttingen_, Bd. xiii.;
+    _Ges. math. Werke_, pp. 254-269; translated by Clifford, _Collected
+    Mathematical Papers_.
+
+  [11] Cf. _Gesamm. math. und phys. Werke_, vol. i. (Leipzig, 1894).
+
+  [12] _Wiss. Abh._ vol. ii. pp. 610, 618 (1866, 1868).
+
+  [13] _Mind_, O.S., vols. i. and iii.; _Vorträge und Reden_, vol. ii.
+    pp. 1, 256.
+
+  [14] His papers are "Saggio di interpretazione della geometria
+    non-Euclidea," _Giornale di matematiche_, vol. vi. (1868); "Teoria
+    fondamentale degli spazii di curvatura costante," _Annali di
+    matematica_, vol. ii. (1868-1869). Both were translated into French
+    by J. Hoüel, _Annales scientifiques de l'École Normale supérieure_,
+    vol. vi. (1869).
+
+  [15] Beltrami shows also that this definition agrees with that of
+    Gauss.
+
+  [16] "Sur la théorie des foyers," _Nouv. Ann._ vol. xii.
+
+  [17] _Math. Annalen_, iv. vi., 1871-1872.
+
+  [18] For an investigation of these and similar properties, see
+    Whitehead, _Universal Algebra_ (Cambridge, 1898), bk. vi. ch. ii. The
+    polar form was independently discovered by Simon Newcomb in 1877.
+
+  [19] For an analysis of Leibnitz's ideas on space, cf. B. Russell,
+    _The Philosophy of Leibnitz_, chs. viii.-x.
+
+  [20] Cf. Hon. Bertrand Russell, "Is Position in Time and Space
+    Absolute or Relative?" _Mind_, n.s. vol. 10 (1901), and A.N.
+    Whitehead, "Mathematical Concepts of the Material World," _Phil.
+    Trans._ (1906), p. 205.
+
+  [21] Cf. _Critique of Pure Reason_, 1st section: "Of Space,"
+    conclusion A, Max Müller's translation.
+
+  [22] Cf. Ernst Mach, _Erkenntniss und Irrtum_ (Leipzig); the relevant
+    chapters are translated by T.J. McCormack, _Space and Geometry_
+    (London, 1906); also A. Meinong, _Über die Stellung der
+    Gegenstandstheorie im System der Wissenschaften_ (Leipzig, 1907).
+
+  [23] Cf. Russell, _Principles of Mathematics_, § 352 (Cambridge,
+    1903).
+
+  [24] Cf. A.N. Whitehead, _The Axioms of Projective Geometry_, § 3
+    (Cambridge, 1906).
+
+  [25] Cf. Russell, _Princ. of Math._, ch. i.
+
+  [26] Cf. Russell, _loc. cit._, and G. Frege, "Über die Grundlagen der
+    Géométrie," _Jahresber. der Deutsch. Math. Ver._ (1906).
+
+  [27] This formulation--though not in respect to number--is in all
+    essentials that of M. Pieri, cf. "I principii della Geometria di
+    Posizione," _Accad. R. di Torino_ (1898); also cf. Whitehead, _loc.
+    cit._
+
+  [28] Cf. G. Peano, "Sui fondamenti della Geometria," p. 73, _Rivista
+    di matematica_, vol. iv. (1894), and D. Hilbert, _Grundlagen der
+    Geometrie_ (Leipzig, 1899); and R.F. Moulton, "A Simple
+    non-Desarguesian Plane Geometry," _Trans. Amer. Math. Soc._, vol.
+    iii. (1902).
+
+  [29] Cf. "Sui postulati fondamentali della geometria projettiva,"
+    _Giorn. di matematica_, vol. xxx. (1891); also of Pieri, _loc. cit._,
+    and Whitehead, _loc. cit._
+
+  [30] Cf. Hilbert, _loc. cit._; for a fuller exposition of Hilbert's
+    proof cf. K.T. Vahlen, _Abstrakte Geometrie_ (Leipzig, 1905), also
+    Whitehead, _loc. cit._
+
+  [31] Cf. H. Wiener, _Jahresber. der Deutsch. Math. Ver._ vol. i.
+    (1890); and F. Schur, "Über den Fundamentalsatz der projectiven
+    Geometrie," _Math. Ann._ vol. li. (1899).
+
+  [32] Cf. Hilbert, _loc. cit._, and Whitehead, _loc. cit._
+
+  [33] Cf. Dedekind, _Stetigkeit und irrationale Zahlen_ (1872).
+
+  [34] Cf. v. Staudt, _Geometrie der Lage_ (1847).
+
+  [35] Cf. Pasch, _Vorlesungen über neuere Geometrie_ (Leipzig, 1882),
+    a classic work; also Fiedler, _Die darstellende Geometrie_ (1st ed.,
+    1871, 3rd ed., 1888); Clebsch, _Vorlesungen über Geometrie_, vol.
+    iii.; Hilbert, _loc. cit._; F. Schur, _Math. Ann. Bd._ lv. (1902);
+    Vahlen, _loc. cit._; Whitehead, _loc. cit._
+
+  [36] Cf. _loc. cit._
+
+  [37] Cf. _I Principii di geometria_ (Turin, 1889) and "Sui fondamenti
+    della geometria," _Rivista di mat._ vol. iv. (1894).
+
+  [38] Cf. _loc. cit._
+
+  [39] Cf. Vailati, _Rivista di mat._ vol. iv. and Russell, _loc. cit._
+    § 376.
+
+  [40] Cf. O. Veblen, "On the Projective Axioms of Geometry," _Trans.
+    Amer. Math. Soc._ vol. iii. (1902).
+
+  [41] Cf. P. Stäckel and F. Engel, _Die Theorie der Parallellinien von
+    Euklid bis auf Gauss_ (Leipzig, 1895).
+
+  [42] Cf. Pasch, _loc. cit._, and R. Bonola, "Sulla introduzione degli
+    enti improprii in geometria projettive," _Giorn. di mat._ vol.
+    xxxviii. (1900); and Whitehead, _Axioms of Descriptive Geometry_
+    (Cambridge, 1907).
+
+  [43] The original idea (confined to this particular case) of ideal
+    points is due to von Staudt (_loc. cit._).
+
+  [44] Cf. _Critique_, "Trans. Aesth." Sect. I.
+
+  [45] Cf. _loc. cit._
+
+  [46] Cf. _Über die Grundlagen der Geometrie_ (Leipzig, Ber., 1890);
+    and _Theorie der Transformationsgruppen_ (Leipzig, 1893), vol. iii.
+
+  [47] Cf. A. Cayley, "A Sixth Memoir on Quantics," _Trans. Roy. Soc._,
+    1859, and _Coll. Papers_, vol. ii.; and F. Klein, _Math. Ann._ vol.
+    iv., 1871.
+
+  [48] Cf. _loc. cit._
+
+  [49] For similar deductions from a third set of axioms, suggested in
+    essence by Peano, Riv. mat. vol. iv. _loc. cit._ cf. Whitehead, _Desc.
+    Geom. loc. cit._
+
+  [50] Cf. H. Poincaré, _La Science et l'hypothčse_, ch. iii.
+
+
+
+
+
+
+
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+<pre>
+
+The Project Gutenberg EBook of Encyclopaedia Britannica, 11th Edition,
+Volume 11, Slice 6, by Various
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+Title: Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6
+       "Geodesy" to "Geometry"
+
+Author: Various
+
+Release Date: September 17, 2011 [EBook #37461]
+
+Language: English
+
+Character set encoding: ISO-8859-1
+
+*** START OF THIS PROJECT GUTENBERG EBOOK ENCYCLOPAEDIA BRITANNICA ***
+
+
+
+
+Produced by Marius Masi, Don Kretz and the Online
+Distributed Proofreading Team at http://www.pgdp.net
+
+
+
+
+
+
+</pre>
+
+
+
+<table border="0" cellpadding="10" style="background-color: #dcdcdc; color: #696969; " summary="Transcriber's note">
+<tr>
+<td style="width:25%; vertical-align:top">
+Transcriber&rsquo;s note:
+</td>
+<td class="norm">
+A few typographical errors have been corrected. They
+appear in the text <span class="correction" title="explanation will pop up">like this</span>, and the
+explanation will appear when the mouse pointer is moved over the marked
+passage. Sections in Greek will yield a transliteration
+when the pointer is moved over them, and words using diacritic characters in the
+Latin Extended Additional block, which may not display in some fonts or browsers, will
+display an unaccented version. <br /><br />
+<a name="artlinks">Links to other EB articles:</a> Links to articles residing in other EB volumes will
+be made available when the respective volumes are introduced online.
+</td>
+</tr>
+</table>
+<div style="padding-top: 3em; ">&nbsp;</div>
+
+<h2>THE ENCYCLOP&AElig;DIA BRITANNICA</h2>
+
+<h2>A DICTIONARY OF ARTS, SCIENCES, LITERATURE AND GENERAL INFORMATION</h2>
+
+<h3>ELEVENTH EDITION</h3>
+<div style="padding-top: 3em; ">&nbsp;</div>
+
+<hr class="full" />
+<h3>VOLUME XI SLICE VI<br /><br />
+Geodesy to Geometry</h3>
+<hr class="full" />
+<div style="padding-top: 3em; ">&nbsp;</div>
+
+<p class="center1" style="font-size: 150%; font-family: 'verdana';">Articles in This Slice</p>
+<table class="reg" style="width: 90%; font-size: 90%; border: gray 2px solid;" cellspacing="8" summary="Contents">
+
+<tr><td class="tcl"><a href="#ar1">GEODESY</a></td> <td class="tcl"><a href="#ar11">GEOFFROY, ÉTIENNE FRANĒOIS</a></td></tr>
+<tr><td class="tcl"><a href="#ar2">GEOFFREY</a> (Martel)</td> <td class="tcl"><a href="#ar12">GEOFFROY, JULIEN LOUIS</a></td></tr>
+<tr><td class="tcl"><a href="#ar3">GEOFFREY</a> (Plantagenet)</td> <td class="tcl"><a href="#ar13">GEOFFROY SAINT-HILAIRE, ÉTIENNE</a></td></tr>
+<tr><td class="tcl"><a href="#ar4">GEOFFREY</a> (duke of Brittany)</td> <td class="tcl"><a href="#ar14">GEOFFROY SAINT-HILAIRE, ISIDORE</a></td></tr>
+<tr><td class="tcl"><a href="#ar5">GEOFFREY</a> (archbishop of York)</td> <td class="tcl"><a href="#ar15">GEOGRAPHY</a></td></tr>
+<tr><td class="tcl"><a href="#ar6">GEOFFREY DE MONTBRAY</a></td> <td class="tcl"><a href="#ar16">GEOID</a></td></tr>
+<tr><td class="tcl"><a href="#ar7">GEOFFREY OF MONMOUTH</a></td> <td class="tcl"><a href="#ar17">GEOK-TEPE</a></td></tr>
+<tr><td class="tcl"><a href="#ar8">GEOFFREY OF PARIS</a></td> <td class="tcl"><a href="#ar18">GEOLOGY</a></td></tr>
+<tr><td class="tcl"><a href="#ar9">GEOFFREY THE BAKER</a></td> <td class="tcl"><a href="#ar19">GEOMETRICAL CONTINUITY</a></td></tr>
+<tr><td class="tcl"><a href="#ar10">GEOFFRIN, MARIE THÉRČSE RODET</a></td> <td class="tcl"><a href="#ar20">GEOMETRY</a></td></tr>
+</table>
+
+<hr class="art" />
+<p><span class="pagenum"><a name="page607" id="page607"></a>607</span></p>
+<p><span class="bold">GEODESY<a name="ar1" id="ar1"></a></span> (from the Gr. <span class="grk" title="gź">&#947;&#8134;</span>, the earth, and <span class="grk" title="daiein">&#948;&#945;&#943;&#949;&#953;&#957;</span>, to divide),
+the science of surveying (<i>q.v.</i>) extended to large tracts of country,
+having in view not only the production of a system of maps of
+very great accuracy, but the determination of the curvature of
+the surface of the earth, and eventually of the figure and
+dimensions of the earth. This last, indeed, may be the sole
+object in view, as was the case in the operations conducted in
+Peru and in Lapland by the celebrated French astronomers
+P. Bouguer, C.M. de la Condamine, P.L.M. de Maupertuis,
+A.C. Clairault and others; and the measurement of the meridian
+<span class="pagenum"><a name="page608" id="page608"></a>608</span>
+arc of France by P.F.A. Méchain and J.B.J. Delambre had
+for its end the determination of the true length of the &ldquo;metre&rdquo;
+which was to be the legal standard of length of France (see
+<span class="sc"><a href="#artlinks">Earth, Figure of the</a></span>).</p>
+
+<p>The basis of every extensive survey is an accurate triangulation,
+and the operations of geodesy consist in the measurement, by
+theodolites, of the angles of the triangles; the measurement of
+one or more sides of these triangles on the ground; the determination
+by astronomical observations of the azimuth of the whole
+network of triangles; the determination of the actual position
+of the same on the surface of the earth by observations, first for
+latitude at some of the stations, and secondly for longitude; the
+determination of altitude for all stations.</p>
+
+<p>For the computation, the points of the actual surface of the
+earth are imagined as projected along their plumb lines on the
+mathematical figure, which is given by the stationary sea-level,
+and the extension of the sea through the continents by a system
+of imaginary canals. For many purposes the mathematical
+surface is assumed to be a plane; in other cases a sphere of
+radius 6371 kilometres (20,900,000 ft.). In the case of extensive
+operations the surface must be considered as a compressed
+ellipsoid of rotation, whose minor axis coincides with the earth&rsquo;s
+axis, and whose compression, flattening, or ellipticity is about
+1/298.</p>
+
+<p class="pt2 center"><i>Measurement of Base Lines.</i></p>
+
+<div class="condensed">
+<p>To determine by actual measurement on the ground the length of a
+side of one of the triangles (&ldquo;base line&rdquo;), wherefrom to infer the
+lengths of all the other sides in the triangulation, is not the least
+difficult operation of a trigonometrical survey. When the problem
+is stated thus&mdash;To determine the number of times that a certain
+standard or unit of length is contained between two finely marked
+points on the surface of the earth at a distance of some miles asunder,
+so that the error of the result may be pronounced to lie between
+certain very narrow limits,&mdash;then the question demands very
+serious consideration. The representation of the unit of length by
+means of the distance between two fine lines on the surface of a bar
+of metal at a certain temperature is never itself free from uncertainty
+and probable error, owing to the difficulty of knowing at any moment
+the precise temperature of the bar; and the transference of this
+unit, or a multiple of it, to a measuring bar will be affected not
+only with errors of observation, but with errors arising from uncertainty
+of temperature of both bars. If the measuring bar be not
+self-compensating for temperature, its expansion must be determined
+by very careful experiments. The thermometers required for this
+purpose must be very carefully studied, and their errors of division
+and index error determined.</p>
+
+<p>In order to avoid the difficulty in exactly determining the temperature
+of a bar by the mercury thermometer, F.W. Bessel introduced
+in 1834 near Königsberg a compound bar which constituted a
+metallic thermometer.<a name="fa1a" id="fa1a" href="#ft1a"><span class="sp">1</span></a> A zinc bar is laid on an iron bar two toises
+long, both bars being perfectly planed and in free contact, the zinc
+bar being slightly shorter and the two bars rigidly united at one end.
+As the temperature varies, the difference of the lengths of the bars,
+as perceived by the other end, also varies, and affords a quantitative
+correction for temperature variations, which is applied to reduce the
+length to standard temperature. During the measurement of the
+base line the bars were not allowed to come into contact, the interval
+being measured by the insertion of glass wedges. The results of the
+comparisons of four measuring rods with one another and with the
+standards were elaborately computed by the method of least-squares.
+The probable error of the measured length of 935 toises (about
+6000 ft.) has been estimated as 1/863500 or 1.2 &mu; (&mu; denoting a
+millionth). With this apparatus fourteen base lines were measured
+in Prussia and some neighbouring states; in these cases a somewhat
+higher degree of accuracy was obtained.</p>
+
+<p>The principal triangulation of Great Britain and Ireland has seven
+base lines: five have been measured by steel chains, and two,
+more exactly, by the compensation bars of General T.F. Colby, an
+apparatus introduced in 1827-1828 at Lough Foyle in Ireland. Ten
+base lines were measured in India in 1831-1869 by the same apparatus.
+This is a system of six compound-bars self-correcting for temperature.
+The bars may be thus described: Two bars, one of brass and the
+other of iron, are laid in parallelism side by side, firmly united at
+their centres, from which they may freely expand or contract; at
+the standard temperature they are of the same length. Let AB be
+one bar, A&prime;B&prime; the other; draw lines through the corresponding
+extremities AA&prime; (to P) and BB&prime; (to Q), and make A&prime;P = B&prime;Q, AA&prime;
+being equal to BB&prime;. If the ratio A&prime;P/AP equals the ratio of the coefficients
+of expansion of the bars A&prime;B&prime; and AB, then, obviously,
+the distance PQ is constant (or nearly so). In the actual instrument
+P and Q are finely engraved dots 10 ft. apart. In practice the bars,
+when aligned, are not in contact, an interval of 6 in. being allowed
+between each bar and its neighbour. This distance is accurately
+measured by an ingenious micrometrical arrangement constructed
+on exactly the same principle as the bars themselves.</p>
+
+<p>The last base line measured in India had a length of 8913 ft. In
+consequence of some suspicion as to the accuracy of the compensation
+apparatus, the measurement was repeated four times, the operations
+being conducted so as to determine the actual values of the probable
+errors of the apparatus. The direction of the line (which is at Cape
+Comorin) is north and south. In two of the measurements the brass
+component was to the west, in the others to the east; the differences
+between the individual measurements and the mean of the four were
++0.0017, &minus;0.0049, &minus;0.0015, +0.0045 ft. These differences are
+very small; an elaborate investigation of all sources of error shows
+that the probable error of a base line in India is on the average
+±2.8 &mu;. These compensation bars were also used by Sir Thomas
+Maclear in the measurement of the base line in his extension of
+Lacaille&rsquo;s arc at the Cape. The account of this operation will be
+found in a volume entitled <i>Verification and Extension of Lacaille&rsquo;s
+Arc of Meridian at the Cape of Good Hope</i>, by Sir Thomas Maclear,
+published in 1866. A rediscussion has been given by Sir David
+Gill in his <i>Report on the Geodetic Survey of South Africa, &amp;c., 1896</i>.</p>
+
+<p>A very simple base apparatus was employed by W. Struve in his
+triangulations in Russia from 1817 to 1855. This consisted of four
+wrought-iron bars, each two toises (rather more than 13 ft.) long;
+one end of each bar is terminated in a small steel cylinder presenting
+a slightly convex surface for contact, the other end carries a contact
+lever rigidly connected with the bar. The shorter arm of the lever
+terminates below in a polished hemisphere, the upper and longer
+arm traversing a vertical divided arc. In measuring, the plane end
+of one bar is brought into contact with the short arm of the contact
+lever (pushed forward by a weak spring) of the next bar. Each bar
+has two thermometers, and a level for determining the inclination
+of the bar in measuring. The manner of transferring the end of a
+bar to the ground is simply this: under the end of the bar a stake
+is driven very firmly into the ground, carrying on its upper surface
+a disk, capable of movement in the direction of the measured line
+by means of slow-motion screws. A fine mark on this disk is
+brought vertically under the end of the bar by means of a theodolite
+which is planted at a distance of 25 ft. from the stake in a direction
+perpendicular to the base. Struve investigated for each base the
+probable errors of the measurement arising from each of these seven
+causes: Alignment, inclination, comparisons with standards, readings
+of index, personal errors, uncertainties of temperature, and the
+probable errors of adopted rates of expansion. He found that
+±0.8 &mu; was the mean of the probable errors of the seven bases
+measured by him. The Austro-Hungarian apparatus is similar;
+the distance of the rods is measured by a slider, which rests on one
+of the ends of each rod. Twenty-two base lines were measured in
+1840-1899.</p>
+
+<p>General Carlos Ibańez employed in 1858-1879, for the measurement
+of nine base lines in Spain, two apparatus similar to the
+apparatus previously employed by Porro in Italy; one is complicated,
+the other simplified. The first, an apparatus of the brothers Brunner
+of Paris, was a thermometric combination of two bars, one of platinum
+and one of brass, in length 4 metres, furnished with three levels and
+four thermometers. Suppose A, B, C three micrometer microscopes
+very firmly supported at intervals of 4 metres with their axes vertical,
+and aligned in the plane of the base line by means of a transit
+instrument, their micrometer screws being in the line of measurement.
+The measuring bar is brought under say A and B, and those micrometers
+read; the bar is then shifted and brought under B and C. By
+repetition of this process, the reading of a micrometer indicating the
+end of each position of the bar, the measurement is made.</p>
+
+<p>Quite similar apparatus (among others) has been employed by the
+French and Germans. Since, however, it only permitted a distance
+of about 300 m. to be measured daily, Ibańez introduced a simplification;
+the measuring rod being made simply of steel, and provided
+with inlaid mercury thermometers. This apparatus was used in
+Switzerland for the measurement of three base lines. The accuracy
+is shown by the estimated probable errors: ±0.2 &mu; to ±0.8 &mu;.
+The distance measured daily amounts at least to 800 m.</p>
+
+<p>A greater daily distance can be measured with the same accuracy
+by means of Bessel&rsquo;s apparatus; this permits the ready measurement
+of 2000 m. daily. For this, however, it is important to notice
+that a large staff and favourable ground are necessary. An important
+improvement was introduced by Edward Jäderin of Stockholm,
+who measures with stretched wires of about 24 metres long;
+these wires are about 1.65 mm. in diameter, and when in use are
+stretched by an accurate spring balance with a tension of 10 kg.<a name="fa2a" id="fa2a" href="#ft2a"><span class="sp">2</span></a>
+The nature of the ground has a very trifling effect on this method.
+The difficulty of temperature determinations is removed by employing
+wires made of invar, an alloy of steel (64%) and nickel (36%)
+which has practically no linear expansion for small thermal changes
+<span class="pagenum"><a name="page609" id="page609"></a>609</span>
+at ordinary temperatures; this alloy was discovered in 1896 by
+Benōit and Guillaume of the International Bureau of Weights and
+Measures at Breteuil. Apparently the future of base-line measurements
+rests with the invar wires of the Jäderin apparatus; next
+comes Porro&rsquo;s apparatus with invar bars 4 to 5 metres long.</p>
+
+<p>Results have been obtained in the United States, of great importance
+in view of their accuracy, rapidity of determination and
+economy. For the measurement of the arc of meridian in longitude
+98° E., in 1900, nine base lines of a total length of 69.2 km. were
+measured in six months. The total cost of one base was $1231.
+At the beginning and at the end of the field-season a distance of
+exactly 100 m. was measured with R.S. Woodward&rsquo;s &ldquo;5-m. ice-bar&rdquo;
+(invented in 1891); by means of the remeasurement of this
+length the standardization of the apparatus was done under the same
+conditions as existed in the case of the base measurements. For
+the measurements there were employed two steel tapes of 100 m.
+long, provided with supports at distances of 25 m., two of 50 m.,
+and the duplex apparatus of Eimbeck, consisting of four 5-m. rods.
+Each base was divided into sections of about 1000 m.; one of these,
+the &ldquo;test kilometre,&rdquo; was measured with all the five apparatus,
+the others only with two apparatus, mostly tapes. The probable
+error was about ±0.8 &mu;, and the day&rsquo;s work a distance of about
+2000 m. Each of the four rods of the duplex apparatus consists of
+two bars of brass and steel. Mercury thermometers are inserted
+in both bars; these serve for the measurement of the length of the
+base lines by each of the bars, as they are brought into their consecutive
+positions, the contact being made by an elastic-sliding
+contact. The length of the base lines may be calculated for each
+bar only, and also by the supposition that both bars have the same
+temperature. The apparatus thus affords three sets of results,
+which mutually control themselves, and the contact adjustments
+permit rapid work. The same device has been applied to the older
+bimetallic-compensating apparatus of Bache-Würdemann (six
+bases, 1847-1857) and of Schott. There was also employed a single
+rod bimetallic apparatus on F. Porro&rsquo;s principle, constructed by the
+brothers Repsold for some base lines. Excellent results have been
+more recently obtained with invar tapes.</p>
+
+<p>The following results show the lengths of the same German base
+lines as measured by different apparatus:</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcl" colspan="4">&nbsp;</td> <td class="tcc">metres.</td></tr>
+<tr><td class="tcl">Base at Berlin</td> <td class="tcc">1864</td> <td class="tcc">Apparatus of</td> <td class="tcl">Bessel</td> <td class="tcr">2336·3920</td></tr>
+<tr><td class="tcl"> &emsp;&emsp; &rdquo; &emsp;&emsp; &rdquo;</td> <td class="tcc">1880</td> <td class="tcc">&rdquo;</td> <td class="tcl">Brunner</td> <td class="tcr">·3924</td></tr>
+<tr><td class="tcl">Base at Strehlen</td> <td class="tcc">1854</td> <td class="tcc">&rdquo;</td> <td class="tcl">Bessel</td> <td class="tcr">2762·5824</td></tr>
+<tr><td class="tcl"> &emsp;&emsp; &rdquo; &emsp;&emsp; &rdquo;</td> <td class="tcc">1879</td> <td class="tcc">&rdquo;</td> <td class="tcl">Brunner</td> <td class="tcr">·5852</td></tr>
+<tr><td class="tcl">Old base at Bonn</td> <td class="tcc">1847</td> <td class="tcc">&rdquo;</td> <td class="tcl">Bessel</td> <td class="tcr">2133·9095</td></tr>
+<tr><td class="tcl"> &emsp;&emsp; &rdquo; &emsp;&emsp; &rdquo;</td> <td class="tcc">1892</td> <td class="tcc">&rdquo;</td> <td class="tcc">&rdquo;</td> <td class="tcr">·9097</td></tr>
+<tr><td class="tcl">New base at Bonn</td> <td class="tcc">1892</td> <td class="tcc">&rdquo;</td> <td class="tcc">&rdquo;</td> <td class="tcr">2512·9612</td></tr>
+<tr><td class="tcl"> &emsp;&emsp; &rdquo; &emsp;&emsp; &rdquo;</td> <td class="tcc">1892</td> <td class="tcc">&rdquo;</td> <td class="tcl">Brunner</td> <td class="tcr">·9696</td></tr>
+</table>
+
+<p>It is necessary that the altitude above the level of the sea of every
+part of a base line be ascertained by spirit levelling, in order that
+the measured length may be reduced to what it would have been
+had the measurement been made on the surface of the sea, produced
+in imagination. Thus if l be the length of a measuring bar, h its
+height at any given position in the measurement, r the radius of
+the earth, then the length radially projected on to the level of the
+sea is l(1 &minus; h/r). In the Salisbury Plain base line the reduction to
+the level of the sea is &minus;0.6294 ft.</p>
+
+<table class="flt" style="float: right; width: 250px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:203px; height:347px" src="images/img609.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 1.</span></td></tr></table>
+
+<p>The total number of base lines measured in Europe up to the
+present time is about one hundred and ten, nineteen of which do
+not exceed in length 2500 metres, or about 1½ miles, and three&mdash;one
+in France, the others in Bavaria&mdash;exceed
+19,000 metres. The question
+has been frequently discussed whether
+or not the advantage of a long base is
+sufficiently great to warrant the expenditure
+of time that it requires, or
+whether as much precision is not obtainable
+in the end by careful triangulation
+from a short base. But the answer
+cannot be given generally; it must
+depend on the circumstances of each
+particular case. With Jäderin&rsquo;s apparatus,
+provided with invar wires, bases
+of 20 to 30 km. long are obtained without
+difficulty.</p>
+
+<p>In working away from a base line ab,
+stations c, d, e, f are carefully selected so
+as to obtain from well-shaped triangles
+gradually increasing sides. Before, however,
+finally leaving the base line, it is
+usual to verify it by triangulation thus:
+during the measurement two or more
+points, as p, q (fig. 1), are marked in the
+base in positions such that the lengths of
+the different segments of the line are
+known; then, taking suitable external stations, as h, k, the angles of
+the triangles bhp, phq, hqk, kqa are measured. From these angles
+can be computed the ratios of the segments, which must agree, if all
+operations are correctly performed, with the ratios resulting from
+the measures. Leaving the base line, the sides increase up to 10,
+30 or 50 miles occasionally, but seldom reaching 100 miles. The
+triangulation points may either be natural objects presenting themselves
+in suitable positions, such as church towers; or they may be
+objects specially constructed in stone or wood on mountain tops
+or other prominent ground. In every case it is necessary that the
+precise centre of the station be marked by some permanent mark.
+In India no expense is spared in making permanent the principal
+trigonometrical stations&mdash;costly towers in masonry being erected.
+It is essential that every trigonometrical station shall present a fine
+object for observation from surrounding stations.</p>
+
+<p class="pt2 center"><i>Horizontal Angles.</i></p>
+
+<p>In placing the theodolite over a station to be observed from, the
+first point to be attended to is that it shall rest upon a perfectly
+solid foundation. The method of obtaining this desideratum must
+depend entirely on the nature of the ground; the instrument must
+if possible be supported on rock, or if that be impossible a solid
+foundation must be obtained by digging. When the theodolite is
+required to be raised above the surface of the ground in order to
+command particular points, it is necessary to build two scaffolds,&mdash;the
+outer one to carry the observatory, the inner one to carry the
+instrument,&mdash;and these two edifices must have no point of contact.
+Many cases of high scaffolding have occurred on the English Ordnance
+Survey, as for instance at Thaxted church, where the tower, 80 ft.
+high, is surmounted by a spire of 90 ft. The scaffold for the observatory
+was carried from the base to the top of the spire; that
+for the instrument was raised from a point of the spire 140 ft. above
+the ground, having its bearing upon timbers passing through the
+spire at that height. Thus the instrument, at a height of 178 ft.
+above the ground, was insulated, and not affected by the action of
+the wind on the observatory.</p>
+
+<p>At every station it is necessary to examine and correct the adjustments
+of the theodolite, which are these: the line of collimation
+of the telescope must be perpendicular to its axis of rotation; this
+axis perpendicular to the vertical axis of the instrument; and the
+latter perpendicular to the plane of the horizon. The micrometer
+microscopes must also measure correct quantities on the divided
+circle or circles. The method of observing is this. Let A, B, C ...
+be the stations to be observed taken in order of azimuth; the
+telescope is first directed to A and the cross-hairs of the telescope
+made to bisect the object presented by A, then the microscopes or
+verniers of the horizontal circle (also of the vertical circle if necessary)
+are read and recorded. The telescope is then turned to B, which
+is observed in the same manner; then C and the other stations.
+Coming round by continuous motion to A, it is again observed, and
+the agreement of this second reading with the first is some test of
+the stability of the instrument. In taking this round of angles&mdash;or
+&ldquo;arc,&rdquo; as it is called on the Ordnance Survey&mdash;it is desirable
+that the interval of time between the first and second observations
+of A should be as small as may be consistent with due care. Before
+taking the next arc the horizontal circle is moved through 20° or
+30°; thus a different set of divisions of the circle is used in each
+arc, which tends to eliminate the errors of division.</p>
+
+<p>It is very desirable that all arcs at a station should contain one
+point in common, to which all angular measurements are thus
+referred,&mdash;the observations on each arc commencing and ending
+with this point, which is on the Ordnance Survey called the &ldquo;referring
+object.&rdquo; It is usual for this purpose to select, from among the
+points which have to be observed, that one which affords the best
+object for precise observation. For mountain tops a &ldquo;referring
+object&rdquo; is constructed of two rectangular plates of metal in the
+same vertical plane, their edges parallel and placed at such a distance
+apart that the light of the sky seen through appears as a vertical line
+about 10&Prime; in width. The best distance for this object is from
+1 to 2 miles.</p>
+
+<p>This method seems at first sight very advantageous; but if,
+however, it be desired to attain the highest accuracy, it is better,
+as shown by General Schreiber of Berlin in 1878, to measure only
+single angles, and as many of these as possible between the directions
+to be determined. Division-errors are thus more perfectly eliminated,
+and errors due to the variation in the stability, &amp;c., of the instruments
+are diminished. This method is rapidly gaining precedence.</p>
+
+<p>The theodolites used in geodesy vary in pattern and in size&mdash;the
+horizontal circles ranging from 10 in. to 36 in. in diameter. In
+Ramsden&rsquo;s 36-in. theodolite the telescope has a focal length of
+36 in. and an aperture of 2.5 in., the ordinarily used magnifying
+power being 54; this last, however, can of course be changed at the
+requirements of the observer or of the weather. The probable
+error of a single observation of a fine object with this theodolite
+is about 0&Prime;.2. Fig. 2 represents an altazimuth theodolite of an
+improved pattern used on the Ordnance Survey. The horizontal
+circle of 14-in. diameter is read by three micrometer microscopes;
+the vertical circle has a diameter of 12 in., and is read by two microscopes.
+In the great trigonometrical survey of India the theodolites
+used in the more important parts of the work have been of 2 and
+3 ft. diameter&mdash;the circle read by five equidistant microscopes.
+Every angle is measured twice in each position of the zero of the
+horizontal circle, of which there are generally ten; the entire
+<span class="pagenum"><a name="page610" id="page610"></a>610</span>
+number of measures of an angle is never less than 20. An examination
+of 1407 angles showed that the probable error of an observed
+angle is on the average ±0&Prime;.28.</p>
+
+<p>For the observations of very distant stations it is usual to employ
+a heliotrope (from the Gr. <span class="grk" title="hźlios">&#7973;&#955;&#953;&#959;&#962;</span>, sun; <span class="grk" title="tropos">&#964;&#961;&#972;&#960;&#959;&#962;</span>, a turn), invented by
+Gauss at Göttingen in 1821. In its simplest form this is a plane
+mirror, 4, 6, or 8 in. in diameter, capable of rotation round a horizontal
+and a vertical axis. This mirror is placed at the station to be observed,
+and in fine weather it is kept so directed that the rays of the
+sun reflected by it strike the distant observing telescope. To the
+observer the heliotrope presents the appearance of a star of the
+first or second magnitude, and is generally a pleasant object for
+observing.</p>
+
+<p>Observations at night, with the aid of light-signals, have been
+repeatedly made, and with good results, particularly in France
+by General Franēois Perrier, and more recently in the United
+States by the Coast and Geodetic Survey; the signal employed
+being an acetylene bicycle-lamp, with a lens 5 in. in diameter.
+Particularly noteworthy are the trigonometrical connexions of
+Spain and Algeria, which were carried out in 1879 by Generals
+Ibańez and Perrier (over a distance of 270 km.), of Sicily and Malta
+in 1900, and of the islands of Elba and Sardinia in 1902 by Dr
+Guarducci (over distances up to 230 km.); in these cases artificial
+light was employed: in the first case electric light and in the two
+others acetylene lamps.</p>
+
+<table class="nobctr" style="clear: both;" summary="Illustration">
+<tr><td class="figcenter"><img style="width:431px; height:692px" src="images/img610a.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 2.</span>&mdash;Altazimuth Theodolite.</td></tr></table>
+
+<p class="pt2 center"><i>Astronomical Observations.</i></p>
+
+<p>The direction of the meridian is determined either by a theodolite
+or a portable transit instrument. In the former case the operation
+consists in observing the angle between a terrestrial object&mdash;generally
+a mark specially erected and capable of illumination at night&mdash;and
+a close circumpolar star at its greatest eastern or western
+azimuth, or, at any rate, when very near that position. If the
+observation be made t minutes of time before or after the time of
+greatest azimuth, the azimuth then will differ from its maximum
+value by (450t)² sin 1&Prime; sin 2&delta;/sin z, in seconds of angle, omitting
+smaller terms, &delta; being the star&rsquo;s declination and z its zenith distance.
+The collimation and level errors are very carefully determined
+before and after these observations, and it is usual to arrange the
+observations by the reversal of the telescope so that collimation
+error shall disappear. If b, c be the level and collimation errors,
+the correction to the circle reading is b cot z ± c cosec z, b being
+positive when the west end of the axis is high. It is clear that any
+uncertainty as to the real state of the level will produce a corresponding
+uncertainty in the resulting value of the azimuth,&mdash;an
+uncertainty which increases with the latitude and is very large
+in high latitudes. This may be partly remedied by observing in
+connexion with the star its reflection in mercury. In determining
+the value of &ldquo;one division&rdquo; of a level tube, it is necessary to bear
+in mind that in some the value varies considerably with the temperature.
+By experiments on the level of Ramsden&rsquo;s 3-foot theodolite,
+it was found that though at the ordinary temperature of 66° the
+value of a division was about one second, yet at 32° it was about
+five seconds.</p>
+
+<p>In a very excellent portable transit used on the Ordnance Survey,
+the uprights carrying the telescope are constructed of mahogany,
+each upright being built of several pieces glued and screwed together;
+the base, which is a solid and heavy plate of iron, carries a reversing
+apparatus for lifting the telescope out of its bearings, reversing it
+and letting it down again. Thus is avoided the change of temperature
+which the telescope would incur by being lifted by the hands
+of the observer. Another form of transit is the German diagonal
+form, in which the rays of light after passing through the object-glass
+are turned by a total reflection prism through one of the transverse
+arms of the telescope, at the extremity of which arm is the
+eye-piece. The unused half of the ordinary telescope being cut away
+is replaced by a counterpoise. In this instrument there is the
+advantage that the observer without moving the position of his eye
+commands the whole meridian, and that the level may remain on
+the pivots whatever be the elevation of the telescope. But there is
+the disadvantage that the flexure of the transverse axis causes a
+variable collimation error depending on the zenith distance of the
+star to which it is directed; and moreover it has been found that in
+some cases the personal error of an observer is not the same in the
+two positions of the telescope.</p>
+
+<p>To determine the direction of the meridian, it is well to erect two
+marks at nearly equal angular distances on either side of the north
+meridian line, so that the pole star crosses the vertical of each mark
+a short time before and after attaining its greatest eastern and
+western azimuths.</p>
+
+<p>If now the instrument, perfectly levelled, is adjusted to have its
+centre wire on one of the marks, then when elevated to the star,
+the star will traverse the wire, and its exact position in the field at
+any moment can be measured by the micrometer wire. Alternate
+observations of the star and the terrestrial mark, combined with
+careful level readings and reversals of the instrument, will enable
+one, even with only one mark, to determine the direction of the
+meridian in the course of an hour with a probable error of less than
+a second. The second mark enables one to complete the station
+more rapidly and gives a check upon the work. As an instance,
+at Findlay Seat, in latitude 57° 35&prime;, the resulting azimuths of the
+two marks were 177° 45&prime; 37&Prime;.29 ± 0&Prime;.20 and 182° 17&prime; 15&Prime;.61 ± 0&Prime;.13,
+while the angle between the two marks directly measured by a
+theodolite was found to be 4° 31&prime; 37&Prime;.43 ± 0&Prime;.23.</p>
+
+<table class="flt" style="float: right; width: 260px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:208px; height:207px" src="images/img610b.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 3.</span></td></tr></table>
+
+<p>We now come to the consideration of the determination of time
+with the transit instrument. Let fig. 3 represent the sphere stereographically
+projected on the plane of
+the horizon,&mdash;ns being the meridian,
+we the prime vertical, Z, P the zenith
+and the pole. Let p be the point in
+which the production of the axis of
+the instrument meets the celestial
+sphere, S the position of a star when
+observed on a wire whose distance
+from the collimation centre is c. Let
+a be the azimuthal deviation, namely,
+the angle wZp, b the level error so
+that Zp = 90° &minus; b. Let also the hour
+angle corresponding to p be 90° &minus; n,
+and the declination of the same = m,
+the star&rsquo;s declination being &delta;, and the
+latitude &phi;. Then to find the hour
+angle ZPS = &tau; of the star when observed, in the triangles pPS, pPZ
+we have, since pPS = 90 + &tau; &minus; n,</p>
+
+<table class="reg" style="clear: both;" summary="poem"><tr><td> <div class="poemr">
+<p> &emsp;&ensp; &minus; Sin c = sin m sin &delta; + cos m cos &delta; sin (n &minus; &tau;),</p>
+<p> &emsp;&emsp; Sin m = sin b sin &phi; &minus; cos b cos &phi; sin a,</p>
+<p>Cos m sin n = sin b cos &phi; + cos b sin &phi; sin a.</p>
+</div> </td></tr></table>
+
+<p class="noind">And these equations solve the problem, however large be the errors
+of the instrument. Supposing, as usual, a, b, m, n to be small,
+we have at once &tau; = n + c sec &delta; + m tan &delta;, which is the correction to
+the observed time of transit. Or, eliminating m and n by means
+of the second and third equations, and putting z for the zenith
+distance of the star, t for the observed time of transit, the corrected
+time is t + (a sin z + b cos z + c) / cos &delta;. Another very convenient form
+for stars near the zenith is &tau; = b sec &phi; + c sec &delta; + m (tan &delta; &minus; tan &phi;).</p>
+
+<p>Suppose that in commencing to observe at a station the error of the
+chronometer is not known; then having secured for the instrument
+a very solid foundation, removed as far as possible level and collimation
+errors, and placed it by estimation nearly in the meridian,
+let two stars differing considerably in declination be observed&mdash;the
+instrument not being reversed between them. From these two
+stars, neither of which should be a close circumpolar star, a good
+approximation to the chronometer error can be obtained; thus
+<span class="pagenum"><a name="page611" id="page611"></a>611</span>
+let &epsilon;<span class="su">1</span>, &epsilon;<span class="su">2</span>, be the apparent clock errors given by these stars if &delta;<span class="su">1</span>, &delta;<span class="su">2</span>
+be their declinations the real error is</p>
+
+<p class="center">&epsilon; = &epsilon;<span class="su">1</span> + (&epsilon;<span class="su">1</span> &minus; &epsilon;<span class="su">2</span>) (tan &phi; &minus; tan &delta;<span class="su">1</span>) / (tan &delta;<span class="su">1</span> &minus; tan &delta;<span class="su">2</span>).</p>
+
+<p class="noind">Of course this is still only approximate, but it will enable the observer
+(who by the help of a table of natural tangents can compute &epsilon; in a
+few minutes) to find the meridian by placing at the proper time,
+which he now knows approximately, the centre wire of his instrument
+on the first star that passes&mdash;not near the zenith.</p>
+
+<p>The transit instrument is always reversed at least once in the
+course of an evening&rsquo;s observing, the level being frequently read and
+recorded. It is necessary in most instruments to add a correction
+for the difference in size of the pivots.</p>
+
+<p>The transit instrument is also used in the prime vertical for the
+determination of latitudes. In the preceding figure let q be the point
+in which the northern extremity of the axis of the instrument
+produced meets the celestial sphere. Let nZq be the azimuthal
+deviation = a, and b being the level error, Zq = 90° &minus; b; let also
+nPq = &tau; and Pq = &psi;. Let S&prime; be the position of a star when observed
+on a wire whose distance from the collimation centre is c, positive
+when to the south, and let h be the observed hour angle of the star,
+viz. ZPS&prime;. Then the triangles qPS&prime;, gPZ give</p>
+
+<table class="reg" style="clear: both;" summary="poem"><tr><td> <div class="poemr">
+<p> &emsp;&ensp; &minus;Sin c = sin &delta; cos &psi; &minus; cos &delta; sin &psi; cos (h + &tau;),</p>
+<p> &emsp;&emsp; Cos &psi; = sin b sin &phi; + cos b cos &phi; cos a,</p>
+<p>Sin &psi; sin &tau; = cos b sin a.</p>
+</div> </td></tr></table>
+
+<p>Now when a and b are very small, we see from the last two equations
+that &psi; = &phi; &minus; b, a = &tau; sin &psi;, and if we calculate &phi;&prime; by the formula
+cot &phi;&prime; = cot &delta; cos h, the first equation leads us to this result&mdash;</p>
+
+<p class="center">&phi; = &phi;&prime; + (a sin z + b cos z + c) / cos z,</p>
+
+<p class="noind">the correction for instrumental error being very similar to that
+applied to the observed time of transit in the case of meridian
+observations. When a is not very small and z is small, the formulae
+required are more complicated.</p>
+
+<table class="flt" style="float: right; width: 360px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:312px; height:644px" src="images/img611.jpg" alt="" /></td></tr>
+<tr><td class="caption1"><span class="sc">Fig. 4.</span>&mdash;Zenith Telescope constructed
+for the International Stations at Mizusawa, Carloforte, Gaithersburg and
+Ukiah, by Hermann Wanschaff, Berlin.</td></tr></table>
+
+<p>The method of determining latitude by transits in the prime
+vertical has the disadvantage of being a somewhat slow process,
+and of requiring a very precise knowledge of the time, a disadvantage
+from which the zenith telescope is free. In principle this instrument
+is based on the proposition
+that when the meridian
+zenith distances of
+two stars at their upper
+culminations&mdash;one being
+to the north and the other
+to the south of the zenith&mdash;are
+equal, the latitude
+is the mean of their
+declinations; or, if the
+zenith distance of a star
+culminating to the south
+of the zenith be Z, its
+declination being &delta;, and
+that of another culminating
+to the north with
+zenith distance Z&prime; and
+declination &delta;&prime;, then clearly
+the latitude is ½(&delta; + &delta;&prime;) +
+½(Z &minus; Z&prime;). Now the zenith
+telescope does away with
+the divided circle, and
+substitutes the measurement
+micrometrically of
+the quantity Z&prime; &minus; Z.</p>
+
+<p>In fig. 4 is shown a
+zenith telescope by H.
+Wanschaff of Berlin,
+which is the type used
+(according to the Central
+Bureau at Potsdam) since
+about 1890 for the determination
+of the variations
+of latitude due to different,
+but as yet imperfectly
+understood, influences.
+The instrument is supported
+on a strong tripod,
+fitted with levelling
+screws; to this tripod is
+fixed the azimuth circle
+and a long vertical steel
+axis. Fitting on this axis
+is a hollow axis which
+carries on its upper end a
+short transverse horizontal
+axis with a level. This
+latter carries the telescope, which, supported at the centre of its
+length, is free to rotate in a vertical plane. The telescope is thus
+mounted eccentrically with respect to the vertical axis around
+which it revolves. Two extremely sensitive levels are attached to
+the telescope, which latter carries a micrometer in its eye-piece,
+with a screw of long range for measuring differences of zenith distance.
+Two levels are employed for controlling and increasing the
+accuracy. For this instrument stars are selected in pairs, passing
+north and south of the zenith, culminating within a few minutes
+of time and within about twenty minutes (angular) of zenith distance
+of each other. When a pair of stars is to be observed, the
+telescope is set to the mean of the zenith distances and in the plane
+of the meridian. The first star on passing the central meridional
+wire is bisected by the micrometer; then the telescope is rotated
+very carefully through 180° round the vertical axis, and the second
+star on passing through the field is bisected by the micrometer on
+the centre wire. The micrometer has thus measured the difference
+of the zenith distances, and the calculation to get the latitude is
+most simple. Of course it is necessary to read the level, and the
+observations are not necessarily confined to the centre wire. In
+fact if n, s be the north and south readings of the level for the south
+star, n&prime;, s&prime; the same for the north star, l the value of one division
+of the level, m the value of one division of the micrometer, r, r&prime; the
+refraction corrections, &mu;, &mu;&prime; the micrometer readings of the south
+and north star, the micrometer being supposed to read from the
+zenith, then, supposing the observation made on the centre wire,&mdash;</p>
+
+<p class="center">&phi; = ½ (&delta; + &delta;&prime;) + ½ (&mu; &minus; mu&prime;)m + ¼ (n + n&prime; &minus; s &minus; s&prime;)l + ½ (r &minus; r&prime;).</p>
+
+<p>It is of course of the highest importance that the value m of the
+screw be well determined. This is done most effectually by observing
+the vertical movement of a close circumpolar star when at its greatest
+azimuth.</p>
+
+<p>In a single night with this instrument a very accurate result,
+say with a probable error of about 0&Prime;.2, could be obtained for
+latitude from, say, twenty pair of stars; but when the latitude is
+required to be obtained with the highest possible precision, two
+nights at least are necessary. The weak point of the zenith telescope
+lies in the circumstance that its requirements prevent the selection
+of stars whose positions are well fixed; very frequently it is necessary
+to have the declinations of the stars selected for this instrument
+specially observed at fixed observatories. The zenith telescope is
+made in various sizes from 30 to 54 in. in focal length; a 30-in.
+telescope is sufficient for the highest purposes and is very portable.
+The net observation probable-error for one pair of stars is only
+±0&Prime;.1.</p>
+
+<p>The zenith telescope is a particularly pleasant instrument to
+work with, and an observer has been known (a sergeant of Royal
+Engineers, on one occasion) to take every star in his list during
+eleven hours on a stretch, namely, from 6 o&rsquo;clock <span class="scs">P.M.</span> until 5 <span class="scs">A.M.</span>,
+and this on a very cold November night on one of the highest points
+of the Grampians. Observers accustomed to geodetic operations
+attain considerable powers of endurance. Shortly after the commencement
+of the observations on one of the hills in the Isle of Skye
+a storm carried away the wooden houses of the men and left the
+observatory roofless. Three observatory roofs were subsequently
+demolished, and for some time the observatory was used without a
+roof, being filled with snow every night and emptied every morning.
+Quite different, however, was the experience of the same party when
+on the top of Ben Nevis, 4406 ft. high. For about a fortnight the
+state of the atmosphere was unusually calm, so much so, that a
+lighted candle could often be carried between the tents of the men
+and the observatory, whilst at the foot of the hill the weather was
+wild and stormy.</p>
+
+<p>The determination of the difference of longitude between two
+stations A and B resolves itself into the determination of the local
+time at each of the stations, and the comparison by signals of the
+clocks at A and B. Whenever telegraphic lines are available these
+comparisons are made by telegraphy. A small and delicately-made
+apparatus introduced into the mechanism of an astronomical clock
+or chronometer breaks or closes by the action of the clock an electric
+circuit every second. In order to record the minutes as well as
+seconds, one second in each minute, namely that numbered 0 or 60,
+is omitted. The seconds are recorded on a chronograph, which
+consists of a cylinder revolving uniformly at the rate of one revolution
+per minute covered with white paper, on which a pen having a slow
+movement in the direction of the axis of the cylinder describes a
+continuous spiral. This pen is deflected through the agency of an
+electromagnet every second, and thus the seconds of the clock are
+recorded on the chronograph by offsets from the spiral curve. An
+observer having his hand on a contact key in the same circuit can
+record in the same manner his observed times of transits of stars.
+The method of determination of difference of longitude is, therefore,
+virtually as follows. After the necessary observations for instrumental
+corrections, which are recorded only at the station of observation,
+the clock at A is put in connexion with the circuit so as to
+write on both chronographs, namely, that at A and that at B.
+Then the clock at B is made to write on both chronographs. It is
+clear that by this double operation one can eliminate the effect of the
+small interval of time consumed in the transmission of signals, for
+the difference of longitude obtained from the one chronograph
+will be in excess by as much as that obtained from the other will be
+in defect. The determination of the personal errors of the observers
+in this delicate operation is a matter of the greatest importance,
+as therein lies probably the chief source of residual error.</p>
+
+<p><span class="pagenum"><a name="page612" id="page612"></a>612</span></p>
+
+<p>These errors can nevertheless be almost entirely avoided by using
+the impersonal micrometer of Dr Repsold (Hamburg, 1889). In
+this device there is a movable micrometer wire which is brought by
+hand into coincidence with the star and moved along with it; at
+fixed points there are electrical contacts, which replace the fixed
+wires. Experiments at the Geodetic Institute and Central Bureau
+at Potsdam in 1891 gave the following personal equations in the case
+of four observers:&mdash;</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcc">&nbsp;</td> <td class="tcc">Older Procedure.</td> <td class="tcc">New Procedure.</td></tr>
+
+<tr><td class="tcc">A &minus; B</td> <td class="tcc">&minus;0<span class="sp">s</span>.108</td> <td class="tcc">&minus;0<span class="sp">s</span>.004</td></tr>
+<tr><td class="tcc">A &minus; G</td> <td class="tcc">&minus;0<span class="sp">s</span>.314</td> <td class="tcc">&minus;0<span class="sp">s</span>.035</td></tr>
+<tr><td class="tcc">A &minus; S</td> <td class="tcc">&minus;0<span class="sp">s</span>.184</td> <td class="tcc">&minus;0<span class="sp">s</span>.027</td></tr>
+<tr><td class="tcc">B &minus; G</td> <td class="tcc">&minus;0<span class="sp">s</span>.225</td> <td class="tcc">+0<span class="sp">s</span>.013</td></tr>
+<tr><td class="tcc">B &minus; S</td> <td class="tcc">&minus;0<span class="sp">s</span>.086</td> <td class="tcc">&minus;0<span class="sp">s</span>.023</td></tr>
+<tr><td class="tcc">G &minus; S</td> <td class="tcc">+0<span class="sp">s</span>.109</td> <td class="tcc">&minus;0<span class="sp">s</span>.006</td></tr>
+</table>
+
+<p>These results show that in the later method the personal equation
+is small and not so variable; and consequently the repetition of
+longitude determinations with exchanged observers and apparatus
+entirely eliminates the constant errors, the probable error of such
+determinations on ten nights being scarcely ±0<span class="sp">s</span>.01.</p>
+
+<p class="pt2 center"><i>Calculation of Triangulation.</i></p>
+
+<p>The surface of Great Britain and Ireland is uniformly covered by
+triangulation, of which the sides are of various lengths from 10 to
+111 miles. The largest triangle has one angle at Snowdon in Wales,
+another on Slieve Donard in Ireland, and a third at Scaw Fell in
+Cumberland; each side is over a hundred miles and the spherical
+excess is 64&Prime;. The more ordinary method of triangulation is, however,
+that of chains of triangles, in the direction of the meridian and
+perpendicular thereto. The principal triangulations of France,
+Spain, Austria and India are so arranged. Oblique chains of triangles
+are formed in Italy, Sweden and Norway, also in Germany
+and Russia, and in the United States. Chains are composed sometimes
+merely of consecutive plain triangles; sometimes, and more
+frequently in India, of combinations of triangles forming consecutive
+polygonal figures. In this method of triangulating, the sides of the
+triangles are generally from 20 to 30 miles in length&mdash;seldom exceeding
+40.</p>
+
+<p>The inevitable errors of observation, which are inseparable from
+all angular as well as other measurements, introduce a great difficulty
+into the calculation of the sides of a triangulation. Starting from a
+given base in order to get a required distance, it may generally be
+obtained in several different ways&mdash;that is, by using different sets
+of triangles. The results will certainly differ one from another,
+and probably no two will agree. The experience of the computer
+will then come to his aid, and enable him to say which is the most
+trustworthy result; but no experience or ability will carry him
+through a large network of triangles with anything like assurance.
+The only way to obtain trustworthy results is to employ the method
+of least squares. We cannot here give any illustration of this method
+as applied to general triangulation, for it is most laborious, even for
+the simplest cases.</p>
+
+<p>Three stations, projected on the surface of the sea, give a spherical
+or spheroidal triangle according to the adoption of the sphere or
+the ellipsoid as the form of the surface. A spheroidal triangle differs
+from a spherical triangle, not only in that the curvatures of the sides
+are different one from another, but more especially in this that,
+while in the spherical triangle the normals to the surface at the angular
+points meet at the centre of the sphere, in the spheroidal triangle
+the normals at the angles A, B, C meet the axis of revolution of the
+spheroid in three different points, which we may designate &alpha;, &beta;, &gamma;
+respectively. Now the angle A of the triangle as measured by a
+theodolite is the inclination of the planes BA&alpha; and CA&alpha;, and the angle
+at B is that contained by the planes AB&beta; and CB&beta;. But the planes
+AB&alpha; and AB&beta; containing the line AB in common cut the surface in
+two distinct plane curves. In order, therefore, that a spheroidal
+triangle may be exactly defined, it is necessary that the nature of the
+lines joining the three vertices be stated. In a mathematical point
+of view the most natural definition is that the sides be geodetic or
+shortest lines. C.C.G. Andrae, of Copenhagen, has also shown
+that other lines give a less convenient computation.</p>
+
+<p>K.F. Gauss, in his treatise, <i>Disquisitiones generales circa superficies
+curvas</i>, entered fully into the subject of geodetic (or geodesic)
+triangles, and investigated expressions for the angles of a geodetic
+triangle whose sides are given, not certainly finite expressions, but
+approximations inclusive of small quantities of the fourth order, the
+side of the triangle or its ratio to the radius of the nearly spherical
+surface being a small quantity of the first order. The terms of the
+fourth order, as given by Gauss for any surface in general, are very
+complicated even when the surface is a spheroid. If we retain small
+quantities of the second order only, and put <span class="got">A</span>, <span class="got">B</span>, <span class="got">C</span> for the angles
+of the geodetic triangle, while A, B, C are those of a plane triangle
+having sides equal respectively to those of the geodetic triangle,
+then, &sigma; being the area of the plane triangle and <span class="got">a</span>, <span class="got">b</span>, <span class="got">c</span> the measures
+of curvature at the angular points,</p>
+
+<table class="reg" summary="poem"><tr><td> <div class="poemr">
+<p><span class="got">A</span> = A + &sigma;(2<span class="got">a</span> + <span class="got">b</span> + <span class="got">c</span>) / 12,</p>
+<p><span class="got">B</span> = B + &sigma;(<span class="got">a</span> + 2<span class="got">b</span> + <span class="got">c</span>) / 12,</p>
+<p><span class="got">C</span> = C + &sigma;(<span class="got">a</span> + <span class="got">b</span> + 2<span class="got">c</span>) / 12.</p>
+</div> </td></tr></table>
+
+<p class="noind">For the sphere <span class="got">a</span> = <span class="got">b</span> = <span class="got">c</span>, and making this simplification, we obtain the
+theorem previously given by A.M. Legendre. With the terms of the
+fourth order, we have (after Andrae):</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2"><span class="got">A</span> &minus; A =</td> <td>&epsilon;</td>
+<td rowspan="2">+</td> <td>&sigma;</td>
+<td rowspan="2">k <span class="f150">(</span></td> <td>m² &minus; a²</td>
+<td rowspan="2">k +</td> <td><span class="got">a</span> &minus; k</td>
+<td rowspan="2"><span class="f150">)</span>,</td></tr>
+<tr><td class="denom">3</td> <td class="denom">3</td>
+<td class="denom">20</td> <td class="denom">4k</td></tr></table>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2"><span class="got">B</span> &minus; B =</td> <td>&epsilon;</td>
+<td rowspan="2">+</td> <td>&sigma;</td>
+<td rowspan="2">k <span class="f150">(</span></td> <td>m² &minus; b²</td>
+<td rowspan="2">k +</td> <td><span class="got">b</span> &minus; k</td>
+<td rowspan="2"><span class="f150">)</span>,</td></tr>
+<tr><td class="denom">3</td> <td class="denom">3</td>
+<td class="denom">20</td> <td class="denom">4k</td></tr></table>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2"><span class="got">C</span> &minus; C =</td> <td>&epsilon;</td>
+<td rowspan="2">+</td> <td>&sigma;</td>
+<td rowspan="2">k <span class="f150">(</span></td> <td>m² &minus; c²</td>
+<td rowspan="2">k +</td> <td><span class="got">c</span> &minus; k</td>
+<td rowspan="2"><span class="f150">)</span>,</td></tr>
+<tr><td class="denom">3</td> <td class="denom">3</td>
+<td class="denom">20</td> <td class="denom">4k</td></tr></table>
+
+<p class="noind">in which &epsilon; = &sigma;k {1 + (m²k / 8)}, 3m² = a² + b² + c², 3k = <span class="got">a</span> + <span class="got">b</span> + <span class="got">c</span>. For the
+ellipsoid of rotation the measure of curvature is equal to 1/&rho;n,
+&rho; and n being the radii of curvature of the meridian and perpendicular.</p>
+
+<p>It is rarely that the terms of the fourth order are required. As a
+rule spheroidal triangles are calculated as spherical (after Legendre),
+<i>i.e.</i> like plane triangles with a decrease of each angle of about &epsilon;/3;
+&epsilon; must, however, be calculated for each triangle separately with its
+mean measure of curvature k.</p>
+
+<p>The geodetic line being the shortest that can be drawn on any
+surface between two given points, we may be conducted to its most
+important characteristics by the following considerations: let p, q
+be adjacent points on a curved surface; through s the middle point
+of the chord pq imagine a plane drawn perpendicular to pq, and let
+S be any point in the intersection of this plane with the surface;
+then pS + Sq is evidently least when sS is a minimum, which is
+when sS is a normal to the surface; hence it follows that of all
+plane curves on the surface joining p, q, when those points are indefinitely
+near to one another, that is the shortest which is made
+by the normal plane. That is to say, the osculating plane at any
+point of a geodetic line contains the normal to the surface at that
+point. Imagine now three points in space, A, B, C, such that AB =
+BC = c; let the direction cosines of AB be l, m, n, those of BC l&rsquo;,
+m&prime;, n&prime;, then x, y, z being the co-ordinates of B, those of A and C will
+be respectively&mdash;</p>
+
+<table class="reg" summary="poem"><tr><td> <div class="poemr">
+<p>x &minus; cl : y &minus; cm : z &minus; cn</p>
+<p>x + cl&prime; : y + cm&prime; : z + cn&prime;.</p>
+</div> </td></tr></table>
+
+<p class="noind">Hence the co-ordinates of the middle point M of AC are x + ½c(l&prime; &minus; l),
+y + ½c(m&prime; &minus; m), z + ½c(n&prime; &minus; n), and the direction cosines of BM are
+therefore proportional to l&prime; &minus; l: m&prime; &minus; m: n&prime; &minus; n. If the angle made
+by BC with AB be indefinitely small, the direction cosines of BM
+are as &delta;l : &delta;m : &delta;n. Now if AB, BC be two contiguous elements of
+a geodetic, then BM must be a normal to the surface, and since &delta;l,
+&delta;m, &delta;n are in this case represented by &delta;(dx/ds), &delta;(dy/ds), &delta;(dz/ds),
+and if the equation of the surface be u = 0, we have</p>
+
+<table class="math0" summary="math">
+<tr><td>d²x</td>
+<td rowspan="2"><span class="f200">/</span></td> <td>du</td>
+<td rowspan="2">=</td> <td>d²y</td>
+<td rowspan="2"><span class="f200">/</span></td> <td>du</td>
+<td rowspan="2">=</td> <td>d²z</td>
+<td rowspan="2"><span class="f200">/</span></td> <td>du</td>
+<td rowspan="2">,</td></tr>
+<tr><td class="denom">ds²</td> <td class="denom">dx</td>
+<td class="denom">ds²</td> <td class="denom">dy</td>
+<td class="denom">ds²</td> <td class="denom">dz</td></tr></table>
+
+<p class="noind">which, however, are equivalent to only one equation. In the case
+of the spheroid this equation becomes</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">y</td> <td>d²x</td>
+<td rowspan="2">&minus;</td> <td>d²y</td>
+<td rowspan="2">= 0,</td></tr>
+<tr><td class="denom">ds²</td> <td class="denom">ds²</td></tr></table>
+
+<p class="noind">which integrated gives ydx &minus; xdy = Cds. This again may be put in
+the form r sin a = C, where a is the azimuth of the geodetic at any
+point&mdash;the angle between its direction and that of the meridian&mdash;and
+r the distance of the point from the axis of revolution.</p>
+
+<p>From this it may be shown that the azimuth at A of the geodetic
+joining AB is not the same as the astronomical azimuth at A of B
+or that determined by the vertical plane A&alpha;B. Generally speaking,
+the geodetic lies between the two plane section curves joining A and
+B which are formed by the two vertical planes, supposing these points
+not far apart. If, however, A and B are nearly in the same latitude,
+the geodetic may cross (between A and B) that plane curve which
+lies nearest the adjacent pole of the spheroid. The condition of
+crossing is this. Suppose that for a moment we drop the consideration
+of the earth&rsquo;s non-sphericity, and draw a perpendicular from
+the pole C on AB, meeting it in S between A and B. Then A being
+that point which is nearest the pole, the geodetic will cross the plane
+curve if AS be between ¼AB and <span class="spp">3</span>&frasl;<span class="suu">8</span>AB. If AS lie between this last
+value and ½AB, the geodetic will lie wholly to the north of both
+plane curves, that is, supposing both points to be in the northern
+hemisphere.</p>
+
+<p>The difference of the azimuths of the vertical section AB and of
+the geodetic AB, <i>i.e.</i> the astronomical and geodetic azimuths, is
+very small for all observable distances, being approximately:&mdash;</p>
+
+<p>Geod. azimuth = Astr. azimuth &minus;1/12 [e²/(1 &minus; e²)] [(s²/&rho;n (cos²&phi; sin 2&alpha; + (s/4a) | sin 2&phi; sin &alpha;)],
+in which: e and a are the numerical eccentricity
+and semi-major axis respectively of the meridian ellipse, &phi; and &alpha; are
+the latitude and azimuth at A, s = AB, and &rho; and n are the radii of
+curvature of the meridian and perpendicular at A. For s = 100
+kilometres, only the first term is of moment; its value is 0&Prime;.028
+cos² &phi; sin 2&alpha;, and it lies well within the errors of observation. If we
+imagine the geodetic AB, it will generally trisect the angles between
+the vertical sections at A and B, so that the geodetic at A is near
+<span class="pagenum"><a name="page613" id="page613"></a>613</span>
+the vertical section AB, and at B near the section BA.<a name="fa3a" id="fa3a" href="#ft3a"><span class="sp">3</span></a> The
+greatest distance of the vertical sections one from another is
+e²s³ cos² &phi;<span class="su">0</span> sin 2&alpha;<span class="su">0</span>/16a², in which &phi;<span class="su">0</span> and &alpha;<span class="su">0</span> are the mean latitude
+and azimuth respectively of the middle point of AB. For the value
+s = 64 kilometres, the maximum distance is 3 mm.</p>
+
+<p>An idea of the course of a longer geodetic line may be gathered
+from the following example. Let the line be that joining Cadiz and
+St Petersburg, whose approximate positions are&mdash;</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcc">Cadiz.</td> <td class="tcc">St Petersburg.</td></tr>
+<tr><td class="tcl">Lat. &emsp; 36° 22&prime; N.</td> <td class="tcc">59° 56&prime; N.</td></tr>
+<tr><td class="tcl">Long. 6° &emsp; 18&prime; W.</td> <td class="tcc">30° 17&prime; E.</td></tr>
+</table>
+
+<p class="noind">If G be the point on the geodetic corresponding to F on that one
+of the plane curves which contains the normal at Cadiz (by &ldquo;corresponding&rdquo;
+we mean that F and G are on a meridian) then G is to
+the north of F; at a quarter of the whole distance from Cadiz GF
+is 458 ft., at half the distance it is 637 ft., and at three-quarters it is
+473 ft. The azimuth of the geodetic at Cadiz differs 20&Prime; from that
+of the vertical plane, which is the astronomical azimuth.</p>
+
+<p>The azimuth of a geodetic line cannot be observed, so that the
+line does not enter of necessity into practical geodesy, although
+many formulae connected with its use are of great simplicity and
+elegance. The geodetic line has always held a more important place
+in the science of geodesy among the mathematicians of France,
+Germany and Russia than has been assigned to it in the operations
+of the English and Indian triangulations. Although the observed
+angles of a triangulation are not geodetic angles, yet in the calculation
+of the distance and reciprocal bearings of two points which
+are far apart, and are connected by a long chain of triangles, we may
+fall upon the geodetic line in this manner:&mdash;</p>
+
+<p>If A, Z be the points, then to start the calculation from A, we
+obtain by some preliminary calculation the approximate azimuth
+of Z, or the angle made by the direction of Z with the side AB or
+AC of the first triangle. Let P<span class="su">1</span> be the point where this line intersects
+BC; then, to find P<span class="su">2</span>, where the line cuts the next triangle
+side CD, we make the angle BP<span class="su">1</span>P<span class="su">2</span> such that BP<span class="su">1</span>P<span class="su">2</span> + BP<span class="su">1</span>A = 180°.
+This fixes P<span class="su">2</span>, and P<span class="su">3</span> is fixed by a repetition of the same process;
+so for P<span class="su">4</span>, P<span class="su">5</span> .... Now it is clear that the points P<span class="su">1</span>, P<span class="su">2</span>, P<span class="su">3</span> so computed
+are those which would be actually fixed by an observer with
+a theodolite, proceeding in the following manner. Having set the
+instrument up at A, and turned the telescope in the direction of
+the computed bearing, an assistant places a mark P<span class="su">1</span> on the line
+BC, adjusting it till bisected by the cross-hairs of the telescope at
+A. The theodolite is then placed over P<span class="su">1</span>, and the telescope turned
+to A; the horizontal circle is then moved through 180°. The
+assistant then places a mark P<span class="su">2</span> on the line CD, so as to be bisected
+by the telescope, which is then moved to P<span class="su">2</span>, and in the same manner
+P<span class="su">3</span> is fixed. Now it is clear that the series of points P<span class="su">1</span>, P<span class="su">2</span>, P<span class="su">3</span>
+approaches to the geodetic line, for the plane of any two consecutive
+elements P<span class="su">n&minus;1</span> P<span class="su">n</span>, P<span class="su">n</span> P<span class="su">n+1</span> contains the normal at P<span class="su">n</span>.</p>
+
+<p>If the objection be raised that not the geodetic azimuths but the
+astronomical azimuths are observed, it is necessary to consider that
+the observed vertical sections do not correspond to points on the
+sea-level but to elevated points. Since the normals of the ellipsoid
+of rotation do not in general intersect, there consequently arises an
+influence of the height on the azimuth. In the case of the measurement
+of the azimuth from A to B, the instrument is set to a point A&prime;
+over the surface of the ellipsoid (the sea-level), and it is then adjusted
+to a point B&prime;, also over the surface, say at a height h&prime;. The vertical
+plane containing A&prime; and B&prime; also contains A but not B: it must
+therefore be rotated through a small azimuth in order to contain B.
+The correction amounts approximately to &minus;e²h&prime; cos²&phi; sin 2&alpha;/2a;
+in the case of h&prime; = 1000 m., its value is 0&Prime;.108 cos²&phi; sin 2&alpha;.</p>
+
+<p>This correction is therefore of greater importance in the case of
+observed azimuths and horizontal angles than in the previously
+considered case of the astronomical and the geodetic azimuths. The
+observed azimuths and horizontal angles must therefore also be
+corrected in the case, where it is required to dispense with geodetic
+lines.</p>
+
+<p>When the angles of a triangulation have been adjusted by the
+method of least squares, and the sides are calculated, the next
+process is to calculate the latitudes and longitudes of all the stations
+starting from one given point. The calculated latitudes, longitudes
+and azimuths, which are designated geodetic latitudes, longitudes
+and azimuths, are not to be confounded with the observed latitudes,
+longitudes and azimuths, for these last are subject to somewhat
+large errors. Supposing the latitudes of a number of stations in the
+triangulation to be observed, practically the mean of these determines
+the position in latitude of the network, taken as a whole. So the
+orientation or general azimuth of the whole is inferred from all the
+azimuth observations. The triangulation is then supposed to be
+projected on a spheroid of given elements, representing as nearly as
+one knows the real figure of the earth. Then, taking the latitude
+of one point and the direction of the meridian there as given&mdash;obtained,
+namely, from the astronomical observations there&mdash;one
+can compute the latitudes of all the other points with any degree of
+precision that may be considered desirable. It is necessary to employ
+for this purpose formulae which will give results true even for the
+longest distances to the second place of decimals of seconds, otherwise
+there will arise an accumulation of errors from imperfect calculation
+which should always be avoided. For very long distances, eight
+places of decimals should be employed in logarithmic calculations;
+if seven places only are available very great care will be required to
+keep the last place true. Now let &phi;, &phi;&prime; be the latitudes of two stations
+A and B; &alpha;, &alpha;<span class="sp">*</span> their mutual azimuths counted from north by east
+continuously from 0° to 360°; &omega; their difference of longitude
+measured from west to east; and s the distance AB.</p>
+
+<p>First compute a latitude &phi;<span class="su">1</span> by means of the formula &phi;<span class="su">1</span> = &phi;
++ (s cos &alpha;)/&rho;, where &rho; is the radius of curvature of the meridian at the
+latitude &phi;; this will require but four places of logarithms. Then,
+in the first two of the following, five places are sufficient&mdash;</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&epsilon; =</td> <td>s²</td>
+<td rowspan="2">sin &alpha; cos a, &emsp; &eta; =</td> <td>s²</td>
+<td rowspan="2">sin² &alpha; tan &phi;<span class="su">1</span>,</td></tr>
+<tr><td class="denom">2&rho;n</td> <td class="denom">2&rho;n</td></tr></table>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&phi;&prime; &minus; &phi; =</td> <td>s</td>
+<td rowspan="2">cos (&alpha; &minus; <span class="spp">2</span>&frasl;<span class="suu">3</span>&epsilon;) &minus; &eta;,</td></tr>
+<tr><td class="denom">rho<span class="su">0</span></td></tr></table>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&omega; =</td> <td>s sin (alpha &minus; <span class="spp">1</span>&frasl;<span class="suu">3</span>&epsilon;)</td>
+<td rowspan="2">,</td></tr>
+<tr><td class="denom">n cos (&phi;&prime; + <span class="spp">1</span>&frasl;<span class="suu">3</span>&eta;)</td></tr></table>
+
+<p class="center">&alpha;<span class="sp">*</span> &minus; &alpha; = &omega; sin (&phi;&prime; + <span class="spp">2</span>&frasl;<span class="suu">3</span>&eta;) &minus; &epsilon; + 180°.</p>
+
+<p class="noind">Here n is the normal or radius of curvature perpendicular to the
+meridian; both n and &rho; correspond to latitude &phi;<span class="su">1</span>, and &rho;<span class="su">0</span> to latitude
+½(&phi; + &phi;&prime;). For calculations of latitude and longitude, tables of the
+logarithmic values of &rho; sin 1&Prime;, n sin 1&Prime;, and 2 n &rho; sin 1&Prime; are necessary.
+The following table contains these logarithms for every ten minutes
+of latitude from 52° to 53° computed with the elements a = 20926060
+and a : b = 295 : 294 :&mdash;</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcc allb">Lat.</td> <td class="tcc allb">Log. 1/&rho; sin 1&Prime;.</td> <td class="tcc allb">Log. 1/n sin 1&Prime;.</td> <td class="tcc allb">Log. 1/2&rho;n sin 1&Prime;.</td></tr>
+
+<tr><td class="tcr lb rb">° &ensp; &prime;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td></tr>
+
+<tr><td class="tcr lb rb">52 0</td> <td class="tcr rb">7.9939434</td> <td class="tcr rb">7.9928231</td> <td class="tcr rb">0.37131</td></tr>
+<tr><td class="tcr lb rb">10</td> <td class="tcr rb">9309</td> <td class="tcr rb">8190</td> <td class="tcr rb">29</td></tr>
+<tr><td class="tcr lb rb">20</td> <td class="tcr rb">9185</td> <td class="tcr rb">8148</td> <td class="tcr rb">28</td></tr>
+<tr><td class="tcr lb rb">30</td> <td class="tcr rb">9060</td> <td class="tcr rb">8107</td> <td class="tcr rb">26</td></tr>
+<tr><td class="tcr lb rb">40</td> <td class="tcr rb">8936</td> <td class="tcr rb">8065</td> <td class="tcr rb">24</td></tr>
+<tr><td class="tcr lb rb">50</td> <td class="tcr rb">8812</td> <td class="tcr rb">8024</td> <td class="tcr rb">23</td></tr>
+<tr><td class="tcr lb rb bb">53 0</td> <td class="tcr rb bb">8688</td> <td class="tcr rb bb">7982</td> <td class="tcr rb bb">22</td></tr>
+</table>
+
+<p>The logarithm in the last column is that required also for the
+calculation of spherical excesses, the spherical excess of a triangle
+being expressed by a b sin C/(2&rho;n) sin 1&Prime;.</p>
+
+<p>It is frequently necessary to obtain the co-ordinates of one point
+with reference to another point; that is, let a perpendicular arc be
+drawn from B to the meridian of A meeting it in P, then, &alpha; being
+the azimuth of B at A, the co-ordinates of B with reference to A are</p>
+
+<p class="center">AP = s cos (&alpha; &minus; <span class="spp">2</span>&frasl;<span class="suu">3</span>&epsilon;), BP = s sin (&alpha; &minus; <span class="spp">1</span>&frasl;<span class="suu">3</span>&epsilon;),</p>
+
+<p class="noind">where &epsilon; is the spherical excess of APB, viz. s² sin &alpha; cos &alpha; multiplied
+by the quantity whose logarithm is in the fourth column of the above
+table.</p>
+
+<p>If it be necessary to determine the geographical latitude and
+longitude as well as the azimuths to a greater degree of accuracy
+than is given by the above formulae, we make use of the following
+formula: given the latitude &phi; of A, and the azimuth &alpha; and the
+distance s of B, to determine the latitude &phi;&prime; and longitude &omega; of B,
+and the back azimuth &alpha;&prime;. Here it is understood that &alpha;&prime; is symmetrical
+to &alpha;, so that &alpha;<span class="sp">*</span> + &alpha;&prime; = 360°.</p>
+
+<p class="noind">Let</p>
+
+<p class="center">&theta; = s&Delta; / a, where &Delta; = (1 &minus; e² sin² &phi;)<span class="sp">1/2</span></p>
+
+<p class="noind">and</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&xi; =</td> <td>e² &theta;²</td>
+<td rowspan="2">cos² &phi; sin 2&alpha;, &emsp; &xi;&prime; =</td> <td>e² &theta;³</td>
+<td rowspan="2">cos² &phi; cos² &alpha;;</td></tr>
+<tr><td class="denom">4 (1 &minus; e²)</td> <td class="denom">6 (1 &minus; e²)</td></tr></table>
+
+<p>&xi;, &xi;&prime; are always very minute quantities even for the longest distances;
+then, putting &kappa; = 90° &minus; &phi;,</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">tan</td> <td>&alpha;&prime; + &xi; &minus; &omega;</td>
+<td rowspan="2">=</td> <td>sin ½(&kappa; &minus; &theta; &minus; &xi;&prime;)</td>
+<td rowspan="2">cot</td> <td>&alpha;</td></tr>
+<tr><td class="denom">2</td> <td class="denom">sin ½(&kappa; + &theta; + &xi;&prime;)</td>
+<td class="denom">2</td></tr></table>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">tan</td> <td>&alpha;&prime; + &xi; &minus; &omega;</td>
+<td rowspan="2">=</td> <td>cos ½(&kappa; &minus; &theta; &minus; &xi;&prime;)</td>
+<td rowspan="2">cot</td> <td>&alpha;</td></tr>
+<tr><td class="denom">2</td> <td class="denom">cos ½(&kappa; + &theta; + &xi;&prime;)</td>
+<td class="denom">2</td></tr></table>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&phi;&prime; &minus; &phi; =</td> <td>s sin ½(&alpha;&prime; + &xi; &minus; &alpha;)</td>
+<td rowspan="2"><span class="f150">(</span> 1 +</td> <td>&theta;²</td>
+<td rowspan="2">cos²</td> <td>&alpha;&prime; &minus; &alpha;</td>
+<td rowspan="2"><span class="f150">)</span>;</td></tr>
+<tr><td class="denom">&rho;<span class="su">0</span> sin ½(&alpha;&prime; + &xi; + &alpha;)</td> <td class="denom">12</td>
+<td class="denom">2</td></tr></table>
+
+<p class="noind">here &rho;<span class="su">0</span> is the radius of curvature of the meridian for the mean
+latitude ½(&phi; + &phi;&prime;). These formulae are approximate only, but they
+are sufficiently precise even for very long distances.</p>
+
+<p>For lines of any length the formulae of F.W. Bessel (<i>Astr. Nach.</i>,
+1823, iv. 241) are suitable.</p>
+
+<p>If the two points A and B be defined by their geographical
+<span class="pagenum"><a name="page614" id="page614"></a>614</span>
+co-ordinates, we can accurately calculate the corresponding astronomical
+azimuths, <i>i.e.</i> those of the vertical section, and then proceed,
+in the case of not too great distances, to determine the length and
+the azimuth of the shortest lines. For <i>any</i> distances recourse must
+again be made to Bessel&rsquo;s formula.<a name="fa4a" id="fa4a" href="#ft4a"><span class="sp">4</span></a></p>
+
+<p>Let &alpha;, &alpha;&prime; be the mutual azimuths of two points A, B on a spheroid,
+k the chord line joining them, &mu;, &mu;&prime; the angles made by the chord
+with the normals at A and B, &phi;, &phi;&prime;, &omega; their latitudes and difference of
+longitude, and (x² + y²)/a² + z² b² = 1 the equation of the surface;
+then if the plane xz passes through A the co-ordinates of A and B
+will be</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcl">x = (a/&Delta;) cos &phi;,</td> <td class="tcl">x&prime; = (a/&Delta;&rsquo;) cos &phi;&prime; cos &omega;,</td></tr>
+
+<tr><td class="tcl">y = 0</td> <td class="tcl">y&prime; = (a/&Delta;&rsquo;) cos &phi;&prime; sin &omega;,</td></tr>
+
+<tr><td class="tcl">z = (a/&Delta;) (1 &minus; e²) sin &phi;,</td> <td class="tcl">z&prime; = (a/&Delta;&prime;) (1 &minus; e²) sin &phi;&prime;,</td></tr>
+</table>
+
+<p class="noind">where &Delta; = (1 &minus; e² sin² &phi;)<span class="sp">1/2</span>, &Delta;&prime; = (1 &minus; e² sin² &phi;&prime;)<span class="sp">1/2</span>, and e is the eccentricity.
+Let f, g, h be the direction cosines of the normal to that
+plane which contains the normal at A and the point B, and whose
+inclinations to the meridian plane of A is = &alpha;; let also l, m, n and
+l&rsquo;, m&rsquo;, n&rsquo; be the direction cosines of the normal at A, and of the
+tangent to the surface at A which lies in the plane passing through
+B, then since the first line is perpendicular to each of the other two
+and to the chord k, whose direction cosines are proportional to
+x&prime; &minus; x, y&prime; &minus; y, z&prime; &minus; z, we have these three equations</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcr">f (x&prime; &minus; x) + gy&prime; + h (z&prime; &minus; z) = 0</td></tr>
+
+<tr><td class="tcr">fl + gm + hn = 0</td></tr>
+
+<tr><td class="tcr">fl&prime; + gm&prime; + hn&prime; = 0.</td></tr>
+</table>
+
+<p>Eliminate f, g, h from these equations, and substitute</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcl">l = cos &phi;</td> <td class="tcl">l&prime; = &minus; sin &phi; cos &alpha;</td></tr>
+
+<tr><td class="tcl">m = 0</td> <td class="tcl">m&prime; = sin &alpha;</td></tr>
+
+<tr><td class="tcl">n = sin &phi;</td> <td class="tcl">n&prime; = cos &phi; cos &alpha;,</td></tr>
+</table>
+
+<p class="noind">and we get</p>
+
+<p class="center">(x&prime; &minus; x) sin &phi; + y&prime; cot &alpha; &minus; (z&prime; &minus; z) cos &phi; = 0.</p>
+
+<p class="noind">The substitution of the values of x, z, x&prime;, y&prime;, z&prime; in this equation will
+give immediately the value of cot &alpha;; and if we put &zeta;, &zeta;&rsquo; for the
+corresponding azimuths on a sphere, or on the supposition e = 0,
+the following relations exist</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">cot &alpha; &minus; cot &zeta; = e²</td> <td>cos &phi; Q</td></tr>
+<tr><td class="denom">cos &phi;&prime; &Delta;</td></tr></table>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">cot &alpha;&prime; &minus; cot &zeta;&prime; = &minus;e²</td> <td>cos &phi;&prime; Q</td>
+<td rowspan="2"></td></tr>
+<tr><td class="denom">cos &phi; &Delta;&prime;</td></tr></table>
+
+<p class="center">&Delta;&prime; sin &phi; &minus; &Delta; sin &phi;&prime; = Q sin &omega;.</p>
+
+<p>If from B we let fall a perpendicular on the meridian plane of A,
+and from A let fall a perpendicular on the meridian plane of B,
+then the following equations become geometrically evident:</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcl">k sin &mu; sin &alpha; = (a/&Delta;&prime;) cos &phi;&prime; sin &omega;</td></tr>
+<tr><td class="tcl">k sin &mu;&prime; sin &alpha;&prime; = (a/&Delta;) cos &phi; sin &omega;.</td></tr>
+</table>
+
+<p>Now in any surface u = 0 we have</p>
+
+<p class="center">k² = (x&prime; &minus; x)² + (y&prime; &minus; y)² + (z&prime; &minus; z)²</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&minus;cos &mu; = <span class="f150">[</span> (x&prime; &minus; x)</td> <td>du</td>
+<td rowspan="2">+ (y&prime; &minus; y)</td> <td>du</td>
+<td rowspan="2">+ (z&prime; &minus; z)</td> <td>du</td>
+<td rowspan="2"><span class="f150">] /</span> k <span class="f150">(</span></td> <td>du²</td>
+<td rowspan="2">+</td> <td>du²</td>
+<td rowspan="2">+</td> <td>du²</td>
+<td rowspan="2"><span class="f150">)</span></td> <td><span class="sp">1/2</span></td></tr>
+<tr><td class="denom">dx</td> <td class="denom">dy</td>
+<td class="denom">dz</td> <td class="denom">dx²</td>
+<td class="denom">dy²</td> <td class="denom">dz²</td></tr></table>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">cos &mu;&prime; = <span class="f150">[</span> (x&prime; &minus; x)</td> <td>du</td>
+<td rowspan="2">+ (y&prime; &minus; y)</td> <td>du</td>
+<td rowspan="2">+ (z&prime; &minus; z)</td> <td>du</td>
+<td rowspan="2"><span class="f150">] /</span> k <span class="f150">(</span></td> <td>du²</td>
+<td rowspan="2">+</td> <td>du²</td>
+<td rowspan="2">+</td> <td>du²</td>
+<td rowspan="2"><span class="f150">)</span></td> <td><span class="sp">1/2</span></td> <td rowspan="2">.</td></tr>
+<tr><td class="denom">dx&prime;</td> <td class="denom">dy&prime;</td>
+<td class="denom">dz&prime;</td> <td class="denom">dx&prime;²</td>
+<td class="denom">dy&prime;²</td> <td class="denom">dz&prime;²</td></tr></table>
+
+<p class="noind">In the present case, if we put</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">1 &minus;</td> <td>xx&prime;</td>
+<td rowspan="2">&minus;</td> <td>zz&prime;</td>
+<td rowspan="2">= U,</td></tr>
+<tr><td class="denom">a²</td> <td class="denom">b²</td></tr></table>
+
+<p class="noind">then</p>
+
+<table class="math0" summary="math">
+<tr><td>k²</td>
+<td rowspan="2">= 2U &minus; e² <span class="f150">(</span></td> <td>z&prime; &minus; z</td>
+<td rowspan="2"><span class="f150">)</span></td> <td>²</td></tr>
+<tr><td class="denom">a²</td> <td class="denom">b</td></tr></table>
+
+<p class="center">cos &mu; = (a/k) &Delta;U;  cos &mu;&prime; = (a/k) &Delta;&prime;U.</p>
+
+<p class="noind">Let u be such an angle that</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcr">(1 &minus; e²)<span class="sp">½</span> sin &phi; = &Delta; sin u</td></tr>
+
+<tr><td class="tcr">cos &phi; = &Delta; cos u,</td></tr>
+</table>
+
+<p class="noind">then on expressing x, x&prime;, z, z&prime; in terms of u and u&prime;,</p>
+
+<p class="center">U = 1 &minus; cos u cos u&prime; cos &omega; &minus; sin u sin u&prime;;</p>
+
+<p class="noind">also, if v be the third side of a spherical triangle, of which two
+sides are ½&pi; &minus; u and ½&pi; &minus; u&prime; and the included angle &omega;, using a subsidiary
+angle &psi; such that</p>
+
+<p class="center">sin &psi; sin ½v = e sin ½ (u&prime; &minus; u) cos ½ (u&prime; + u),</p>
+
+<p class="noind">we obtain finally the following equations:&mdash;</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcr">k</td> <td class="tcl">= 2a cos &psi; sin ½v</td></tr>
+<tr><td class="tcr">cos &mu;</td> <td class="tcl">= &Delta; sec &psi; sin ½v</td></tr>
+<tr><td class="tcr">cos &mu;&prime;</td> <td class="tcl">= &Delta;&prime; sec &psi; sin ½v</td></tr>
+<tr><td class="tcr">sin &mu; sin &alpha;</td> <td class="tcl">= (a/k) cos u&prime; sin &omega;</td></tr>
+<tr><td class="tcr">sin &mu;&prime; sin &alpha;&prime;</td> <td class="tcl">= (a/k) cos u sin &omega;.</td></tr>
+</table>
+
+<p>These determine rigorously the distance, and the mutual zenith
+distances and azimuths, of any two points on a spheroid whose
+latitudes and difference of longitude are given.</p>
+
+<p>By a series of reductions from the equations containing &zeta;, &zeta;&prime; it
+may be shown that</p>
+
+<p class="center">&alpha; + &alpha;&prime; = &zeta; + &zeta;&prime; + ¼e<span class="sp">4</span>&omega; (&phi;&prime; &minus; &phi;)² cos<span class="sp">4</span> &phi;<span class="su">0</span> sin &phi;<span class="su">0</span> + ...,</p>
+
+<p class="noind">where &phi;<span class="su">0</span> is the mean of &phi; and &phi;&prime;, and the higher powers of e are
+neglected. A short computation will show that the small quantity
+on the right-hand side of this equation cannot amount even to
+the thousandth part of a second for k &lt; 0.1a, which is, practically
+speaking, zero; consequently the sum of the azimuths &alpha; + &alpha;&prime; on the
+spheroid is equal to the sum of the spherical azimuths, whence
+follows this very important theorem (known as Dalby&prime;s theorem).
+If &phi;, &phi;&prime; be the latitudes of two points on the surface of a spheroid, &omega;
+their difference of longitude, &alpha;, &alpha;&prime; their reciprocal azimuths,</p>
+
+<p class="center">tan ½&omega; = cot ½ (&alpha; + &alpha;&prime;) {cos ½ (&phi;&prime; &minus; &phi;) / sin ½ (&phi;&prime; + &phi;)}.</p>
+
+<p>The computation of the geodetic from the astronomical azimuths
+has been given above. From k we can now compute the length s
+of the vertical section, and from this the shortest length. The
+difference of length of the geodetic line and either of the plane
+curves is</p>
+
+<p class="center">e<span class="sp">4</span>s<span class="sp">5</span> cos<span class="sp">4</span> &phi;<span class="su">0</span> sin² 2&alpha;<span class="su">0</span>/360 a<span class="sp">4</span>.</p>
+
+<p class="noind">At least this is an approximate expression. Supposing s = 0.1a,
+this quantity would be less than one-hundredth of a millimetre.
+The line s is now to be calculated as a circular arc with a mean radius r
+along AB. If &phi;<span class="su">0</span> = ½ (&phi; + &phi;&prime;), &alpha;<span class="su">0</span> = ½ (180° + &alpha; &minus; &alpha;&prime;), &Delta;<span class="su">0</span> = (1 &minus; e² sin² &phi;<span class="su">0</span>)<span class="sp">1/2</span>,
+then 1/r = &Delta;<span class="su">0</span>/a [1 + (e²/(1 &minus; e²) cos² &phi;<span class="su">0</span> cos² &alpha;<span class="su">0</span>], and approximately sin (s/2r) =
+k/2r. These formulae give, in the case of k = 0.1a, values certain to
+eight logarithmic decimal places. An excellent series of formulae
+for the solution of the problem, to determine the azimuths, chord
+and distance along the surface from the geographical co-ordinates,
+was given in 1882 by Ch. M. Schols (<i>Archives Néerlandaises</i>, vol. xvii.).</p>
+
+<p class="pt2 center"><i>Irregularities of the Earth&rsquo;s Surface.</i></p>
+
+<p>In considering the effect of unequal distribution of matter in the
+earth&rsquo;s crust on the form of the surface, we may simplify the matter
+by disregarding the considerations of rotation and eccentricity.
+In the first place, supposing the earth a sphere covered with a film of
+water, let the density &rho; be a function of the distance from the centre
+so that surfaces of equal density are concentric spheres. Let now a
+disturbance of the arrangement of matter take place, so that the
+density is no longer to be expressed by &rho;, a function of r only, but is
+expressed by &rho; + &rho;&prime;, where &rho;&prime; is a function of three co-ordinates &theta;, &phi;, r.
+Then &rho;&prime; is the density of what may be designated disturbing matter;
+it is positive in some places and negative in others, and the whole
+quantity of matter whose density is &rho;&prime; is zero. The previously
+spherical surface of the sea of radius a now takes a new form. Let
+P be a point on the disturbed surface, P&prime; the corresponding point
+vertically below it on the undisturbed surface, PP&prime; = N. The
+knowledge of N over the whole surface gives us the form of the
+disturbed or actual surface of the sea; it is an equipotential surface,
+and if V be the potential at P of the disturbing matter &rho;&prime;, M the
+mass of the earth (the attraction-constant is assumed equal to unity)</p>
+
+<table class="math0" summary="math">
+<tr><td>M</td>
+<td rowspan="2">+ V = C =</td> <td>M</td>
+<td rowspan="2">&minus;</td> <td>M</td>
+<td rowspan="2">N + V.</td></tr>
+<tr><td class="denom">a + N</td> <td class="denom">a</td>
+<td class="denom">a²</td></tr></table>
+
+<p class="noind">As far as we know, N is always a very small quantity, and we have
+with sufficient approximation N = 3V/4&pi;&delta;a, where &delta; is the mean
+density of the earth. Thus we have the disturbance in elevation
+of the sea-level expressed in terms of the potential of the disturbing
+matter. If at any point P the value of N remain constant when we
+pass to any adjacent point, then the actual surface is there parallel
+to the ideal spherical surface; as a rule, however, the normal at P is
+inclined to that at P&prime;, and astronomical observations have shown
+that this inclination, the deflection or deviation, amounting
+ordinarily to one or two seconds, may in some cases exceed 10&Prime;,
+or, as at the foot of the Himalayas, even 60&Prime;. By the expression
+&ldquo;mathematical figure of the earth&rdquo; we mean the surface of the sea
+produced in imagination so as to percolate the continents. We
+see then that the effect of the uneven distribution of matter in the
+crust of the earth is to produce small elevations and depressions on
+the mathematical surface which would be otherwise spheroidal.
+No geodesist can proceed far in his work without encountering the
+irregularities of the mathematical surface, and it is necessary that
+he should know how they affect his astronomical observations. The
+whole of this subject is dealt with in his usual elegant manner by
+Bessel in the <i>Astronomische Nachrichten</i>, Nos. 329, 330, 331, in a
+paper entitled &ldquo;Ueber den Einfluss der Unregelmässigkeiten der
+Figur der Erde auf geodätische Arbeiten, &amp;c.&rdquo; But without entering
+into further details it is not difficult to see how local attraction at
+any station affects the determinations of latitude, longitude and
+azimuth there.</p>
+
+<p>Let there be at the station an attraction to the north-east throwing
+the zenith to the south-west, so that it takes in the celestial sphere a
+position Z&prime;, its undisturbed position being Z. Let the rectangular
+components of the displacement ZZ&prime; be &xi; measured southwards
+<span class="pagenum"><a name="page615" id="page615"></a>615</span>
+and &eta; measured westwards. Now the great circle joining Z&prime; with
+the pole of the heavens P makes there an angle with the meridian
+PZ = &eta; cosec PZ&prime; = &eta; sec &phi;, where &phi; is the latitude of the station.
+Also this great circle meets the horizon in a point whose distance
+from the great circle PZ is &eta; sec &phi; sin &phi; = &eta; tan &phi;. That is, a meridian
+mark, fixed by observations of the pole star, will be placed that
+amount to the east of north. Hence the observed latitude requires
+the correction &xi;; the observed longitude a correction &eta; sec &phi;; and
+any observed azimuth a correction &eta; tan &phi;. Here it is supposed
+that azimuths are measured from north by east, and longitudes
+eastwards. The horizontal angles are also influenced by the deflections
+of the plumb-line, in fact, just as if the direction of the vertical
+axis of the theodolite varied by the same amount. This influence,
+however, is slight, so long as the sights point almost horizontally
+at the objects, which is always the case in the observation of distant
+points.</p>
+
+<p>The expression given for N enables one to form an approximate
+estimate of the effect of a compact mountain in raising the sea-level.
+Take, for instance, Ben Nevis, which contains about a couple of
+cubic miles; a simple calculation shows that the elevation produced
+would only amount to about 3 in. In the case of a mountain mass
+like the Himalayas, stretching over some 1500 miles of country with
+a breadth of 300 and an average height of 3 miles, although it is difficult
+or impossible to find an expression for V, yet we may ascertain
+that an elevation amounting to several hundred feet may exist
+near their base. The geodetical operations, however, rather negative
+this idea, for it was shown by Colonel Clarke (<i>Phil. Mag.</i>, 1878)
+that the form of the sea-level along the Indian arc departs but slightly
+from that of the mean figure of the earth. If this be so, the action
+of the Himalayas must be counteracted by subterranean tenuity.</p>
+
+<p>Suppose now that A, B, C, ... are the stations of a network of
+triangulation projected on or lying on a spheroid of semiaxis major
+and eccentricity a, e, this spheroid having its axis parallel to the axis
+of rotation of the earth, and its surface coinciding with the mathematical
+surface of the earth at A. Then basing the calculations
+on the observed elements at A, the calculated latitudes, longitudes
+and directions of the meridian at the other points will be the true
+latitudes, &amp;c., of the points as projected on the spheroid. On
+comparing these geodetic elements with the corresponding astronomical
+determinations, there will appear a system of differences
+which represent the inclinations, at the various points, of the actual
+irregular surface to the surface of the spheroid of reference. These
+differences will suggest two things,&mdash;first, that we may improve the
+agreement of the two surfaces, by not restricting the spheroid of
+reference by the condition of making its surface coincide with the
+mathematical surface of the earth at A; and secondly, by altering
+the form and dimensions of the spheroid. With respect to the first
+circumstance, we may allow the spheroid two degrees of freedom,
+that is, the normals of the surfaces at A may be allowed to separate
+a small quantity, compounded of a meridional difference and a
+difference perpendicular to the same. Let the spheroid be so placed
+that its normal at A lies to the north of the normal to the earth&rsquo;s
+surface by the small quantity &xi; and to the east by the quantity &eta;.
+Then in starting the calculation of geodetic latitudes, longitudes and
+azimuths from A, we must take, not the observed elements &phi;, &alpha;,
+but for &phi;, &phi; + &xi;, and for &alpha;, &alpha; + &eta; tan &phi;, and zero longitude must be
+replaced by &eta; sec &phi;. At the same time suppose the elements of the
+spheroid to be altered from a, e to a + da, e + de. Confining our
+attention at first to the two points A, B, let (&phi;&prime;), (&alpha;&prime;), (&omega;) be the
+numerical elements at B as obtained in the first calculation, viz.
+before the shifting and alteration of the spheroid; they will now
+take the form</p>
+
+<table class="reg" summary="poem"><tr><td> <div class="poemr">
+<p>(&phi;&prime;) + f&xi; + g&eta; + hda + kde,</p>
+<p>(&alpha;&prime;) + f&prime;&xi; + g&prime;&eta; + h&prime;da + k&prime;de,</p>
+<p>&omega; + f&Prime;&xi; + g&Prime;&eta; + h&Prime;da + k&Prime;de,</p>
+</div> </td></tr></table>
+
+<p class="noind">where the coefficients f, g, ... &amp;c. can be numerically calculated.
+Now these elements, corresponding to the projection of B on the
+spheroid of reference, must be equal severally to the astronomically
+determined elements at B, corrected for the inclination of the surfaces
+there. If &xi;&prime;, &eta;&prime; be the components of the inclination at that
+point, then we have</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcr">&xi;&prime;</td> <td class="tcl">= (&phi;&prime;) &minus; &phi;&prime; + f&xi; + g&eta; + hda + kde,</td></tr>
+
+<tr><td class="tcr">&eta;&prime; tan &phi;&prime;</td> <td class="tcl">= (&alpha;&prime;) &minus; &alpha;&prime; + f&prime;&xi; + g&prime;&eta; + h&prime;da + k&prime;de,</td></tr>
+
+<tr><td class="tcr">&eta;&prime; sec &phi;&prime;</td> <td class="tcl">= (&omega;) &minus; &omega; + f&Prime;&xi; + g&Prime;&eta; + h&Prime;da + k&Prime;de,</td></tr>
+</table>
+
+<p class="noind">where &phi;&prime;, &alpha;&prime;, &omega; are the observed elements at B. Here it appears
+that the observation of longitude gives no additional information,
+but is available as a check upon the azimuthal observations.</p>
+
+<p>If now there be a number of astronomical stations in the triangulation,
+and we form equations such as the above for each point,
+then we can from them determine those values of &xi;, &eta;, da, de, which
+make the quantity &xi;² + &eta;² + &xi;&prime;² + &eta;&prime;² + ... a minimum. Thus we
+obtain that spheroid which best represents the surface covered by the
+triangulation.</p>
+
+<p>In the <i>Account of the Principal Triangulation of Great Britain and
+Ireland</i> will be found the determination, from 75 equations, of the
+spheroid best representing the surface of the British Isles. Its
+elements are a = 20927005 ± 295 ft., b : a &minus; b = 280 ± 8; and it is so
+placed that at Greenwich Observatory &xi; = 1&Prime;.864, &eta; = &minus;0&Prime;.546.</p>
+
+<p>Taking Durham Observatory as the origin, and the tangent plane
+to the surface (determined by &xi; = &minus;0&Prime;.664, &eta; = &minus;4&Prime;.117) as the plane
+of x and y, the former measured northwards, and z measured vertically
+downwards, the equation to the surface is</p>
+
+<p class="center">.99524953 x² + .99288005 y² + .99763052 z² &minus; 0.00671003xz &minus; 41655070z = 0.</p>
+
+<p class="pt2 center"><i>Altitudes.</i></p>
+
+<p>The precise determination of the altitude of his station is a matter
+of secondary importance to the geodesist; nevertheless it is usual
+to observe the zenith distances of all trigonometrical points. Of
+great importance is a knowledge of the height of the base for its reduction
+to the sea-level. Again the height of a station does influence
+a little the observation of terrestrial angles, for a vertical line at B
+does not lie generally in the vertical plane of A (see above). The
+height above the sea-level also influences the geographical latitude,
+inasmuch as the centrifugal force is increased and the magnitude and
+direction of the attraction of the earth are altered, and the effect
+upon the latitude is a very small term expressed by the formula
+h (g&prime; &minus; g) sin 2 &phi;/ag, where g, g&prime; are the values of gravity at the equator
+and at the pole. This is h sin 2 &phi;/5820 seconds, h being in metres,
+a quantity which may be neglected, since for ordinary mountain
+heights it amounts to only a few hundredths of a second. We
+can assume this amount as joined with the northern component of
+the plumb-line perturbations.</p>
+
+<p>The uncertainties of terrestrial refraction render it impossible to
+determine accurately by vertical angles the heights of distant points.
+Generally speaking, refraction is greatest at about daybreak; from
+that time it diminishes, being at a minimum for a couple of hours
+before and after mid-day; later in the afternoon it again increases.
+This at least is the general march of the phenomenon, but it is by
+no means regular. The vertical angles measured at the station on
+Hart Fell showed on one occasion in the month of September a
+refraction of double the average amount, lasting from 1 <span class="scs">P.M.</span> to 5 <span class="scs">P.M.</span>
+The mean value of the coefficient of refraction k determined from a
+very large number of observations of terrestrial zenith distances in
+Great Britain is .0792 ± .0047; and if we separate those rays which
+for a considerable portion of their length cross the sea from those
+which do not, the former give k = .0813 and the latter k = .0753.
+These values are determined from high stations and long distances;
+when the distance is short, and the rays graze the ground, the amount
+of refraction is extremely uncertain and variable. A case is noted
+in the Indian survey where the zenith distance of a station 10.5 miles
+off varied from a depression of 4&prime; 52&Prime;.6 at 4.30 <span class="scs">P.M.</span> to an elevation
+of 2&prime; 24&Prime;.0 at 10.50 <span class="scs">P.M.</span></p>
+
+<p>If h, h&prime; be the heights above the level of the sea of two stations,
+90° + &delta;, 90° + &delta;&prime; their mutual zenith distances (&delta; being that observed
+at h), s their distance apart, the earth being regarded as a sphere of
+radius = a, then, with sufficient precision,</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">h&prime; &minus; h = s tan <span class="f150">(</span> s</td> <td>1 &minus; 2k</td>
+<td rowspan="2">&minus; &delta;<span class="f150">)</span>, &emsp; h &minus; h&prime; = s tan <span class="f150">(</span> s</td> <td>1 &minus; 2k</td>
+<td rowspan="2">&minus; &delta;&prime;<span class="f150">)</span>.</td></tr>
+<tr><td class="denom">2a</td> <td class="denom">2a</td></tr></table>
+
+<p class="noind">If from a station whose height is h the horizon of the sea be observed
+to have a zenith distance 90° + &delta;, then the above formula gives for h
+the value</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">h =</td> <td>a</td>
+<td rowspan="2">&nbsp;</td> <td>tan² &delta;</td>
+<td rowspan="2"></td></tr>
+<tr><td class="denom">2</td> <td class="denom">1 &minus; 2k</td></tr></table>
+
+<p>Suppose the depression &delta; to be n minutes, then h = 1.054n² if
+the ray be for the greater part of its length crossing the sea; if
+otherwise, h = 1.040n². To take an example: the mean of eight
+observations of the zenith distance of the sea horizon at the top of
+Ben Nevis is 91° 4&prime; 48&Prime;, or &delta; = 64.8; the ray is pretty equally disposed
+over land and water, and hence h = 1.047n² = 4396 ft. The
+actual height of the hill by spirit-levelling is 4406 ft., so that the error
+of the height thus obtained is only 10 ft.</p>
+
+<p>The determination of altitudes by means of spirit-levelling is
+undoubtedly the most exact method, particularly in its present
+development as precise-levelling, by which there have been determined
+in all civilized countries close-meshed nets of elevated points
+covering the entire land.</p>
+</div>
+<div class="author">(A. R. C; F. R. H.)</div>
+
+<hr class="foot" /> <div class="note">
+
+<p><a name="ft1a" id="ft1a" href="#fa1a"><span class="fn">1</span></a> An arrangement acting similarly had been previously introduced
+by Borda.</p>
+
+<p><a name="ft2a" id="ft2a" href="#fa2a"><span class="fn">2</span></a> <i>Geodetic Survey of South Africa</i>, vol. iii. (1905), p. viii; <i>Les Nouveaux
+Appareils pour la mesure rapide des bases géod.</i>, par J. René Benoīt
+et Ch. Éd. Guillaume (1906).</p>
+
+<p><a name="ft3a" id="ft3a" href="#fa3a"><span class="fn">3</span></a> See a paper &ldquo;On the Course of Geodetic Lines on the Earth&rsquo;s
+Surface&rdquo; in the <i>Phil. Mag.</i> 1870; Helmert, <i>Theorien der höheren
+Geodäsie</i>, 1. 321.</p>
+
+<p><a name="ft4a" id="ft4a" href="#fa4a"><span class="fn">4</span></a> Helmert, Theorien der höheren Geodäsie, 1. 232, 247.</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">GEOFFREY<a name="ar2" id="ar2"></a></span>, surnamed <span class="sc">Martel</span> (1006-1060), count of Anjou,
+son of the count Fulk Nerra (<i>q.v.</i>) and of the countess Hildegarde
+or Audegarde, was born on the 14th of October 1006. During his
+father&rsquo;s lifetime he was recognized as suzerain by Fulk l&rsquo;Oison
+(&ldquo;the Gosling&rdquo;), count of Vendōme, the son of his half-sister
+Adela. Fulk having revolted, he confiscated the countship,
+which he did not restore till 1050. On the 1st of January 1032
+he married Agnes, widow of William the Great, duke of Aquitaine,
+and taking arms against William the Fat, eldest son and successor
+of William the Great, defeated him and took him prisoner at
+Mont-Couėr near Saint-Jouin-de-Marnes on the 20th of September
+1033. He then tried to win recognition as dukes of Aquitaine for
+the sons of his wife Agnes by William the Great, who were still
+minors, but Fulk Nerra promptly took up arms to defend his
+suzerain William the Fat, from whom he held the Loudunois and
+<span class="pagenum"><a name="page616" id="page616"></a>616</span>
+Saintonge in fief against his son. In 1036 Geoffrey Martel had to
+liberate William the Fat, on payment of a heavy ransom, but the
+latter having died in 1038, and the second son of William the
+Great, Odo, duke of Gascony, having fallen in his turn at the
+siege of Mauzé (10th of March 1039) Geoffrey made peace with his
+father in the autumn of 1039, and had his wife&rsquo;s two sons recognized
+as dukes. About this time, also, he had interfered in the
+affairs of Maine, though without much result, for having sided
+against Gervais, bishop of Le Mans, who was trying to make
+himself guardian of the young count of Maine, Hugh, he had been
+beaten and forced to make terms with Gervais in 1038. In 1040
+he succeeded his father in Anjou and was able to conquer Touraine
+(1044) and assert his authority over Maine (see <span class="sc"><a href="#artlinks">Anjou</a></span>). About
+1050 he repudiated Agnes, his first wife, and married Grécie, the
+widow of Bellay, lord of Montreuil-Bellay (before August 1052),
+whom he subsequently left in order to marry Adela, daughter of a
+certain Count Odo. Later he returned to Grécie, but again left
+her to marry Adelaide the German. When, however, he died on
+the 14th of November 1060, at the monastery of St Nicholas at
+Angers, he left no children, and transmitted the countship to
+Geoffrey the Bearded, the eldest of his nephews (see ANJOU).</p>
+
+<div class="condensed">
+<p>See Louis Halphen, <i>Le Comté d&rsquo;Anjou au XI<span class="sp">e</span> sičcle</i> (Paris, 1906).
+A summary biography is given by Célestin Port, <i>Dictionnaire
+historique, géographique et biographique de Maine-et-Loire</i> (3 vols.,
+Paris-Angers, 1874-1878), vol. ii. pp. 252-253, and a sketch of the
+wars by Kate Norgate, <i>England under the Angevin Kings</i> (2 vols.,
+London, 1887), vol. i. chs. iii. iv.</p>
+</div>
+<div class="author">(L. H.*)</div>
+
+
+<hr class="art" />
+<p><span class="bold">GEOFFREY,<a name="ar3" id="ar3"></a></span> surnamed <span class="sc">Plantagenet</span> [or <span class="sc">Plantegenet</span>]
+(1113-1151), count of Anjou, was the son of Count Fulk the Young
+and of Eremburge (or Arembourg of La Flčche); he was born on
+the 24th of August 1113. He is also called &ldquo;le bel&rdquo; or &ldquo;the
+handsome,&rdquo; and received the surname of Plantagenet from the
+habit which he is said to have had of wearing in his cap a sprig of
+broom (<i>genźt</i>). In 1127 he was made a knight, and on the 2nd of
+June 1129 married Matilda, daughter of Henry I. of England, and
+widow of the emperor Henry V. Some months afterwards he
+succeeded to his father, who gave up the countship when he
+definitively went to the kingdom of Jerusalem. The years of his
+government were spent in subduing the Angevin barons and in
+conquering Normandy (see <span class="sc"><a href="#artlinks">Anjou</a></span>). In 1151, while returning
+from the siege of Montreuil-Bellay, he took cold, in consequence of
+bathing in the Loir at Chāteau-du-Loir, and died on the 7th of
+September. He was buried in the cathedral of Le Mans. By his
+wife Matilda he had three sons: Henry Plantagenet, born at Le
+Mans on Sunday, the 5th of March 1133; Geoffrey, born at
+Argentan on the 1st of June 1134; and William Long-Sword, born
+on the 22nd of July 1136.</p>
+
+<div class="condensed">
+<p>See Kate Norgate, <i>England under the Angevin Kings</i> (2 vols.,
+London, 1887), vol. i. chs. v.-viii.; Célestin Port, <i>Dictionnaire
+historique, géographique et biographique de Maine-et-Loire</i> (3 vols.,
+Paris-Angers, 1874-1878), vol. ii. pp. 254-256. A history of
+Geoffrey le Bel has yet to be written; there is a biography of him
+written in the 12th century by Jean, a monk of Marmoutier, <i>Historia
+Gaufredi, ducis Normannorum et comitis Andegavorum</i>, published by
+Marchegay et Salmon; &ldquo;Chroniques des comtes d&rsquo;Anjou&rdquo; (<i>Société
+de l&rsquo;histoire de France</i>, Paris, 1856), pp. 229-310.</p>
+</div>
+<div class="author">(L. H.*)</div>
+
+
+<hr class="art" />
+<p><span class="bold">GEOFFREY<a name="ar4" id="ar4"></a></span> (1158-1186), duke of Brittany, fourth son of the
+English king Henry II. and his wife Eleanor of Aquitaine, was
+born on the 23rd of September 1158. In 1167 Henry suggested a
+marriage between Geoffrey and Constance (d. 1201), daughter and
+heiress of Conan IV., duke of Brittany (d. 1171); and Conan not
+only assented, perhaps under compulsion, to this proposal, but
+surrendered the greater part of his unruly duchy to the English
+king. Having received the homage of the Breton nobles,
+Geoffrey joined his brothers, Henry and Richard, who, in alliance
+with Louis VII. of France, were in revolt against their father; but
+he made his peace in 1174, afterwards helping to restore order in
+Brittany and Normandy, and aiding the new French king, Philip
+Augustus, to crush some rebellious vassals. In July 1181 his
+marriage with Constance was celebrated, and practically the
+whole of his subsequent life was spent in warfare with his brother
+Richard. In 1183 he made peace with his father, who had come
+to Richard&rsquo;s assistance; but a fresh struggle soon broke out for
+the possession of Anjou, and Geoffrey was in Paris treating for
+aid with Philip Augustus, when he died on the 19th of August
+1186. He left a daughter, Eleanor, and his wife bore a
+posthumous son, the unfortunate Arthur.</p>
+
+
+<hr class="art" />
+<p><span class="bold">GEOFFREY<a name="ar5" id="ar5"></a></span> (<i>c.</i> 1152-1212), archbishop of York, was a bastard
+son of Henry II., king of England. He was distinguished from
+his legitimate half-brothers by his consistent attachment and
+fidelity to his father. He was made bishop of Lincoln at the age
+of twenty-one (1173); but though he enjoyed the temporalities
+he was never consecrated and resigned the see in 1183. He then
+became his father&rsquo;s chancellor, holding a large number of lucrative
+benefices in plurality. Richard nominated him archbishop of
+York in 1189, but he was not consecrated till 1191, or enthroned
+till 1194. Geoffrey, though of high character, was a man of
+uneven temper; his history in chiefly one of quarrels, with the
+see of Canterbury, with the chancellor <span class="correction" title="amended from Willian">William</span> Longchamp, with
+his half-brothers Richard and John, and especially with his
+canons at York. This last dispute kept him in litigation before
+Richard and the pope for many years. He led the clergy in their
+refusal to be taxed by John and was forced to fly the kingdom in
+1207. He died in Normandy on the 12th of December 1212.</p>
+
+<div class="condensed">
+<p>See Giraldus Cambrensis, <i>Vita Galfridi</i>; Stubbs&rsquo;s prefaces to
+<i>Roger de Hoveden</i>, vols. iii. and iv. (Rolls Series).</p>
+</div>
+<div class="author">(H. W. C. D.)</div>
+
+
+<hr class="art" />
+<p><span class="bold">GEOFFREY DE MONTBRAY<a name="ar6" id="ar6"></a></span> (d. 1093), bishop of Coutances
+(<i>Constantiensis</i>), a right-hand man of William the Conqueror, was
+a type of the great feudal prelate, warrior and administrator at
+need. He knew, says Orderic, more about marshalling mailed
+knights than edifying psalm-singing clerks. Obtaining, as a young
+man, in 1048, the see of Coutances, by his brother&rsquo;s influence
+(see <span class="sc"><a href="#artlinks">Mowbray</a></span>), he raised from his fellow nobles and from their
+Sicilian spoils funds for completing his cathedral, which was
+consecrated in 1056. With bishop Odo, a warrior like himself,
+he was on the battle-field of Hastings, exhorting the Normans to
+victory; and at William&rsquo;s coronation it was he who called on
+them to acclaim their duke as king. His reward in England was a
+mighty fief scattered over twelve counties. He accompanied
+William on his visit to Normandy (1067), but, returning, led a
+royal force to the relief of Montacute in September 1069. In 1075
+he again took the field, leading with Bishop Odo a vast host
+against the rebel earl of Norfolk, whose stronghold at Norwich
+they besieged and captured.</p>
+
+<p>Meanwhile the Conqueror had invested him with important
+judicial functions. In 1072 he had presided over the great
+Kentish suit between the primate and Bishop Odo, and about the
+same time over those between the abbot of Ely and his despoilers,
+and between the bishop of Worcester and the abbot of Ely, and
+there is some reason to think that he acted as a Domesday
+commissioner (1086), and was placed about the same time in
+charge of Northumberland. The bishop, who attended the
+Conqueror&rsquo;s funeral, joined in the great rising against William
+Rufus next year (1088), making Bristol, with which (as
+Domesday shows) he was closely connected and where he had
+built a strong castle, his base of operations. He burned Bath and
+ravaged Somerset, but had submitted to the king before the end
+of the year. He appears to have been at Dover with William in
+January 1090, but, withdrawing to Normandy, died at Coutances
+three years later. In his fidelity to Duke Robert he seems to
+have there held out for him against his brother Henry, when the
+latter obtained the Cotentin.</p>
+
+<div class="condensed">
+<p>See E.A. Freeman, <i>Norman Conquest</i> and <i>William Rufus</i>; J.H.
+Round, <i>Feudal England</i>; and, for original authorities, the works of
+Orderic Vitalis and William of Poitiers, and of Florence of Worcester;
+the Anglo-Saxon Chronicle; William of Malmesbury&rsquo;s <i>Gesta pontificum</i>,
+and Lanfranc&rsquo;s works, ed. Giles; Domesday Book.</p>
+</div>
+<div class="author">(J. H. R.)</div>
+
+
+<hr class="art" />
+<p><span class="bold">GEOFFREY OF MONMOUTH<a name="ar7" id="ar7"></a></span> (d. 1154), bishop of St Asaph
+and writer on early British history, was born about the year 1100.
+Of his early life little is known, except that he received a liberal
+education under the eye of his paternal uncle, Uchtryd, who was
+at that time archdeacon, and subsequently bishop, of Llandaff.
+In 1129 Geoffrey appears at Oxford among the witnesses of an
+Oseney charter. He subscribes himself Geoffrey Arturus;
+from this we may perhaps infer that he had already begun his
+experiments in the manufacture of Celtic mythology. A first
+edition of his <i>Historia Britonum</i> was in circulation by the year
+<span class="pagenum"><a name="page617" id="page617"></a>617</span>
+1139, although the text which we possess appears to date from
+1147. This famous work, which the author has the audacity
+to place on the same level with the histories of William of
+Malmesbury and Henry of Huntingdon, professes to be a translation
+from a Celtic source; &ldquo;a very old book in the British
+tongue&rdquo; which Walter, archdeacon of Oxford, had brought
+from Brittany. Walter the archdeacon is a historical personage;
+whether his book has any real existence may be fairly questioned.
+There is nothing in the matter or the style of the <i>Historia</i> to
+preclude us from supposing that Geoffrey drew partly upon
+confused traditions, partly on his own powers of invention, and
+to a very slight degree upon the accepted authorities for early
+British history. His chronology is fantastic and incredible;
+William of Newburgh justly remarks that, if we accepted the
+events which Geoffrey relates, we should have to suppose that
+they had happened in another world. William of Newburgh
+wrote, however, in the reign of Richard I. when the reputation
+of Geoffrey&rsquo;s work was too well established to be shaken by such
+criticisms. The fearless romancer had achieved an immediate
+success. He was patronized by Robert, earl of Gloucester, and
+by two bishops of Lincoln; he obtained, about 1140, the archdeaconry
+of Llandaff &ldquo;on account of his learning&rdquo;; and in
+1151 was promoted to the see of St Asaph.</p>
+
+<p>Before his death the <i>Historia Britonum</i> had already become a
+model and a quarry for poets and chroniclers. The list of
+imitators begins with Geoffrey Gaimar, the author of the <i>Estorie
+des Engles</i> (<i>c.</i> 1147), and Wace, whose <i>Roman de Brut</i> (1155) is
+partly a translation and partly a free paraphrase of the <i>Historia</i>.
+In the next century the influence of Geoffrey is unmistakably
+attested by the <i>Brut</i> of Layamon, and the rhyming English
+chronicle of Robert of Gloucester. Among later historians who
+were deceived by the <i>Historia Britonum</i> it is only needful to
+mention Higdon, Hardyng, Fabyan (1512), Holinshed (1580)
+and John Milton. Still greater was the influence of Geoffrey
+upon those writers who, like Warner in <i>Albion&rsquo;s England</i> (1586),
+and Drayton in <i>Polyolbion</i> (1613), deliberately made their
+accounts of English history as poetical as possible. The stories
+which Geoffrey preserved or invented were not infrequently
+a source of inspiration to literary artists. The earliest English
+tragedy, <i>Gorboduc</i> (1565), the <i>Mirror for Magistrates</i> (1587), and
+Shakespeare&rsquo;s Lear, are instances in point. It was, however,
+the Arthurian legend which of all his fabrications attained the
+greatest vogue. In the work of expanding and elaborating this
+theme the successors of Geoffrey went as far beyond him as he
+had gone beyond Nennius; but he retains the credit due to the
+founder of a great school. Marie de France, who wrote at the
+court of Henry II., and Chrétien de Troyes, her French contemporary,
+were the earliest of the avowed romancers to take
+up the theme. The succeeding age saw the Arthurian story
+popularized, through translations of the French romances, as
+far afield as Germany and Scandinavia. It produced in England
+the <i>Roman du Saint Graal</i> and the <i>Roman de Merlin</i>, both from
+the pen of Robert de Borron; the <i>Roman de Lancelot</i>; the <i>Roman
+de Tristan</i>, which is attributed to a fictitious Lucas de Gast. In
+the reign of Edward IV. Sir Thomas Malory paraphrased and
+arranged the best episodes of these romances in English prose.
+His <i>Morte d&rsquo;Arthur</i>, printed by Caxton in 1485, epitomizes the
+rich mythology which Geoffrey&rsquo;s work had first called into life,
+and gave the Arthurian story a lasting place in the English
+imagination. The influence of the <i>Historia Britonum</i> may be
+illustrated in another way, by enumerating the more familiar
+of the legends to which it first gave popularity. Of the twelve
+books into which it is divided only three (Bks. IX., X., XI.) are
+concerned with Arthur. Earlier in the work, however, we have
+the adventures of Brutus; of his follower Corineus, the vanquisher
+of the Cornish giant Goemagol (Gogmagog); of Locrinus and
+his daughter Sabre (immortalized in Milton&rsquo;s <i>Comus</i>); of Bladud
+the builder of Bath; of Lear and his daughters; of the three
+pairs of brothers, Ferrex and Porrex, Brennius and Belinus,
+Elidure and Peridure. The story of Vortigern and Rowena
+takes its final form in the <i>Historia Britonum</i>; and Merlin makes
+his first appearance in the prelude to the Arthur legend. Besides
+the <i>Historia Britonum</i> Geoffrey is also credited with a <i>Life of
+Merlin</i> composed in Latin verse. The authorship of this work
+has, however, been disputed, on the ground that the style is distinctly
+superior to that of the <i>Historia</i>. A minor composition, the
+<i>Prophecies of Merlin</i>, was written before 1136, and afterwards incorporated
+with the <i>Historia</i>, of which it forms the seventh book.</p>
+
+<div class="condensed">
+<p>For a discussion of the manuscripts of Geoffrey&rsquo;s work, see Sir
+T.D. Hardy&rsquo;s <i>Descriptive Catalogue</i> (Rolls Series), i. pp. 341 ff. The
+<i>Historia Britonum</i> has been critically edited by San Marte (Halle,
+1854). There is an English translation by J.A. Giles (London, 1842).
+The <i>Vita Merlini</i> has been edited by F. Michel and T. Wright (Paris,
+1837). See also the <i>Dublin Univ. Magazine</i> for April 1876, for an
+article by T. Gilray on the literary influence of Geoffrey; G. Heeger&rsquo;s
+<i>Trojanersage der Britten</i> (1889); and La Borderie&rsquo;s <i>Études historiques
+bretonnes</i> (1883).</p>
+</div>
+<div class="author">(H. W. C. D.)</div>
+
+
+<hr class="art" />
+<p><span class="bold">GEOFFREY OF PARIS<a name="ar8" id="ar8"></a></span> (d. <i>c.</i> 1320), French chronicler, was
+probably the author of the <i>Chronique métrique de Philippe le
+Bel, or Chronique rimée de Geoffroi de Paris</i>. This work, which
+deals with the history of France from 1300 to 1316, contains
+7918 verses, and is valuable as that of a writer who had a personal
+knowledge of many of the events which he relates. Various short
+historical poems have also been attributed to Geoffrey, but there
+is no certain information about either his life or his writings.</p>
+
+<div class="condensed">
+<p>The <i>Chronique</i> was published by J.A. Buchon in his <i>Collection des
+chroniques</i>, tome ix. (Paris, 1827), and it has also been printed in
+tome xxii. of the <i>Recueil des historiens des Gaules et de la France</i>
+(Paris, 1865). See G. Paris, <i>Histoire de la littérature franēaise au
+moyen āge</i> (Paris, 1890); and A. Molinier, <i>Les Sources de l&rsquo;histoire de
+France</i>, tome iii. (Paris, 1903).</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">GEOFFREY THE BAKER<a name="ar9" id="ar9"></a></span> (d. <i>c.</i> 1360), English chronicler,
+is also called Walter of Swinbroke, and was probably a secular
+clerk at Swinbrook in Oxfordshire. He wrote a <i>Chronicon
+Angliae temporibus Edwardi II. et Edwardi III.</i>, which deals
+with the history of England from 1303 to 1356. From the beginning
+until about 1324 this work is based upon Adam Murimuth&rsquo;s
+<i>Continuatio chronicarum</i>, but after this date it is valuable and
+interesting, containing information not found elsewhere, and
+closing with a good account of the battle of Poitiers. The author
+obtained his knowledge about the last days of Edward II. from
+William Bisschop, a companion of the king&rsquo;s murderers, Thomas
+Gurney and John Maltravers. Geoffrey also wrote a <i>Chroniculum</i>
+from the creation of the world until 1336, the value of
+which is very slight. His writings have been edited with notes
+by Sir E.M. Thompson as the <i>Chronicon Galfridi le Baker de
+Swynebroke</i> (Oxford, 1889). Some doubt exists concerning
+Geoffrey&rsquo;s share in the compilation of the <i>Vita et mors Edwardi
+II.</i>, usually attributed to Sir Thomas de la More, or Moor, and
+printed by Camden in his <i>Anglica scripta</i>. It has been maintained
+by Camden and others that More wrote an account of Edward&rsquo;s
+reign in French, and that this was translated into Latin by
+Geoffrey and used by him in compiling his <i>Chronicon</i>. Recent
+scholarship, however, asserts that More was no writer, and that
+the <i>Vita et mors</i> is an extract from Geoffrey&rsquo;s <i>Chronicon</i>, and
+was attributed to More, who was the author&rsquo;s patron. In the
+main this conclusion substantiates the verdict of Stubbs, who
+has published the <i>Vita et mors</i> in his <i>Chronicles of the reigns of
+Edward I. and Edward II.</i> (London, 1883). The manuscripts
+of Geoffrey&rsquo;s works are in the Bodleian library at Oxford.</p>
+
+
+<hr class="art" />
+<p><span class="bold">GEOFFRIN, MARIE THÉRČSE RODET<a name="ar10" id="ar10"></a></span> (1699-1777), a
+Frenchwoman who played an interesting part in French literary
+and artistic life, was born in Paris in 1699. She married, on the
+19th of July 1713, Pierre Franēois Geoffrin, a rich manufacturer
+and lieutenant-colonel of the National Guard, who died in 1750.
+It was not till Mme Geoffrin was nearly fifty years of age that we
+begin to hear of her as a power in Parisian society. She had
+learned much from Mme de Tencin, and about 1748 began to
+gather round her a literary and artistic circle. She had every
+week two dinners, on Monday for artists, and on Wednesday for
+her friends the Encyclopaedists and other men of letters. She
+received many foreigners of distinction, Hume and Horace
+Walpole among others. Walpole spent much time in her society
+before he was finally attached to Mme du Deffand, and speaks of
+her in his letters as a model of common sense. She was indeed
+somewhat of a small tyrant in her circle. She had adopted the
+pose of an old woman earlier than necessary, and her coquetry, if
+<span class="pagenum"><a name="page618" id="page618"></a>618</span>
+such it can be called, took the form of being mother and mentor to
+her guests, many of whom were indebted to her generosity for
+substantial help. Although her aim appears to have been to
+have the <i>Encyclopédie</i> in conversation and action around her, she
+was extremely displeased with any of her friends who were so
+rash as to incur open disgrace. Marmontel lost her favour after
+the official censure of <i>Bélisaire</i>, and her advanced views did not
+prevent her from observing the forms of religion. A devoted
+Parisian, Mme Geoffrin rarely left the city, so that her journey to
+Poland in 1766 to visit the king, Stanislas Poniatowski, whom she
+had known in his early days in Paris, was a great event in her life.
+Her experiences induced a sensible gratitude that she had been
+born &ldquo;<i>Franēaise</i>&rdquo; and &ldquo;<i>particuličre</i>.&rdquo; In her last illness her
+daughter, Thérčse, marquise de la Ferté Imbault, excluded her
+mother&rsquo;s old friends so that she might die as a good Christian, a
+proceeding wittily described by the old lady: &ldquo;My daughter is
+like Godfrey de Bouillon, she wished to defend my tomb from
+the infidels.&rdquo; Mme Geoffrin died in Paris on the 6th of October
+1777.</p>
+
+<div class="condensed">
+<p>See <i>Correspondance inédite du roi Stanislas Auguste Poniatowski et
+de Madame Geoffrin</i>, edited by the comte de Mou’ (1875); P. de
+Ségur, <i>Le Royaume de la rue Saint-Honoré, Madame Geoffrin et sa
+fille</i> (1897); A. Tornezy, <i>Un Bureau d&rsquo;esprit au XVIII<span class="sp">e</span> sičcle: le
+salon de Madame Geoffrin</i> (1895); and Janet Aldis, <i>Madame Geoffrin,
+her Salon and her Times, 1750-1777</i> (1905).</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">GEOFFROY, ÉTIENNE FRANĒOIS<a name="ar11" id="ar11"></a></span> (1672-1731), French
+chemist, born in Paris on the 13th of February 1672, was first
+an apothecary and then practised medicine. After studying at
+Montpellier he accompanied Marshal Tallard on his embassy to
+London in 1698 and thence travelled to Holland and Italy.
+Returning to Paris he became professor of chemistry at the
+Jardin du Roi and of pharmacy and medicine at the Collčge de
+France, and dean of the faculty of medicine. He died in Paris on
+the 6th of January 1731. His name is best known in connexion
+with his tables of affinities (<i>tables des rapports</i>), which he presented
+to the French Academy in 1718 and 1720. These were lists,
+prepared by collating observations on the actions of substances
+one upon another, showing the varying degrees of affinity exhibited
+by analogous bodies for different reagents, and they retained
+their vogue for the rest of the century, until displaced by the
+profounder conceptions introduced by C.L. Berthollet. Another
+of his papers dealt with the delusions of the philosopher&rsquo;s stone,
+but nevertheless he believed that iron could be artificially formed
+in the combustion of vegetable matter. His <i>Tractatus de materia
+medica</i>, published posthumously in 1741, was long celebrated.</p>
+
+<p>His brother <span class="sc">Claude Joseph</span>, known as Geoffroy the younger
+(1685-1752), was also an apothecary and chemist who, having a
+considerable knowledge of botany, devoted himself especially to
+the study of the essential oils in plants.</p>
+
+
+<hr class="art" />
+<p><span class="bold">GEOFFROY, JULIEN LOUIS<a name="ar12" id="ar12"></a></span> (1743-1814), French critic, was
+born at Rennes in 1743. He studied in the school of his native
+town and at the Collčge Louis le Grand in Paris. He took orders
+and fulfilled for some time the humble functions of an usher,
+eventually becoming professor of rhetoric at the <i>Collčge Mazarin</i>.
+A bad tragedy, Caton, was accepted at the <i>Théātre Franēais</i>, but
+was never acted. On the death of Élie Fréron in 1776 the other
+collaborators in the <i>Année littéraire</i> asked Geoffroy to succeed him,
+and he conducted the journal until in 1792 it ceased to appear.
+Geoffroy was a bitter critic of Voltaire and his followers, and
+made for himself many enemies. An enthusiastic royalist,
+he published with Fréron&rsquo;s brother-in-law, the abbé Thomas
+Royou (1741-1792), a journal, <i>L&rsquo;Ami du roi</i> (1790-1792),
+which possibly did more harm than good to the king&rsquo;s cause by its
+ill-advised partisanship. During the Terror Geoffroy hid in the
+neighbourhood of Paris, only returning in 1799. An attempt to
+revive the <i>Année littéraire</i> failed, and Geoffroy undertook the
+dramatic feuilleton of the <i>Journal des débats</i>. His scathing
+criticisms had a success of notoriety, but their popularity was
+ephemeral, and the publication of them (5 vols., 1819-1820) as
+<i>Cours de littérature dramatique</i> proved a failure. He was also the
+author of a perfunctory <i>Commentaire</i> on the works of Racine
+prefixed to Lenormant&rsquo;s edition (1808). He died in Paris on the
+27th of February 1814.</p>
+
+
+<hr class="art" />
+<p><span class="bold">GEOFFROY SAINT-HILAIRE, ÉTIENNE<a name="ar13" id="ar13"></a></span> (1772-1844), French
+naturalist, was the son of Jean Gčrard Geoffroy, procurator and
+magistrate of Étampes, Seine-et-Oise, where he was born on the
+15th of April 1772. Destined for the church he entered the
+college of Navarre, in Paris, where he studied natural philosophy
+under M.J. Brisson; and in 1788 he obtained one of the canonicates
+of the chapter of Sainte Croix at Étampes, and also a
+benefice. Science, however, offered him a more congenial career,
+and he gained from his father permission to remain in Paris, and
+to attend the lectures at the Collčge de France and the Jardin des
+Plantes, on the condition that he should also read law. He
+accordingly took up his residence at Cardinal Lemoine&rsquo;s college,
+and there became the pupil and soon the esteemed associate of
+Brisson&rsquo;s friend, the abbé Haüy, the mineralogist. Having,
+before the close of the year 1790, taken the degree of bachelor in
+law, he became a student of medicine, and attended the lectures of
+A.F. de Fourcroy at the Jardin des Plantes, and of L.J.M.
+Daubenton at the Collčge de France. His studies at Paris were at
+length suddenly interrupted, for, in August 1792, Haüy and the
+other professors of Lemoine&rsquo;s college, as also those of the college
+of Navarre, were arrested by the revolutionists as priests, and
+confined in the prison of St Firmin. Through the influence of
+Daubenton and others Geoffroy on the 14th of August obtained
+an order for the release of Haüy in the name of the Academy;
+still the other professors of the two colleges, save C.F. Lhomond,
+who had been rescued by his pupil J.L. Tallien, remained in
+confinement. Geoffroy, foreseeing their certain destruction if
+they remained in the hands of the revolutionists, determined if
+possible to secure their liberty by stratagem. By bribing one of
+the officials at St Firmin, and disguising himself as a commissioner
+of prisons, he gained admission to his friends, and entreated them
+to effect their escape by following him. All, however, dreading
+lest their deliverance should render the doom of their fellow-captives
+the more certain, refused the offer, and one priest only,
+who was unknown to Geoffroy, left the prison. Already on the
+night of the 2nd of September the massacre of the proscribed had
+begun, when Geoffroy, yet intent on saving the life of his friends
+and teachers, repaired to St Firmin. At 4 o&rsquo;clock on the morning
+of the 3rd of September, after eight hours&rsquo; waiting, he by means
+of a ladder assisted the escape of twelve ecclesiastics, not of the
+number of his acquaintance, and then the approach of dawn and
+the discharge of a gun directed at him warned him, his chief
+purpose unaccomplished, to return to his lodgings. Leaving Paris
+he retired to Étampes, where, in consequence of the anxieties of
+which he had lately been the prey, and the horrors which he had
+witnessed, he was for some time seriously ill. At the beginning
+of the winter of 1792 he returned to his studies in Paris, and in
+March of the following year Daubenton, through the interest of
+Bernardin de Saint Pierre, procured him the office of sub-keeper
+and assistant demonstrator of the cabinet of natural history,
+vacant by the resignation of B.G.E. Lacépčde. By a law
+passed in June 1793, Geoffroy was appointed one of the twelve
+professors of the newly constituted museum of natural history,
+being assigned the chair of zoology. In the same year he
+busied himself with the formation of a menagerie at that
+institution.</p>
+
+<p>In 1794 through the introduction of A.H. Tessier he entered
+into correspondence with Georges Cuvier, to whom, after the
+perusal of some of his manuscripts, he wrote: &ldquo;Venez jouer
+parmi nous le rōle de Linné, d&rsquo;un autre législateur de l&rsquo;histoire
+naturelle.&rdquo; Shortly after the appointment of Cuvier as assistant
+at the Muséum d&rsquo;Histoire Naturelle, Geoffroy received him into
+his house. The two friends wrote together five memoirs on
+natural history, one of which, on the classification of mammals,
+puts forward the idea of the subordination of characters upon
+which Cuvier based his zoological system. It was in a paper
+entitled &ldquo;Histoire des Makis, ou singes de Madagascar,&rdquo; written
+in 1795, that Geoffroy first gave expression to his views on &ldquo;the
+unity of organic composition,&rdquo; the influence of which is perceptible
+in all his subsequent writings; nature, he observes,
+presents us with only one plan of construction, the same in
+principle, but varied in its accessory parts.</p>
+
+<p><span class="pagenum"><a name="page619" id="page619"></a>619</span></p>
+
+<p>In 1798 Geoffroy was chosen a member of the great scientific
+expedition to Egypt, and on the capitulation of Alexandria in
+August 1801, he took part in resisting the claim made by the
+British general to the collections of the expedition, declaring that,
+were that demand persisted in, history would have to record
+that he also had burnt a library in Alexandria. Early in January
+1802 Geoffroy returned to his accustomed labours in Paris. He
+was elected a member of the academy of sciences of that city
+in September 1807. In March of the following year the emperor,
+who had already recognized his national services by the award
+of the cross of the legion of honour, selected him to visit the
+museums of Portugal, for the purpose of procuring collections
+from them, and in the face of considerable opposition from the
+British he eventually was successful in retaining them as a
+permanent possession for his country. In 1809, the year after
+his return to France, he was made professor of zoology at the
+faculty of sciences at Paris, and from that period he devoted
+himself more exclusively than before to anatomical study. In
+1818 he gave to the world the first part of his celebrated <i>Philosophie
+anatomique</i>, the second volume of which, published in
+1822, and subsequent memoirs account for the formation of
+monstrosities on the principle of arrest of development, and of
+the attraction of similar parts. When, in 1830, Geoffroy proceeded
+to apply to the invertebrata his views as to the unity of
+animal composition, he found a vigorous opponent in Georges
+Cuvier, and the discussion between them, continued up to the
+time of the death of the latter, soon attracted the attention of
+the scientific throughout Europe. Geoffroy, a synthesist, contended,
+in accordance with his theory of unity of plan in organic
+composition, that all animals are formed of the same elements,
+in the same number, and with the same connexions: homologous
+parts, however they differ in form and size, must remain associated
+in the same invariable order. With Goethe he held that there
+is in nature a law of compensation or balancing of growth, so
+that if one organ take on an excess of development, it is at the
+expense of some other part; and he maintained that, since
+nature takes no sudden leaps, even organs which are superfluous
+in any given species, if they have played an important part in
+other species of the same family, are retained as rudiments,
+which testify to the permanence of the general plan of creation.
+It was his conviction that, owing to the conditions of life, the
+same forms had not been perpetuated since the origin of all
+things, although it was not his belief that existing species are
+becoming modified. Cuvier, who was an analytical observer of
+facts, admitted only the prevalence of &ldquo;laws of co-existence&rdquo;
+or &ldquo;harmony&rdquo; in animal organs, and maintained the absolute
+invariability of species, which he declared had been created
+with a regard to the circumstances in which they were placed,
+each organ contrived with a view to the function it had to
+fulfil, thus putting, in Geoffroy&rsquo;s considerations, the effect for
+the cause.</p>
+
+<p>In July 1840 Geoffroy became blind, and some months later
+he had a paralytic attack. From that time his strength gradually
+failed him. He resigned his chair at the museum in 1841, and
+died at Paris on the 19th of June 1844.</p>
+
+<div class="condensed">
+<p>Geoffroy wrote: <i>Catalogue des mammifčres du Muséum National
+d&rsquo;Histoire Naturelle</i> (1813), not quite completed; <i>Philosophie anatomique</i>&mdash;t.
+i., <i>Des organes respiratoires</i> (1818), and t. ii., <i>Des monstruosités
+humaines</i> (1822); <i>Systčme dentaire des mammifčres et des
+oiseaux</i> (1st pt., 1824); <i>Sur le principe de l&rsquo;unité de composition
+organique</i> (1828); <i>Cours de l&rsquo;histoire naturelle des mammifčres</i>
+(1829); <i>Principes de philosophie zoologique</i> (1830); <i>Études progressives
+d&rsquo;un naturaliste</i> (1835); <i>Fragments biographiques</i> (1832);
+<i>Notions synthétiques, historiques et physiologiques de philosophie
+naturelle</i> (1838), and other works; also part of the <i>Description de
+l&rsquo;Égypte par la commission des sciences</i> (1821-1830); and, with
+Frédéric Cuvier (1773-1838), a younger brother of G. Cuvier, <i>Histoire
+naturelle des mammifčres</i> (4 vols., 1820-1842); besides numerous
+papers on such subjects as the anatomy of marsupials, ruminants
+and electrical fishes, the vertebrate theory of the skull, the opercula
+of fishes, teratology, palaeontology and the influence of surrounding
+conditions in modifying animal forms.</p>
+
+<p>See <i>Vie, travaux, et doctrine scientifique d&rsquo;Étienne Geoffroy Saint-Hilaire,
+par son fils M. Isidore Geoffroy Saint-Hilaire</i> (Paris and
+Strasburg, 1847), to which is appended a list of Geoffroy&rsquo;s works;
+and Joly, in <i>Biog. universelle</i>, t. xvi. (1856).</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">GEOFFROY SAINT-HILAIRE, ISIDORE<a name="ar14" id="ar14"></a></span> (1805-1861), French
+zoologist, son of the preceding, was born at Paris on the 16th of
+December 1805. In his earlier years he showed an aptitude for
+mathematics, but eventually he devoted himself to the study
+of natural history and of medicine, and in 1824 he was appointed
+assistant naturalist to his father. On the occasion of his taking
+the degree of doctor of medicine in September 1829, he read a
+thesis entitled <i>Propositions sur la monstruosité, considérée chez
+l&rsquo;homme et les animaux</i>; and in 1832-1837 was published his
+great teratological work, <i>Histoire générale et particuličre des
+anomalies de l&rsquo;organisation chez l&rsquo;homme et les animaux</i>, 3 vols.
+8vo. with 20 plates. In 1829 he delivered for his father the second
+part of a course of lectures on ornithology, and during the three
+following years he taught zoology at the Athénée, and teratology
+at the École pratique. He was elected a member of the academy
+of sciences at Paris in 1833, was in 1837 appointed to act as
+deputy for his father at the faculty of sciences in Paris, and in
+the following year was sent to Bordeaux to organize a similar
+faculty there. He became successively inspector of the academy
+of Paris (1840), professor of the museum on the retirement of
+his father (1841), inspector-general of the university (1844),
+a member of the royal council for public instruction (1845), and
+on the death of H.M.D. de Blainville, professor of zoology
+at the faculty of sciences (1850). In 1854 he founded the
+Acclimatization Society of Paris, of which he was president.
+He died at Paris on the 10th of November 1861.</p>
+
+<div class="condensed">
+<p>Besides the above-mentioned works, he wrote: <i>Essais de zoologie
+générale</i> (1841); <i>Vie ... d&rsquo;Étienne Geoffroy Saint-Hilaire</i> (1847);
+<i>Acclimatation et domestication des animaux utiles</i> (1849; 4th ed.,
+1861); <i>Lettres sur les substances alimentaires et particuličrement sur
+la viande de cheval</i> (1856); and <i>Histoire naturelle générale des rčgnes
+organiques</i> (3 vols., 1854-1862), which was not quite completed.
+He was the author also of various papers on zoology, comparative
+anatomy and palaeontology.</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">GEOGRAPHY<a name="ar15" id="ar15"></a></span> (Gr. <span class="grk" title="gź">&#947;&#8134;</span>, earth, and <span class="grk" title="graphein">&#947;&#961;&#940;&#966;&#949;&#953;&#957;</span>, to write), the
+exact and organized knowledge of the distribution of phenomena
+on the surface of the earth. The fundamental basis of geography
+is the vertical relief of the earth&rsquo;s crust, which controls all
+mobile distributions. The grander features of the relief of the
+lithosphere or stony crust of the earth control the distribution
+of the hydrosphere or collected waters which gather into the
+hollows, filling them up to a height corresponding to the volume,
+and thus producing the important practical division of the
+surface into land and water. The distribution of the mass of
+the atmosphere over the surface of the earth is also controlled
+by the relief of the crust, its greater or lesser density at the surface
+corresponding to the lesser or greater elevation of the surface.
+The simplicity of the zonal distribution of solar energy on the
+earth&rsquo;s surface, which would characterize a uniform globe, is
+entirely destroyed by the dissimilar action of land and water
+with regard to radiant heat, and by the influence of crust-forms
+on the direction of the resulting circulation. The influence of
+physical environment becomes clearer and stronger when the
+distribution of plant and animal life is considered, and if it is
+less distinct in the case of man, the reason is found in the modifications
+of environment consciously produced by human effort.
+Geography is a synthetic science, dependent for the data with
+which it deals on the results of specialized sciences such as
+astronomy, geology, oceanography, meteorology, biology and
+anthropology, as well as on topographical description. The
+physical and natural sciences are concerned in geography only
+so far as they deal with the forms of the earth&rsquo;s surface, or as
+regards the distribution of phenomena. The distinctive task of
+geography as a science is to investigate the control exercised by
+the crust-forms directly or indirectly upon the various mobile
+distributions. This gives to it unity and definiteness, and renders
+superfluous the attempts that have been made from time to
+time to define the limits which divide geography from geology
+on the one hand and from history on the other. It is essential
+to classify the subject-matter of geography in such a manner as
+to give prominence not only to facts, but to their mutual relations
+and their natural and inevitable order.</p>
+
+<p>The fundamental conception of geography is form, including
+<span class="pagenum"><a name="page620" id="page620"></a>620</span>
+the figure of the earth and the varieties of crustal relief. Hence
+mathematical geography (see <span class="sc"><a href="#artlinks">Map</a></span>), including cartography as
+a practical application, comes first. It merges into physical
+geography, which takes account of the forms of the lithosphere
+(geomorphology), and also of the distribution of the hydrosphere
+and the rearrangements resulting from the workings of solar
+energy throughout the hydrosphere and atmosphere (oceanography
+and climatology). Next follows the distribution of plants
+and animals (biogeography), and finally the distribution of
+mankind and the various artificial boundaries and redistributions
+(anthropogeography). The applications of anthropogeography
+to human uses give rise to political and commercial geography,
+in the elucidation of which all the earlier departments or stages
+have to be considered, together with historical and other purely
+human conditions. The evolutionary idea has revolutionized
+and unified geography as it did biology, breaking down the old
+hard-and-fast partitions between the various departments, and
+substituting the study of the nature and influence of actual
+terrestrial environments for the earlier motive, the discovery
+and exploration of new lands.</p>
+
+<p class="pt2 center sc">History of Geographical Theory</p>
+
+<div class="condensed">
+<p>The earliest conceptions of the earth, like those held by the primitive
+peoples of the present day, are difficult to discover and almost
+impossible fully to grasp. Early generalizations, as far as they were
+made from known facts, were usually expressed in symbolic language,
+and for our present purpose it is not profitable to speculate on the
+underlying truths which may sometimes be suspected in the old
+mythological cosmogonies.</p>
+
+<p>The first definite geographical theories to affect the western world
+were those evolved, or at least first expressed, by the Greeks.<a name="fa1b" id="fa1b" href="#ft1b"><span class="sp">1</span></a>
+The earliest theoretical problem of geography was the
+<span class="sidenote">Early Greek ideas.<br />Flat earth of Homer.</span>
+form of the earth. The natural supposition that the earth
+is a flat disk, circular or elliptical in outline, had in the
+time of Homer acquired a special definiteness by the
+introduction of the idea of the ocean river bounding the whole, an
+application of imperfectly understood observations. Thales of
+Miletus is claimed as the first exponent of the idea of a
+spherical earth; but, although this does not appear to be
+warranted, his disciple Anaximander (<i>c.</i> 580 <span class="scs">B.C.</span>) put
+forward the theory that the earth had the figure of a solid body
+hanging freely in the centre of the hollow sphere of the starry heavens.
+The Pythagorean school of philosophers adopted the theory of a
+spherical earth, but from metaphysical rather than scientific reasons;
+their convincing argument was that a sphere being the most perfect
+solid figure was the only one worthy to circumscribe the dwelling-place
+of man. The division of the sphere into parallel zones and
+some of the consequences of this generalization seem to have presented
+themselves to Parmenides (<i>c.</i> 450 <span class="scs">B.C.</span>); but these ideas did
+not influence the Ionian school of philosophers, who in their treatment
+of geography preferred to deal with facts demonstrable by
+<span class="sidenote">Hecataeus.<br /><br />Herodotus.</span>
+travel rather than with speculations. Thus Hecataeus,
+claimed by H.F. Tozer<a name="fa2b" id="fa2b" href="#ft2b"><span class="sp">2</span></a> as the father of geography on
+account of his <i>Periodos</i>, or general treatise on the earth, did not
+advance beyond the primitive conception of a circular disk. He
+systematized the form of the land within the ring of ocean&mdash;the
+<span class="grk" title="oikoumenź">&#959;&#7984;&#954;&#959;&#965;&#956;&#941;&#957;&#951;</span>, or habitable world&mdash;by recognizing two continents:
+Europe to the north, and Asia to the south of the midland sea.
+Herodotus, equally oblivious of the sphere, criticized and
+ridiculed the circular outline of the <i>oekumene</i>, which he
+knew to be longer from east to west than it was broad from north to
+south. He also pointed out reasons for accepting a division of the
+land into three continents&mdash;Europe, Asia and Africa. Beyond the
+limits of his personal travels Herodotus applied the characteristically
+Greek theory of symmetry to complete, in the unknown, outlines
+<span class="sidenote">The idea of symmetry.</span>
+of lands and rivers analogous to those which had been
+explored. Symmetry was in fact the first geographical
+theory, and the effect of Herodotus&rsquo;s hypothesis that the
+Nile must flow from west to east before turning north in
+order to balance the Danube running from west to east before turning
+south lingered in the maps of Africa down to the time of Mungo
+Park.<a name="fa3b" id="fa3b" href="#ft3b"><span class="sp">3</span></a></p>
+
+<p>To Aristotle (384-322 <span class="scs">B.C.</span>) must be given the distinction of founding
+scientific geography. He demonstrated the sphericity of the
+earth by three arguments, two of which could be tested by observation.
+These were: (1) that the earth must be spherical, because
+<span class="sidenote">Aristotle and the sphere.</span>
+of the tendency of matter to fall together towards a common
+centre; (2) that only a sphere could always throw a
+circular shadow on the moon during an eclipse; and (3)
+that the shifting of the horizon and the appearance of
+new constellations, or the disappearance of familiar stars, as one
+travelled from north to south, could only be explained on the hypothesis
+that the earth was a sphere. Aristotle, too, gave greater
+definiteness to the idea of zones conceived by Parmenides, who had
+pictured a torrid zone uninhabitable by reason of heat, two frigid
+zones uninhabitable by reason of cold, and two intermediate temperate
+zones fit for human occupation. Aristotle defined the temperate
+zone as extending from the tropic to the arctic circle, but there is
+some uncertainty as to the precise meaning he gave to the term
+&ldquo;arctic circle.&rdquo; Soon after his time, however, this conception was
+clearly established, and with so large a generalization the mental
+horizon was widened to conceive of a geography which was a science.
+Aristotle had himself shown that in the southern temperate zone
+winds similar to those of the northern temperate zone should blow,
+but from the opposite direction.</p>
+
+<p>While the theory of the sphere was being elaborated the efforts of
+practical geographers were steadily directed towards ascertaining
+the outline and configuration of the <i>oekumene</i>, or habitable
+world, the only portion of the terrestrial surface known
+<span class="sidenote">Fitting the oekumene to the sphere.</span>
+to the ancients and to the medieval peoples, and still
+retaining a shadow of its old monopoly of geographical
+attention in its modern name of the &ldquo;Old World.&rdquo; The
+fitting of the <i>oekumene</i> to the sphere was the second theoretical
+problem. The circular outline had given way in geographical
+opinion to the elliptical with the long axis lying east and west, and
+Aristotle was inclined to view it as a very long and relatively narrow
+band almost encircling the globe in the temperate zone. His argument
+as to the narrowness of the sea between West Africa and East
+Asia, from the occurrence of elephants at both extremities, is difficult
+to understand, although it shows that he looked on the distribution
+of animals as a problem of geography.</p>
+
+<p>Pythagoras had speculated as to the existence of antipodes, but
+it was not until the first approximately accurate measurements of
+the globe and estimates of the length and breadth of the
+<i>oekumene</i> were made by Eratosthenes (<i>c.</i> 250 <span class="scs">B.C.</span>) that
+<span class="sidenote">Problem of the Antipodes.</span>
+the fact that, as then known, it occupied less than a quarter
+of the surface of the sphere was clearly recognized. It was
+natural, if not strictly logical, that the ocean river should be extended
+from a narrow stream to a world-embracing sea, and here again
+Greek theory, or rather fancy, gave its modern name to the greatest
+feature of the globe. The old instinctive idea of symmetry must
+often have suggested other <i>oekumene</i> balancing the known world
+in the other quarters of the globe. The Stoic philosophers, especially
+Crates of Mallus, arguing from the love of nature for life, placed an
+<i>oekumene</i> in each quarter of the sphere, the three unknown world-islands
+being those of the Antoeci, Perioeci and Antipodes. This
+was a theory not only attractive to the philosophical mind, but
+eminently adapted to promote exploration. It had its opponents,
+however, for Herodotus showed that sea-basins existed cut off from
+the ocean, and it is still a matter of controversy how far the pre-Ptolemaic
+geographers believed in a water-connexion between the
+Atlantic and Indian oceans. It is quite clear that Pomponius Mela
+(<i>c.</i> <span class="scs">A.D.</span> 40), following Strabo, held that the southern temperate zone
+contained a habitable land, which he designated by the name
+<i>Antichthones</i>.</p>
+
+<p>Aristotle left no work on geography, so that it is impossible to
+know what facts he associated with the science of the earth&rsquo;s surface.
+The word geography did not appear before Aristotle,
+the first use of it being in the <span class="grk" title="Peri kosmōn">&#928;&#949;&#961;&#8054; &#954;&#972;&#963;&#956;&#969;&#957;</span>, which is one
+<span class="sidenote">Aristotle&rsquo;s geographical views.</span>
+of the writings doubtfully ascribed to him, and H. Berger
+considers that the expression was introduced by Eratosthenes.<a name="fa4b" id="fa4b" href="#ft4b"><span class="sp">4</span></a>
+Aristotle was certainly conversant with many
+facts, such as the formation of deltas, coast-erosion, and to a certain
+extent the dependence of plants and animals on their physical
+surroundings. He formed a comprehensive theory of the variations
+of climate with latitude and season, and was convinced of the necessity
+of a circulation of water between the sea and rivers, though,
+like Plato, he held that this took place by water rising from the sea
+through crevices in the rocks, losing its dissolved salts in the process.
+He speculated on the differences in the character of races of mankind
+living in different climates, and correlated the political forms of
+communities with their situation on a seashore, or in the neighbourhood
+of natural strongholds.</p>
+
+<p>Strabo (<i>c.</i> 50 <span class="scs">B.C.</span>-<span class="scs">A.D.</span> 24) followed Eratosthenes rather than
+Aristotle, but with sympathies which went out more to the human
+interests than the mathematical basis of geography. He
+<span class="sidenote">Strabo.</span>
+compiled a very remarkable work dealing, in large measure
+from personal travel, with the countries surrounding the Mediterranean.
+He may be said to have set the pattern which was followed
+in succeeding ages by the compilers of &ldquo;political geographies&rdquo;
+<span class="pagenum"><a name="page621" id="page621"></a>621</span>
+dealing less with theories than with facts, and illustrating rather than
+formulating the principles of the science.</p>
+
+<p>Claudius Ptolemaeus (<i>c.</i> <span class="scs">A.D.</span> 150) concentrated in his writings the
+final outcome of all Greek geographical learning, and passed it across
+the gulf of the middle ages by the hands of the Arabs,
+to form the starting-point of the science in modern times.
+<span class="sidenote">Ptolemy.</span>
+His geography was based more immediately on the work of his
+predecessor, Marinus of Tyre, and on that of Hipparchus, the
+follower and critic of Eratosthenes. It was the ambition of Ptolemy
+to describe and represent accurately the surface of the <i>oekumene</i>,
+for which purpose he took immense trouble to collect all existing
+determinations of the latitude of places, all estimates of longitude,
+and to make every possible rectification in the estimates of distances
+by land or sea. His work was mainly cartographical in its aim,
+and theory was as far as possible excluded. The symmetrically
+placed hypothetical islands in the great continuous ocean disappeared,
+and the <i>oekumene</i> acquired a new form by the representation of the
+Indian Ocean as a larger Mediterranean completely cut off by land
+from the Atlantic. The <i>terra incognita</i> uniting Africa and Farther
+Asia was an unfortunate hypothesis which helped to retard exploration.
+Ptolemy used the word <i>geography</i> to signify the description
+of the whole <i>oekumene</i> on mathematical principles, while <i>chorography</i>
+signified the fuller description of a particular region, and
+<i>topography</i> the very detailed description of a smaller locality. He
+introduced the simile that geography represented an artist&rsquo;s sketch
+of a whole portrait, while chorography corresponded to the careful
+and detailed drawing of an eye or an ear.<a name="fa5b" id="fa5b" href="#ft5b"><span class="sp">5</span></a></p>
+
+<p>The Caliph al-Mam&#363;n (<i>c.</i> <span class="scs">A.D.</span> 815), the son and successor of
+H&#257;r&#363;n al-Rash&#299;d, caused an Arabic version of Ptolemy&rsquo;s great
+astronomical work (<span class="grk" title="Suntaxis megistź">&#931;&#973;&#957;&#964;&#945;&#958;&#953;&#962; &#956;&#949;&#947;&#943;&#963;&#964;&#951;</span>) to be made, which is known
+as the <i>Almagest</i>, the word being nothing more than the Gr. <span class="grk" title="megistź">&#956;&#949;&#947;&#943;&#963;&#964;&#951;</span>
+with the Arabic article <i>al</i> prefixed. The geography of Ptolemy was
+also known and is constantly referred to by Arab writers. The
+Arab astronomers measured a degree on the plains of Mesopotamia,
+thereby deducing a fair approximation to the size of the earth.
+The caliph&rsquo;s librarian, Abu Jafar Muhammad Ben Musa, wrote a
+geographical work, now unfortunately lost, entitled <i>Rasm el Arsi</i> (&ldquo;A
+Description of the World&rdquo;), which is often referred to by subsequent
+writers as having been composed on the model of that of Ptolemy.</p>
+
+<p>The middle ages saw geographical knowledge die out in Christendom,
+although it retained, through the Arabic translations of
+Ptolemy, a certain vitality in Islam. The verbal interpretation
+of Scripture led Lactantius (<i>c.</i> <span class="scs">A.D.</span> 320) and
+<span class="sidenote">Geography in the middle ages.</span>
+other ecclesiastics to denounce the spherical theory of the
+earth as heretical. The wretched subterfuge of Cosmas
+(<i>c.</i> <span class="scs">A.D.</span> 550) to explain the phenomena of the apparent
+movements of the sun by means of an earth modelled on the plan
+of the Jewish Tabernacle gave place ultimately to the wheel-maps&mdash;the
+T in an O&mdash;which reverted to the primitive ignorance of the
+times of Homer and Hecataeus.<a name="fa6b" id="fa6b" href="#ft6b"><span class="sp">6</span></a></p>
+
+<p>The journey of Marco Polo, the increasing trade to the East and
+the voyages of the Arabs in the Indian Ocean prepared the way for
+the reacceptance of Ptolemy&rsquo;s ideas when the sealed books of the
+Greek original were translated into Latin by Angelus in 1410.</p>
+
+<p>The old arguments of Aristotle and the old measurements of
+Ptolemy were used by Toscanelli and Columbus in urging a westward
+voyage to India; and mainly on this account did the
+<span class="sidenote">Revival of geography.</span>
+crossing of the Atlantic rank higher in the history of
+scientific geography than the laborious feeling out of the
+coast-line of Africa. But not until the voyage of Magellan shook
+the scales from the eyes of Europe did modern geography begin to
+advance. Discovery had outrun theory; the rush of new facts
+made Ptolemy practically obsolete in a generation, after having been
+the fount and origin of all geography for a millennium.</p>
+
+<p>The earliest evidence of the reincarnation of a sound theoretical
+geography is to be found in the text-books by Peter Apian and
+Sebastian Münster. Apian in his <i>Cosmographicus liber</i>,
+published in 1524, and subsequently edited and added to
+<span class="sidenote">Apianus.</span>
+by Gemma Frisius under the title of <i>Cosmographia</i>, based the whole
+science on mathematics and measurement. He followed Ptolemy
+closely, enlarging on his distinction between geography and chorography,
+and expressing the artistic analogy in a rough diagram.
+This slender distinction was made much of by most subsequent
+writers until Nathanael Carpenter in 1625 pointed out that the
+difference between geography and chorography was simply one of
+degree, not of kind.</p>
+
+<p>Sebastian Münster, on the other hand, in his <i>Cosmographia
+universalis</i> of 1544, paid no regard to the mathematical basis of
+geography, but, following the model of Strabo, described
+<span class="sidenote">Münster.</span>
+the world according to its different political divisions,
+and entered with great zest into the question of the productions
+of countries, and into the manners and costumes of the various
+peoples. Thus early commenced the separation between what were
+long called mathematical and political geography, the one subject
+appealing mainly to mathematicians, the other to historians.</p>
+
+<p>Throughout the 16th and 17th centuries the rapidly accumulating
+store of facts as to the extent, outline and mountain and river
+systems of the lands of the earth were put in order by the generation
+of cartographers of which Mercator was the chief; but the writings
+of Apian and Münster held the field for a hundred years without a
+serious rival, unless the many annotated editions of Ptolemy might
+be so considered. Meanwhile the new facts were the subject of
+original study by philosophers and by practical men without reference
+to classical traditions. Bacon argued keenly on geographical
+matters and was a lover of maps, in which he observed and reasoned
+upon such resemblances as that between the outlines of South
+America and Africa.</p>
+
+<p>Philip Cluver&rsquo;s <i>Introductio in geographiam universam tam veterem
+quam novam</i> was published in 1624. Geography he defined as
+&ldquo;the description of the whole earth, so far as it is known
+to us.&rdquo; It is distinguished from cosmography by dealing
+<span class="sidenote">Cluverius.</span>
+with the earth alone, not with the universe, and from chorography
+and topography by dealing with the whole earth, not with a country
+or a place. The first book, of fourteen short chapters, is concerned
+with the general properties of the globe; the remaining six books
+treat in considerable detail of the countries of Europe and of the
+other continents. Each country is described with particular regard
+to its people as well as to its surface, and the prominence given to
+the human element is of special interest.</p>
+
+<p>A little-known book which appears to have escaped the attention
+of most writers on the history of modern geography was published
+at Oxford in 1625 by Nathanael Carpenter, fellow of
+<span class="sidenote">Carpenter.</span>
+Exeter College, with the title <i>Geographie delineated forth
+in Two Bookes, containing the Sphericall and Topicall parts thereof</i>.
+It is discursive in its style and verbose; but, considering the period
+at which it appeared, it is remarkable for the strong common sense
+displayed by the author, his comparative freedom from prejudice,
+and his firm application of the methods of scientific reasoning to
+the interpretation of phenomena. Basing his work on the principles
+of Ptolemy, he brings together illustrations from the most recent
+travellers, and does not hesitate to take as illustrative examples
+the familiar city of Oxford and his native county of Devon. He
+divides geography into <i>The Spherical Part</i>, or that for the study of
+which mathematics alone is required, and <i>The Topical Part</i>, or the
+description of the physical relations of parts of the earth&rsquo;s surface,
+preferring this division to that favoured by the ancient geographers&mdash;into
+general and special. It is distinguished from other English
+geographical books of the period by confining attention to the
+principles of geography, and not describing the countries of the
+world.</p>
+
+<p>A much more important work in the history of geographical
+method is the <i>Geographia generalis</i> of Bernhard Varenius, a German
+medical doctor of Leiden, who died at the age of twenty-eight
+in 1650, the year of the publication of his book.
+<span class="sidenote">Varenius.</span>
+Although for a time it was lost sight of on the continent, Sir Isaac
+Newton thought so highly of this book that he prepared an annotated
+edition which was published in Cambridge in 1672, with the addition
+of the plates which had been planned by Varenius, but not produced
+by the original publishers. &ldquo;The reason why this great man took
+so much care in correcting and publishing our author was, because
+he thought him necessary to be read by his audience, the young
+gentlemen of Cambridge, while he was delivering lectures on the same
+subject from the Lucasian Chair.&rdquo;<a name="fa7b" id="fa7b" href="#ft7b"><span class="sp">7</span></a> The treatise of Varenius is a
+model of logical arrangement and terse expression; it is a work of
+science and of genius; one of the few of that age which can still be
+studied with profit. The English translation renders the definition
+thus: &ldquo;Geography is that part of <i>mixed mathematics</i> which explains
+the state of the earth and of its parts, depending on quantity, viz.
+its figure, place, magnitude and motion, with the celestial appearances,
+&amp;c. By some it is taken in too limited a sense, for a bare
+description of the several countries; and by others too extensively,
+who along with such a description would have their political constitution.&rdquo;</p>
+
+<p>Varenius was reluctant to include the human side of geography in
+his system, and only allowed it as a concession to custom, and in
+order to attract readers by imparting interest to the sterner details
+of the science. His division of geography was into two parts&mdash;(i.)
+General or universal, dealing with the earth in general, and explaining
+its properties without regard to particular countries; and (ii.) Special
+or particular, dealing with each country in turn from the chorographical
+or topographical point of view. General geography was divided
+into&mdash;(1) the <i>Absolute</i> part, dealing with the form, dimensions,
+position and substance of the earth, the distribution of land and
+water, mountains, woods and deserts, hydrography (including all
+the waters of the earth) and the atmosphere; (2) the <i>Relative</i> part,
+including the celestial properties, <i>i.e.</i> latitude, climate zones, longitude,
+&amp;c.; and (3) the <i>Comparative</i> part, which &ldquo;considers the
+<span class="pagenum"><a name="page622" id="page622"></a>622</span>
+particulars arising from comparing one part with another&rdquo;; but
+under this head the questions discussed were longitude, the situation
+and distances of places, and navigation. Varenius does not treat
+of special geography, but gives a scheme for it under three heads&mdash;(1)
+<i>Terrestrial</i>, including position, outline, boundaries, mountains,
+mines, woods and deserts, waters, fertility and fruits, and living
+creatures; (2) <i>Celestial</i>, including appearance of the heavens and
+the climate; (3) <i>Human</i>, but this was added out of deference to
+popular usage.</p>
+
+<p>This system of geography founded a new epoch, and the book&mdash;translated
+into English, Dutch and French&mdash;was the unchallenged
+standard for more than a century. The framework was capable of
+accommodating itself to new facts, and was indeed far in advance
+of the knowledge of the period. The method included a recognition
+of the causes and effects of phenomena as well as the mere fact of
+their occurrence, and for the first time the importance of the vertical
+relief of the land was fairly recognized.</p>
+
+<p>The physical side of geography continued to be elaborated after
+Varenius&rsquo;s methods, while the historical side was developed separately.
+Both branches, although enriched by new facts, remained
+stationary so far as method is concerned until nearly the end of the
+18th century. The compilation of &ldquo;geography books&rdquo; by uninstructed
+writers led to the pernicious habit, which is not yet wholly
+overcome, of reducing the general or &ldquo;physical&rdquo; part to a few
+pages of concentrated information, and expanding the particular
+or &ldquo;political&rdquo; part by including unrevised travellers&rsquo; stories and
+uncritical descriptions of the various countries of the world. Such
+books were in fact not geography, but merely compressed travel.</p>
+
+<p>The next marked advance in the theory of geography may be
+taken as the nearly simultaneous studies of the physical earth
+carried out by the Swedish chemist, Torbern Bergman,
+acting under the impulse of Linnaeus, and by the German
+<span class="sidenote">Bergman.</span>
+philosopher, Immanuel Kant. Bergman&rsquo;s <i>Physical Description of
+the Earth</i> was published in Swedish in 1766, and translated into
+English in 1772 and into German in 1774. It is a plain, straightforward
+description of the globe, and of the various phenomena
+of the surface, dealing only with definitely ascertained facts in the
+natural order of their relationships, but avoiding any systematic
+classification or even definitions of terms.</p>
+
+<p>The problems of geography had been lightened by the destructive
+criticism of the French cartographer D&rsquo;Anville (who had purged
+the map of the world of the last remnants of traditional
+fact unverified by modern observations) and rendered
+<span class="sidenote">Kant.</span>
+richer by the dawn of the new era of scientific travel, when Kant
+brought his logical powers to bear upon them. Kant&rsquo;s lectures on
+physical geography were delivered in the university of Königsberg
+from 1765 onwards.<a name="fa8b" id="fa8b" href="#ft8b"><span class="sp">8</span></a> Geography appealed to him as a valuable
+educational discipline, the joint foundation with anthropology of
+that &ldquo;knowledge of the world&rdquo; which was the result of reason
+and experience. In this connexion he divided the communication
+of experience from one person to another into two categories&mdash;the
+narrative or historical and the descriptive or geographical; both
+history and geography being viewed as descriptions, the former a
+description in order of time, the latter a description in order of
+space.</p>
+
+<p>Physical geography he viewed as a summary of nature, the basis
+not only of history but also of &ldquo;all the other possible geographies,&rdquo;
+of which he enumerates five, viz. (1) <i>Mathematical geography</i>, which
+deals with the form, size and movements of the earth and its place
+in the solar system; (2) <i>Moral geography</i>, or an account of the
+different customs and characters of mankind according to the region
+they inhabit; (3) <i>Political geography</i>, the divisions according to
+their organized governments; (4) <i>Mercantile geography</i>, dealing
+with the trade in the surplus products of countries; (5) <i>Theological
+geography</i>, or the distribution of religions. Here there is a clear and
+formal statement of the interaction and causal relation of all the
+phenomena of distribution on the earth&rsquo;s surface, including the influence
+of physical geography upon the various activities of mankind
+from the lowest to the highest. Notwithstanding the form of this
+classification, Kant himself treats mathematical geography as preliminary
+to, and therefore not dependent on, physical geography.
+Physical geography itself is divided into two parts: a general,
+which has to do with the earth and all that belongs to it&mdash;water, air
+and land; and a particular, which deals with special products of
+the earth&mdash;mankind, animals, plants and minerals. Particular
+importance is given to the vertical relief of the land, on which the
+various branches of human geography are shown to depend.</p>
+
+<p>Alexander von Humboldt (1769-1859) was the first modern geographer
+to become a great traveller, and thus to acquire an extensive
+stock of first-hand information on which an improved
+system of geography might be founded. The impulse
+<span class="sidenote">Humboldt.</span>
+given to the study of natural history by the example of Linnaeus;
+the results brought back by Sir Joseph Banks, Dr Solander and the
+two Forsters, who accompanied Cook in his voyages of discovery;
+the studies of De Saussure in the Alps, and the lists of desiderata
+in physical geography drawn up by that investigator, combined to
+prepare the way for Humboldt. The theory of geography was
+advanced by Humboldt mainly by his insistence on the great
+principle of the unity of nature. He brought all the &ldquo;observable
+things,&rdquo; which the eager collectors of the previous century had been
+heaping together regardless of order or system, into relation with the
+vertical relief and the horizontal forms of the earth&rsquo;s surface. Thus
+he demonstrated that the forms of the land exercise a directive
+and determining influence on climate, plant life, animal life and on
+man himself. This was no new idea; it had been familiar for
+centuries in a less definite form, deduced from a priori considerations,
+and so far as regards the influence of surrounding circumstances
+upon man, Kant had already given it full expression. Humboldt&rsquo;s
+concrete illustrations and the remarkable power of his personality
+enabled him to enforce these principles in a way that produced
+an immediate and lasting effect. The treatises on physical geography
+by Mrs Mary Somerville and Sir John Herschel (the latter written
+for the eighth edition of the <i>Encyclopaedia Britannica</i>) showed the
+effect produced in Great Britain by the stimulus of Humboldt&rsquo;s work.</p>
+
+<p>Humboldt&rsquo;s contemporary, Carl Ritter (1779-1859), extended and
+disseminated the same views, and in his interpretation of &ldquo;Comparative
+Geography&rdquo; he laid stress on the importance of
+forming conclusions, not from the study of one region by
+<span class="sidenote">Ritter.</span>
+itself, but from the comparison of the phenomena of many places.
+Impressed by the influence of terrestrial relief and climate on human
+movements, Ritter was led deeper and deeper into the study of history
+and archaeology. His monumental <i>Vergleichende Geographie</i>, which
+was to have made the whole world its theme, died out in a wilderness
+of detail in twenty-one volumes before it had covered more of the
+earth&rsquo;s surface than Asia and a portion of Africa. Some of his
+followers showed a tendency to look on geography rather as an
+auxiliary to history than as a study of intrinsic worth.</p>
+
+<p>During the rapid development of physical geography many
+branches of the study of nature, which had been included in the
+cosmography of the early writers, the physiography of
+Linnaeus and even the <i>Erdkunde</i> of Ritter, had been
+<span class="sidenote">Geography as a natural science.</span>
+so much advanced by the labours of specialists that
+their connexion was apt to be forgotten. Thus geology,
+meteorology, oceanography and anthropology developed
+into distinct sciences. The absurd attempt was, and sometimes
+is still, made by geographers to include all natural science in geography;
+but it is more common for specialists in the various detailed
+sciences to think, and sometimes to assert, that the ground of
+physical geography is now fully occupied by these sciences. Political
+geography has been too often looked on from both sides as a mere
+summary of guide-book knowledge, useful in the schoolroom, a poor
+relation of physical geography that it was rarely necessary to
+recognize.</p>
+
+<p>The science of geography, passed on from antiquity by Ptolemy,
+re-established by Varenius and Newton, and systematized by Kant,
+included within itself definite aspects of all those terrestrial phenomena
+which are now treated exhaustively under the heads of geology,
+meteorology, oceanography and anthropology; and the inclusion
+of the requisite portions of the perfected results of these sciences in
+geography is simply the gathering in of fruit matured from the seed
+scattered by geography itself.</p>
+
+<p>The study of geography was advanced by improvements in cartography
+(see <span class="sc"><a href="#artlinks">Map</a></span>), not only in the methods of survey and projection,
+but in the representation of the third dimension by means
+of contour lines introduced by Philippe Buache in 1737, and the
+more remarkable because less obvious invention of isotherms
+introduced by Humboldt in 1817.</p>
+
+<p>The &ldquo;argument from design&rdquo; had been a favourite form of
+reasoning amongst Christian theologians, and, as worked out by
+Paley in his <i>Natural Theology</i>, it served the useful purpose
+of emphasizing the fitness which exists between all the
+<span class="sidenote">The teleological argument in geography.</span>
+inhabitants of the earth and their physical environment.
+It was held that the earth had been created so as to fit
+the wants of man in every particular. This argument was
+tacitly accepted or explicitly avowed by almost every writer on the
+theory of geography, and Carl Ritter distinctly recognized and
+adopted it as the unifying principle of his system. As a student of
+nature, however, he did not fail to see, and as professor of geography
+he always taught, that man was in very large measure conditioned
+by his physical environment. The apparent opposition of the
+observed fact to the assigned theory he overcame by looking upon
+the forms of the land and the arrangement of land and sea as instruments
+of Divine Providence for guiding the destiny as well as for
+supplying the requirements of man. This was the central theme of
+Ritter&rsquo;s philosophy; his religion and his geography were one, and
+the consequent fervour with which he pursued his mission goes far
+to account for the immense influence he acquired in Germany.</p>
+
+<p>The evolutionary theory, more than hinted at in Kant&rsquo;s &ldquo;Physical
+Geography,&rdquo; has, since the writings of Charles Darwin, become the
+unifying principle in geography. The conception of the
+development of the plan of the earth from the first
+<span class="sidenote">The theory of evolution in geography.</span>
+cooling of the surface of the planet throughout the long
+geological periods, the guiding power of environment on
+the circulation of water and of air, on the distribution
+of plants and animals, and finally on the movements of man, give
+to geography a philosophical dignity and a scientific completeness
+<span class="pagenum"><a name="page623" id="page623"></a>623</span>
+which it never previously possessed. The influence of environment
+on the organism may not be quite so potent as it was once believed
+to be, in the writings of Buckle, for instance,<a name="fa9b" id="fa9b" href="#ft9b"><span class="sp">9</span></a> and certainly man,
+the ultimate term in the series, reacts upon and greatly modifies his
+environment; yet the fact that environment does influence all
+distributions is established beyond the possibility of doubt. In
+this way also the position of geography, at the point where physical
+science meets and mingles with mental science, is explained and
+justified. The change which took place during the 19th century
+in the substance and style of geography may be well seen by comparing
+the eight volumes of Malte-Brun&rsquo;s <i>Géographie universelle</i>
+(Paris, 1812-1829) with the twenty-one volumes of Reclus&rsquo;s <i>Géographie
+universelle</i> (Paris, 1876-1895).</p>
+
+<p>In estimating the influence of recent writers on geography it is
+usual to assign to Oscar Peschel (1826-1875) the credit of having
+corrected the preponderance which Ritter gave to the historical
+element, and of restoring physical geography to its old pre-eminence.<a name="fa10b" id="fa10b" href="#ft10b"><span class="sp">10</span></a>
+As a matter of fact, each of the leading modern exponents of theoretical
+geography&mdash;such as Ferdinand von Richthofen, Hermann
+Wagner, Friedrich Ratzel, William M. Davis, A. Penck, A. de
+Lapparent and Elisée Reclus&mdash;has his individual point of view,
+one devoting more attention to the results of geological processes,
+another to anthropological conditions, and the rest viewing the
+subject in various blendings of the extreme lights.</p>
+
+<p>The two conceptions which may now be said to animate the theory
+of geography are the genetic, which depends upon processes of
+origin, and the morphological, which depends on facts of form and
+distribution.</p>
+
+<p class="pt2 center sc">Progress of Geographical Discovery</p>
+
+<p>Exploration and geographical discovery must have started from
+more than one centre, and to deal justly with the matter one ought
+to treat of these separately in the early ages before the whole civilized
+world was bound together by the bonds of modern intercommunication.
+At the least there should be some consideration of four
+separate systems of discovery&mdash;the Eastern, in which Chinese and
+Japanese explorers acquired knowledge of the geography of Asia,
+and felt their way towards Europe and America; the Western, in
+which the dominant races of the Mexican and South American
+plateaus extended their knowledge of the American continent
+before Columbus; the Polynesian, in which the conquering races
+of the Pacific Islands found their way from group to group; and
+the Mediterranean. For some of these we have no certain information,
+and regarding others the tales narrated in the early records
+are so hard to reconcile with present knowledge that they are better
+fitted to be the battle-ground of scholars championing rival theories
+than the basis of definite history. So it has come about that the
+only practicable history of geographical exploration starts from the
+Mediterranean centre, the first home of that civilization which has
+come to be known as European, though its field of activity has long
+since overspread the habitable land of both temperate zones, eastern
+Asia alone in part excepted.</p>
+
+<p>From all centres the leading motives of exploration were probably
+the same&mdash;commercial intercourse, warlike operations, whether
+resulting in conquest or in flight, religious zeal expressed in pilgrimages
+or missionary journeys, or, from the other side, the avoidance
+of persecution, and, more particularly in later years, the
+advancement of knowledge for its own sake. At different times one
+or the other motive predominated.</p>
+
+<p>Before the 14th century <span class="scs">B.C.</span> the warrior kings of Egypt had carried
+the power of their arms southward from the delta of the Nile well-nigh
+to its source, and eastward to the confines of Assyria. The
+hieroglyphic inscriptions of Egypt and the cuneiform inscriptions of
+Assyria are rich in records of the movements and achievements of
+armies, the conquest of towns and the subjugation of peoples; but
+though many of the recorded sites have been identified, their discovery
+by wandering armies was isolated from their subsequent
+history and need not concern us here.</p>
+
+<p>The Phoenicians are the earliest Mediterranean people in the
+consecutive chain of geographical discovery which joins pre-historic
+time with the present. From Sidon, and later from its
+more famous rival Tyre, the merchant adventurers of
+<span class="sidenote">The Phoenicians.</span>
+Phoenicia explored and colonized the coasts of the Mediterranean
+and fared forth into the ocean beyond. They traded also
+on the Red sea, and opened up regular traffic with India as well
+as with the ports of the south and west, so that it was natural for
+Solomon to employ the merchant navies of Tyre in his oversea trade.
+The western emporium known in the scriptures as Tarshish was
+probably situated in the south of Spain, possibly at Cadiz, although
+some writers contend that it was Carthage in North Africa. Still
+more diversity of opinion prevails as to the southern gold-exporting
+port of Ophir, which some scholars place in Arabia, others at one or
+another point on the east coast of Africa. Whether associated
+with the exploitation of Ophir (<i>q.v.</i>) or not the first great voyage of
+African discovery appears to have been accomplished by the Phoenicians
+sailing the Red Sea. Herodotus (himself a notable traveller
+in the 5th century <span class="scs">B.C.</span>) relates that the Egyptian king Necho of
+the XXVIth Dynasty (<i>c.</i> 600 <span class="scs">B.C.</span>) built a fleet on the Red Sea,
+and confided it to Phoenician sailors with the orders to sail southward
+and return to Egypt by the Pillars of Hercules and the Mediterranean
+sea. According to the tradition, which Herodotus quotes
+sceptically, this was accomplished; but the story is too vague to
+be accepted as more than a possibility.</p>
+
+<p>The great Phoenician colony of Carthage, founded before 800 <span class="scs">B.C.</span>,
+perpetuated the commercial enterprise of the parent state, and extended
+the sphere of practical trade to the ocean shores of Africa
+and Europe. The most celebrated voyage of antiquity undertaken
+for the express purpose of discovery was that fitted out by the
+senate of Carthage under the command of Hanno, with the intention
+of founding new colonies along the west coast of Africa. According
+to Pliny, the only authority on this point, the period of the voyage
+was that of the greatest prosperity of Carthage, which may be taken
+as somewhere between 570 and 480 <span class="scs">B.C.</span> The extent of this voyage
+is doubtful, but it seems probable that the farthest point reached
+was on the east-running coast which bounds the Gulf of Guinea
+on the north. Himilco, a contemporary of Hanno, was charged
+with an expedition along the west coast of Iberia northward, and
+as far as the uncertain references to this voyage can be understood,
+he seems to have passed the Bay of Biscay and possibly sighted the
+coast of England.</p>
+
+<p>The sea power of the Greek communities on the coast of Asia
+Minor and in the Archipelago began to be a formidable rival to the
+Phoenician soon after the time of Hanno and Himilco,
+and peculiar interest attaches to the first recorded Greek
+<span class="sidenote">The Greeks.</span>
+voyage beyond the Pillars of Hercules. Pytheas, a
+navigator of the Phocean colony of Massilia (Marseilles), determined
+the latitude of that port with considerable precision by the somewhat
+clumsy method of ascertaining the length of the longest day, and
+when, about 330 <span class="scs">B.C.</span>, he set out on exploration to the northward
+in search of the lands whence came gold, tin and amber, he followed
+this system of ascertaining his position from time to time. If on
+each occasion he himself made the observations his voyage must
+have extended over six years; but it is not impossible that he
+ascertained the approximate length of the longest day in some cases
+by questioning the natives. Pytheas, whose own narrative is not
+preserved, coasted the Bay of Biscay, sailed up the English Channel
+and followed the coast of Britain to its most northerly point. Beyond
+this he spoke of a land called <i>Thule</i>, which, if his estimate of the
+length of the longest day is correct, may have been Shetland, but
+was possibly Iceland; and from some confused statements as to a
+sea which could not be sailed through, it has been assumed that
+Pytheas was the first of the Greeks to obtain direct knowledge of
+the Arctic regions. During this or a second voyage Pytheas entered
+the Baltic, discovered the coasts where amber is obtained and returned
+to the Mediterranean. It does not seem that any maritime
+trade followed these discoveries, and indeed it is doubtful whether
+his contemporaries accepted the truth of Pytheas&rsquo;s narrative;
+Strabo four hundred years later certainly did not, but the critical
+studies of modern scholars have rehabilitated the Massilian explorer.</p>
+
+<p>The Greco-Persian wars had made the remoter parts of Asia
+Minor more than a name to the Greek geographers before the time
+of Alexander the Great, but the campaigns of that conqueror
+<span class="sidenote">Alexander the Great.</span>
+from 329 to 325 <span class="scs">B.C.</span> opened up the greater Asia
+to the knowledge of Europe. His armies crossed the plains
+beyond the Caspian, penetrated the wild mountain passes north-west
+of India, and did not turn back until they had entered on the
+Indo-Gangetic plain. This was one of the few great epochs of
+geographical discovery.</p>
+
+<p>The world was henceforth viewed as a very large place stretching
+far on every side beyond the Midland or Mediterranean Sea, and the
+land journey of Alexander resulted in a voyage of discovery in the
+outer ocean from the mouth of the Indus to that of the Tigris,
+thus opening direct intercourse between Grecian and Hindu civilization.
+The Greeks who accompanied Alexander described with care
+the towns and villages, the products and the aspect of the country.
+The conqueror also intended to open up trade by sea between Europe
+and India, and the narrative of his general Nearchus records this
+famous voyage of discovery, the detailed accounts of the chief
+pilot Onesicritus being lost. At the beginning of October 326 <span class="scs">B.C.</span>
+Nearchus left the Indus with his fleet, and the anchorages sought for
+each night are carefully recorded. He entered the Persian Gulf,
+and rejoined Alexander at Susa, when he was ordered to prepare
+another expedition for the circumnavigation of Arabia. Alexander
+died at Babylon in 323 <span class="scs">B.C.</span>, and the fleet was dispersed without
+making the voyage.</p>
+
+<p>The dynasties founded by Alexander&rsquo;s generals, Seleucus, Antiochus
+and Ptolemy, encouraged the same spirit of enterprise which
+their master had fostered, and extended geographical knowledge
+in several directions. Seleucus Nicator established the Greco-Bactrian
+empire and continued the intercourse with India. Authentic
+information respecting the great valley of the Ganges was supplied
+by Megasthenes, an ambassador sent by Seleucus, who reached the
+remote city of Patali-putra, the modern Patna.</p>
+
+<p>The Ptolemies in Egypt showed equal anxiety to extend the
+bounds of geographical knowledge. Ptolemy Euergetes (247-222 <span class="scs">B.C.</span>)
+<span class="pagenum"><a name="page624" id="page624"></a>624</span>
+rendered the greatest service to geography by the protection and
+<span class="sidenote">The Ptolemies.</span>
+encouragement of Eratosthenes, whose labours gave the first approximate
+knowledge of the true size of the spherical
+earth. The second Euergetes and his successor Ptolemy
+Lathyrus (118-115 <span class="scs">B.C.</span>) furnished Eudoxus with a fleet
+to explore the Arabian sea. After two successful voyages, Eudoxus,
+impressed with the idea that Africa was surrounded by ocean on the
+south, left the Egyptian service, and proceeded to Cadiz and other
+Mediterranean centres of trade seeking a patron who would finance
+an expedition for the purpose of African discovery; and we learn
+from Strabo that the veteran explorer made at least two voyages
+southward along the coast of Africa. The Ptolemies continued to
+send fleets annually from their Red Sea ports of Berenice and Myos
+Hormus to Arabia, as well as to ports on the coasts of Africa and
+India.</p>
+
+<p>The Romans did not encourage navigation and commerce with
+the same ardour as their predecessors; still the luxury of Rome,
+which gave rise to demands for the varied products
+of all the countries of the known world, led to an active
+<span class="sidenote">The Romans.</span>
+trade both by ships and caravans. But it was the military
+genius of Rome, and the ambition for universal empire, which led,
+not only to the discovery, but also to the survey of nearly all Europe,
+and of large tracts in Asia and Africa. Every new war produced
+a new survey and itinerary of the countries which were conquered,
+and added one more to the imperishable roads that led from every
+quarter of the known world to Rome. In the height of their power
+the Romans had surveyed and explored all the coasts of the Mediterranean,
+Italy, Greece, the Balkan Peninsula, Spain, Gaul, western
+Germany and southern Britain. In Africa their empire included
+Egypt, Carthage, Numidia and Mauritania. In Asia they held
+Asia Minor and Syria, had sent expeditions into Arabia, and were
+acquainted with the more distant countries formerly invaded by
+Alexander, including Persia, Scythia, Bactria and India. Roman
+intercourse with India especially led to the extension of geographical
+knowledge.</p>
+
+<p>Before the Roman legions were sent into a new region to extend
+the limits of the empire, it was usual to send out exploring expeditions
+to report as to the nature of the country. It is narrated by Pliny
+and Seneca that the emperor Nero sent out two centurions on such
+a mission towards the source of the Nile (probably about <span class="scs">A.D.</span> 60),
+and that the travellers pushed southwards until they reached vast
+marshes through which they could not make their way either on
+foot or in boats. This seems to indicate that they had penetrated
+to about 9° N. Shortly before <span class="scs">A.D.</span> 79 Hippalus took advantage of
+the regular alternation of the monsoons to make the voyage from
+the Red Sea to India across the open ocean out of sight of land.
+Even though this sea-route was known, the author of the <i>Periplus
+of the Erythraean Sea</i>, published after the time of Pliny, recites the
+old itinerary around the coast of the Arabian Gulf. It was, however,
+in the reigns of Severus and his immediate successors that Roman
+intercourse with India was at its height, and from the writings of
+Pausanias (<i>c.</i> 174) it appears that direct communication between
+Rome and China had already taken place.</p>
+
+<p>After the division of the Roman empire, Constantinople became
+the last refuge of learning, arts and taste; while Alexandria continued
+to be the emporium whence were imported the commodities
+of the East. The emperor Justinian (483-565), in whose reign the
+greatness of the Eastern empire culminated, sent two Nestorian
+monks to China, who returned with eggs of the silkworm concealed
+in a hollow cane, and thus silk manufactures were established in
+the Peloponnesus and the Greek islands. It was also in the reign
+of Justinian that Cosmas Indicopleustes, an Egyptian merchant,
+made several voyages, and afterwards composed his <span class="grk" title="Christianikź
+topographia">&#935;&#961;&#953;&#963;&#964;&#953;&#945;&#957;&#953;&#954;&#8052; &#964;&#959;&#960;&#959;&#947;&#961;&#945;&#966;&#943;&#945;</span> (Christian Topography), containing, in addition to his
+absurd cosmogony, a tolerable description of India.</p>
+
+<p>The great outburst of Mahommedan conquest in the 7th century
+was followed by the Arab civilization, having its centres at Bagdad
+and Cordova, in connexion with which geography again
+received a share of attention. The works of the ancient
+<span class="sidenote">The Arabs.</span>
+Greek geographers were translated into Arabic, and starting with a
+sound basis of theoretical knowledge, exploration once more made
+progress. From the 9th to the 13th century intelligent Arab
+travellers wrote accounts of what they had seen and heard in distant
+lands. The earliest Arabian traveller whose observations have come
+down to us is the merchant Sulaiman, who embarked in the Persian
+Gulf and made several voyages to India and China, in the middle of
+the 9th century. Abu Zaid also wrote on India, and his work is the
+most important that we possess before the epoch-making discoveries
+of Marco Polo. Masudi, a great traveller who knew from personal
+experience all the countries between Spain and China, described the
+plains, mountains and seas, the dynasties and peoples, in his <i>Meadows
+of Gold</i>, an abstract made by himself of his larger work <i>News of the
+Time</i>. He died in 956, and was known, from the comprehensiveness
+of his survey, as the Pliny of the East. Amongst his contemporaries
+were Istakhri, who travelled through all the Mahommedan
+countries and wrote his <i>Book of Climates</i> in 950, and Ibn Haukal,
+whose <i>Book of Roads and Kingdoms</i>, based on the work of Istakhri,
+was written in 976. Idrisi, the best known of the Arabian geographical
+authors, after travelling far and wide in the first half of
+the 12th century, settled in Sicily, where he wrote a treatise descriptive
+of an armillary sphere which he had constructed for Roger II.,
+the Norman king, and in this work he incorporated all accessible
+results of contemporary travel.</p>
+
+<p>The Northmen of Denmark and Norway, whose piratical adventures
+were the terror of all the coasts of Europe, and who established
+themselves in Great Britain and Ireland, in France and
+Sicily, were also geographical explorers in their rough but
+<span class="sidenote">The Northmen.</span>
+practical way during the darkest period of the middle ages.
+All Northmen were not bent on rapine and plunder; many were
+peaceful merchants. Alfred the Great, king of the Saxons in
+England, not only educated his people in the learning of the past
+ages; he inserted in the geographical works he translated many
+narratives of the travel of his own time. Thus he placed on record
+the voyages of the merchant Ulfsten in the Baltic, including particulars
+of the geography of Germany. And in particular he told of
+the remarkable voyage of Other, a Norwegian of Helgeland, who
+was the first authentic Arctic explorer, the first to tell of the rounding
+of the North Cape and the sight of the midnight sun. This voyage
+of the middle of the 9th century deserves to be held in happy memory,
+for it unites the first Norwegian polar explorer with the first English
+collector of travels. Scandinavian merchants brought the products
+of India to England and Ireland. From the 8th to the 11th century
+a commercial route from India passed through Novgorod to the
+Baltic, and Arabian coins found in Sweden, and particularly in
+the island of Gotland, prove how closely the enterprise of the Northmen
+and of the Arabs intertwined. Five-sixths of these coins
+preserved at Stockholm were from the mints of the Samanian
+dynasty, which reigned in Khorasan and Transoxiana from about
+<span class="scs">A.D.</span> 900 to 1000. It was the trade with the East that originally gave
+importance to the city of Visby in Gotland.</p>
+
+<p>In the end of the 9th century Iceland was colonized from Norway;
+and about 985 the intrepid viking, Eric the Red, discovered Greenland,
+and induced some of his Icelandic countrymen to settle on its inhospitable
+shores. His son, Leif Ericsson, and others of his followers
+were concerned in the discovery of the North American coast (see
+<span class="sc"><a href="#artlinks">Vinland</a></span>), which, but for the isolation of Iceland from the centres
+of European awakening, would have had momentous consequences.
+As things were, the importance of this discovery passed unrecognized.
+The story of two Venetians, Nicolo and Antonio Zeno, who gave a
+vague account of voyages in the northern seas in the end of the 13th
+century, is no longer to be accepted as history.</p>
+
+<p>At length the long period of barbarism which accompanied and
+followed the fall of the Roman empire drew to a close in Europe.
+The Crusades had a favourable influence on the intellectual
+state of the Western nations. Interesting regions,
+<span class="sidenote">Close of the dark ages.</span>
+known only by the scant reports of pilgrims, were made
+the objects of attention and study; while religious zeal,
+and the hope of gain, combined with motives of mere curiosity,
+induced several persons to travel by land into remote regions of the
+East, far beyond the countries to which the operations of the crusaders
+extended. Among these was Benjamin of Tudela, who set out from
+Spain in 1160, travelled by land to Constantinople, and having
+visited India and some of the eastern islands, returned to Europe
+by way of Egypt after an absence of thirteen years.</p>
+
+<p>Joannes de Plano Carpini, a Franciscan monk, was the head of
+one of the missions despatched by Pope Innocent to call the chief
+and people of the Tatars to a better mind. He reached
+the headquarters of Batu, on the Volga, in February
+<span class="sidenote">Asiatic journeys.</span>
+1246; and, after some stay, went on to the camp of the
+great khan near Karakorum in central Asia, and returned safely
+in the autumn of 1247. A few years afterwards, a Fleming named
+Rubruquis was sent on a similar mission, and had the merit of being
+the first traveller of this era who gave a correct account of the Caspian
+Sea. He ascertained that it had no outlet. At nearly the same
+time Hayton, king of Armenia, made a journey to Karakorum in
+1254, by a route far to the north of that followed by Carpini and
+Rubruquis. He was treated with honour and hospitality, and
+returned by way of Samarkand and Tabriz, to his own territory.
+The curious narrative of King Hayton was translated by Klaproth.</p>
+
+<p>While the republics of Italy, and above all the state of Venice,
+were engaged in distributing the rich products of India and the Far
+East over the Western world, it was impossible that motives of
+curiosity, as well as a desire of commercial advantage, should not be
+awakened to such a degree as to impel some of the merchants to
+visit those remote lands. Among these were the brothers Polo, who
+traded with the East and themselves visited Tatary. The recital
+of their travels fired the youthful imagination of young Marco Polo,
+son of Nicolo, and he set out for the court of Kublai Khan, with his
+father and uncle, in 1265. Marco remained for seventeen years
+in the service of the Great Khan, and was employed on many
+important missions. Besides what he learnt from his own observation,
+he collected much information from others concerning
+countries which he did not visit. He returned to Europe possessed
+of a vast store of knowledge respecting the eastern parts of the
+world, and, being afterwards made a prisoner by the Genoese, he
+dictated the narrative of his travels during his captivity. The
+work of Marco Polo is the most valuable narrative of travels that
+appeared during the middle ages, and despite a cold reception and
+many denials of the accuracy of the record, its substantial truthfulness
+has been abundantly proved.</p>
+
+<p><span class="pagenum"><a name="page625" id="page625"></a>625</span></p>
+
+<p>Missionaries continued to do useful geographical work. Among
+them were John of Monte Corvino, a Franciscan monk, Andrew of
+Perugia, John Marignioli and Friar Jordanus, who visited the west
+coast of India, and above all Friar Odoric of Pordenone. Odoric
+set out on his travels about 1318, and his journeys embraced parts
+of India, the Malay Archipelago, China and even Tibet, where he
+was the first European to enter Lhasa, not yet a forbidden city.</p>
+
+<p>Ibn Batuta, the great Arab traveller, is separated by a wide space
+of time from his countrymen already mentioned, and he finds his
+proper place in a chronological notice after the days of Marco Polo,
+for he did not begin his wanderings until 1325, his career thus coinciding
+in time with the fabled journeyings of Sir John Mandeville.
+While Arab learning flourished during the darkest ages of European
+ignorance, the last of the Arab geographers lived to see the dawn of
+the great period of the European awakening. Ibn Batuta went by
+land from Tangier to Cairo, then visited Syria, and performed the
+pilgrimages to Medina and Mecca. After exploring Persia, and again
+residing for some time at Mecca, he made a voyage down the Red
+sea to Yemen, and travelled through that country to Aden. Thence
+he visited the African coast, touching at Mombasa and Quiloa, and
+then sailed across to Ormuz and the Persian Gulf. He crossed
+Arabia from Bahrein to Jidda, traversed the Red sea and the desert
+to Syene, and descended the Nile to Cairo. After this he revisited
+Syria and Asia Minor, and crossed the Black sea, the desert from
+Astrakhan to Bokhara, and the Hindu Kush. He was in the service
+of Muhammad Tughluk, ruler of Delhi, about eight years, and was
+sent on an embassy to China, in the course of which the ambassadors
+sailed down the west coast of India to Calicut, and then visited the
+Maldive Islands and Ceylon. Ibn Batuta made the voyage through
+the Malay Archipelago to China, and on his return he proceeded
+from Malabar to Bagdad and Damascus, ultimately reaching Fez,
+the capital of his native country, in November 1349. After a journey
+into Spain he set out once more for Central Africa in 1352, and
+reached Timbuktu and the Niger, returning to Fez in 1353. His
+narrative was committed to writing from his dictation.</p>
+
+<p>The European country which had come the most completely
+under the influence of Arab culture now began to send forth explorers
+to distant lands, though the impulse came not from the
+Moors but from Italian merchant navigators in Spanish
+<span class="sidenote">Spanish exploration.</span>
+service. The peaceful reign of Henry III. of Castile is
+famous for the attempts of that prince to extend the
+diplomatic relations of Spain to the remotest parts of the earth.
+He sent embassies to all the princes of Christendom and to the
+Moors. In 1403 the Spanish king sent a knight of Madrid, Ruy
+Gonzalez de Clavijo, to the distant court of Timur, at Samarkand.
+He returned in 1406, and wrote a valuable narrative of his travels.</p>
+
+<p>Italians continued to make important journeys in the East
+during the 15th century. Among them was Nicolo Conti, who
+passed through Persia, sailed along the coast of Malabar, visited
+Sumatra, Java and the south of China, returned by the Red sea,
+and got home to Venice in 1444 after an absence of twenty-five years.
+He related his adventures to Poggio Bracciolini, secretary to Pope
+Eugenius IV.; and the narrative contains much interesting information.
+One of the most remarkable of the Italian travellers was
+Ludovico di Varthema, who left his native land in 1502. He went
+to Egypt and Syria, and for the sake of visiting the holy cities became
+a Mahommedan. He was the first European who gave an account
+of the interior of Yemen. He afterwards visited and described
+many places in Persia, India and the Malay Archipelago, returning
+to Europe in a Portuguese ship after an absence of five years.</p>
+
+<p>In the 15th century the time was approaching when the discovery
+of the Cape of Good Hope was to widen the scope of geographical
+enterprise. This great event was preceded by the general
+utilization in Europe of the polarity of the magnetic
+<span class="sidenote">Portuguese exploration&mdash;Prince Henry the Navigator.</span>
+needle in the construction of the mariner&rsquo;s compass.
+Portugal took the lead along this new path, and foremost
+among her pioneers stands Prince Henry the Navigator
+(1394-1460), who was a patron both of exploration and
+of the study of geographical theory. The great westward
+projection of the coast of Africa, and the islands to the north-west
+of that continent, were the principal scene of the work of the mariners
+sent out at his expense; but his object was to push onward and
+reach India from the Atlantic. The progress of discovery received
+a check on his death, but only for a time. In 1462 Pedro de Cintra
+extended Portuguese exploration along the African coast and discovered
+Sierra Leone. Fernan Gomez followed in 1469, and opened
+trade with the Gold Coast; and in 1484 Diogo Cćo discovered the
+mouth of the Congo. The king of Portugal next despatched Bartolomeu
+Diaz in 1486 to continue discoveries southwards; while, in the
+following year, he sent Pedro de Covilhćo and Affonso de Payva
+to discover the country of Prester John. Diaz succeeded in rounding
+the southern point of Africa, which he named Cabo Tormentoso&mdash;the
+Cape of Storms&mdash;but King Joćo II., foreseeing the realization of the
+long-sought passage to India, gave it the stimulating and enduring
+name of the Cape of Good Hope. Payva died at Cairo; but Covilhćo,
+having heard that a Christian ruler reigned in the mountains of
+Ethiopia, penetrated into Abyssinia in 1490. He delivered the letter
+which Joćo II. had addressed to Prester John to the Negus Alexander
+of Abyssinia, but he was detained by that prince and never allowed
+to leave the country.</p>
+
+<p>The Portuguese, following the lead of Prince Henry, continued to
+look for the road to India by the Cape of Good Hope. The same
+end was sought by Christopher Columbus, following the
+suggestion of Toscanelli, and under-estimating the diameter
+<span class="sidenote">Columbus.</span>
+of the globe, by sailing due west. The voyages of Columbus
+(1492-1498) resulted in the discovery of the West Indies and North
+America which barred the way to the Far East. In 1493 the pope,
+Alexander VI., issued a bull instituting the famous &ldquo;line of demarcation&rdquo;
+running from N. to S. 100 leagues W. of the Azores, to the
+west of which the Spaniards were authorized to explore and to the
+east of which the Portuguese received the monopoly of discovery.
+The direct line of Portuguese exploration resulted in the discovery
+of the Cape route to India by Vasco da Gama (1498), and in 1500
+to the independent discovery of South America by Pedro Alvarez
+Cabral. The voyages of Columbus and of Vasco da Gama were so
+important that it is unnecessary to detail their results in this place.
+See <span class="sc"><a href="#artlinks">Columbus, Christopher</a></span>; <span class="sc"><a href="#artlinks">Gama, Vasco da</a></span>.</p>
+
+<p>The three voyages of Vasco da Gama (who died on the scene of his
+labours, at Cochin, in 1524) revolutionized the commerce of the
+East. Until then the Venetians held the carrying trade
+<span class="sidenote">Vasco da Gama.</span>
+of India, which was brought by the Persian Gulf and Red
+sea into Syria and Egypt, the Venetians receiving the
+products of the East at Alexandria and Beirut and distributing
+them over Europe. This commerce was a great source of wealth
+to Venice; but after the discovery of the new passage round the
+Cape, and the conquests of the Portuguese, the trade of the East
+passed into other hands.</p>
+
+<p>The discoveries of Columbus awakened a spirit of enterprise in
+Spain which continued in full force for a century; adventurers
+flocked eagerly across the Atlantic, and discovery followed
+discovery in rapid succession. Many of the companions
+<span class="sidenote">Spaniards in America.</span>
+of Columbus continued his work. Vicente Yańez Pinzon
+in 1500 reached the mouth of the Amazon. In the same
+year Alonso de Ojeda, accompanied by Juan de la Cosa, from whose
+maps we learn much of the discoveries of the 16th century navigators,
+and by a Florentine named Amerigo Vespucci, touched the
+coast of South America somewhere near Surinam, following the shore
+as far as the Gulf of Maracaibo. Vespucci afterwards made three
+voyages to the Brazilian coast; and in 1504 he wrote an account
+of his four voyages, which was widely circulated, and became the
+means of procuring for its author at the hands of the cartographer
+Waldseemüller in 1507 the disproportionate distinction of giving his
+name to the whole continent. In 1508 Alonso de Ojeda obtained the
+government of the coast of South America from Cabo de la Vela
+to the Gulf of Darien; Ojeda landed at Cartagena in 1510, and
+sustained a defeat from the natives, in which his lieutenant, Juan
+de la Cosa, was killed. After another reverse on the east side of the
+Gulf of Darien Ojeda returned to Hispaniola and died there. The
+Spaniards in the Gulf of Darien were left by Ojeda under the command
+of Francisco Pizarro, the future conqueror of Peru. After
+suffering much from famine and disease, Pizarro resolved to leave,
+and embarked the survivors in small vessels, but outside the harbour
+they met a ship which proved to be that of Martin Fernandez Enciso,
+Ojeda&rsquo;s partner, coming with provisions and reinforcements. One
+of the crew of Enciso&rsquo;s ship, Vasco Nuńez de Balboa, the future discoverer
+of the Pacific Ocean, induced his commander to form a
+settlement on the other side of the Gulf of Darien. The soldiers
+became discontented and deposed Enciso, who was a man of learning
+and an accomplished cosmographer. His work <i>Suma de Geografia</i>,
+which was printed in 1519, is the first Spanish book which gives an
+account of America. Vasco Nuńez, the new commander, entered
+upon a career of conquest in the neighbourhood of Darien, which
+ended in the discovery of the Pacific Ocean on the 25th of September
+1513. Vasco Nuńez was beheaded in 1517 by Pedrarias de Avila,
+who was sent out to supersede him. This was one of the greatest
+calamities that could have happened to South America; for the
+discoverer of the South sea was on the point of sailing with a little
+fleet into his unknown ocean, and a humane and judicious man would
+probably have been the conqueror of Peru, instead of the cruel and
+ignorant Pizarro. In the year 1519 Panama was founded by
+Pedrarias; and the conquest of Peru by Pizarro followed a few years
+afterwards. Hernan Cortes overran and conquered Mexico from
+1518 to 1521, and the discovery and conquest of Guatemala by
+Alvarado, the invasion of Florida by De Soto, and of Nueva Granada
+by Quesada, followed in rapid succession. The first detailed account
+of the west coast of South America was written by a keenly observant
+old soldier, Pedro de Cieza de Leon, who was travelling in South
+America from 1533 to 1550, and published his story at Seville
+in 1553.</p>
+
+<p>The great desire of the Spanish government at that time was
+to find a westward route to the Moluccas. For this purpose Juan
+Diaz de Solis was despatched in October 1515, and in
+January 1516 he discovered the mouth of the Rio de la
+<span class="sidenote">Pacific Ocean.</span>
+Plata. He was, however, killed by the natives, and his
+ships returned. In the following year the Portuguese Ferdinando
+Magalhćes, familiarly known as Magellan, laid before Charles V.,
+at Valladolid, a scheme for reaching the Spice Islands by sailing
+westward. He started on the 21st of September 1519, entered the
+strait which now bears his name in October 1520, worked his way
+through between Patagonia and Tierra del Fuego, and entered on
+<span class="pagenum"><a name="page626" id="page626"></a>626</span>
+the vast Pacific which he crossed without sighting any of its innumerable
+island groups. This was unquestionably the greatest of
+the voyages which followed from the impulse of Prince Henry, and it
+was rendered possible only by the magnificent courage of the commander
+in spite of rebellion, mutiny and starvation. It was the
+6th of March 1521 when he reached the Ladrone Islands. Thence
+Magellan proceeded to the Philippines, and there his career ended
+in an unimportant encounter with hostile natives. Eventually a
+Biscayan named Sebastian del Cano, sailing home by way of the
+Cape of Good Hope, reached San Lucar in command of the &ldquo;Victoria&rdquo;
+on the 6th of September 1522, with eighteen survivors;
+this one ship of the squadron which sailed on the quest succeeded
+in accomplishing the first circumnavigation of the globe. Del Cano
+was received with great distinction by the emperor, who granted
+him a globe for his crest, and the motto <i>Primus circumdedisti me</i>.</p>
+
+<p>While the Spaniards were circumnavigating the
+world and completing their knowledge of the coasts of
+Central and South America, the Portuguese were actively
+<span class="sidenote">Portuguese in Africa and the East.</span>
+engaged on similar work as regards Africa and the East
+Indies.</p>
+
+<p>With Abyssinia the mission of Covilhćo led to further intercourse.
+In April 1520 Vasco da Gama, as viceroy of the Indies, took a fleet
+into the Red sea, and landed an embassy consisting of Dom Rodriguez
+de Lima and Father Francisco Alvarez, a priest whose detailed narrative
+is the earliest and not the least interesting account we possess
+of Abyssinia. It was not until 1526 that the embassy was dismissed;
+and not many years afterwards the negus entreated the help of the
+Portuguese against Mahommedan invaders, and the viceroy sent an
+expeditionary force, commanded by his brother Cristoforo da Gama,
+with 450 musketeers. Da Gama was taken prisoner and killed, but
+his followers enabled the Christians of Abyssinia to regain their
+power, and a Jesuit mission remained in the country. The Portuguese
+also established a close connexion with the kingdom of Congo
+on the west side of Africa, and obtained much information respecting
+the interior of the continent. Duarte Lopez, a Portuguese settled
+in the country, was sent on a mission to Rome by the king of Congo,
+and Pope Sixtus V. caused him to recount to his chamberlain,
+Felipe Pigafetta, all he had learned during the nine years he had been
+in Africa, from 1578 to 1587. This narrative, under the title of
+<i>Description of the Kingdom of Congo</i>, was published at Rome by
+Pigafetta in 1591. A map was attached on which several great
+equatorial lakes are shown, and the empire of Monomwezi or Unyamwezi
+is laid down. The most valuable work on Africa about
+this time is, however, that written by the Moor Leo Africanus in
+the early part of the 16th century. Leo travelled extensively in
+the north and west of Africa, and was eventually taken by pirates
+and sold to a master who presented him to Pope Leo X. At the
+pope&rsquo;s desire he translated his work on Africa into Italian.</p>
+
+<p>In Further India and the Malay Archipelago the Portuguese
+acquired predominating influence at sea, establishing factories on
+the Malabar coast, in the Persian Gulf, at Malacca, and in the Spice
+Islands, and extending their commercial enterprises from the Red
+sea to China. Their missionaries were received at the court of
+Akbar, and Benedict Goes, a native of the Azores, was despatched
+on a journey overland from Agra to China. He started in 1603,
+and, after traversing the least-known parts of Central Asia, he
+reached the confines of China. He appears to have ascended from
+Kabul to the plateau of the Pamir, and thence onwards by Yarkand,
+Khotan and Aksu. He died on the journey in March 1607; and
+thus, as one of the brethren pronounced his epitaph, &ldquo;seeking
+Cathay he found heaven.&rdquo;</p>
+
+<p>The activity and love of adventure, which became a passion for
+two or three generations in Spain and Portugal, spread to other
+countries. It was the spirit of the age; and England,
+Holland and France were fired by it. English enterprise
+<span class="sidenote">English, Dutch and French.</span>
+was first aroused by John and Sebastian Cabot, father
+and son, who came from Venice and settled at Bristol
+in the time of Henry VII. The Cabots received a patent in 1496,
+empowering them to seek unknown lands; and John Cabot discovered
+Newfoundland and part of the coast of America. Sebastian
+afterwards made a voyage to Rio de la Plata in the service of Spain,
+but he returned to England in 1548 and received a pension from
+Edward VI. At his suggestion a voyage was undertaken for the discovery
+of a north-east passage to Cathay, with Sir Hugh Willoughby
+as captain-general of the fleet and Richard Chancellor as pilot-major.
+They sailed in May 1553, but Willoughby and all his crew
+perished on the Lapland coast. Chancellor, however, was more
+fortunate. He reached the White Sea, performed the journey
+overland to Moscow, where he was well received, and may be said
+to have been the founder of the trade between Russia and England.
+He returned to Archangel and brought his ship back in safety to
+England. On a second voyage, in 1556, Chancellor was drowned;
+and three subsequent voyages, led by Stephen Burrough, Arthur
+Pet and Charles Jackman, in small craft of 50 tons and under,
+carried on an examination of the straits which lead into the Kara
+sea.</p>
+
+<p>The French followed closely on the track of John Cabot, and
+Norman and Breton fishermen frequented the banks of Newfoundland
+at the beginning of the 16th century. In 1524 Francis I. sent
+Giovanni da Verazzano of Florence on an expedition of discovery
+to the coast of North America; and the details of his voyage were
+embodied in a letter addressed by him to the king of France from
+Dieppe, in July 1524. In 1534 Jacques Cartier set out to continue
+the discoveries of Verazzano, and visited Newfoundland and the
+Gulf of St Lawrence. In the following year he made another
+voyage, discovered the island of Anticosti, and ascended the St
+Lawrence to Hochelaga, now Montreal. He returned, after passing
+two winters in Canada; and on another occasion he also failed to
+establish a colony. Admiral de Coligny made several unsuccessful
+endeavours to form a colony in Florida under Jean Ribault
+of Dieppe, René de Laudonničre and others, but the settlers
+were furiously assailed by the Spaniards and the attempt was
+abandoned.</p>
+
+<p>The reign of Elizabeth is famous for the gallant enterprises that
+were undertaken by sea and land to discover and bring to light the
+unknown parts of the earth. The great promoter of
+geographical discovery in the Elizabethan period was
+<span class="sidenote">The Elizabethan era.</span>
+Richard Hakluyt (1553-1616), who was active in the formation
+of the two companies for colonizing Virginia in
+1606; and devoted his life to encouraging and recording similar
+undertakings. He published much, and left many valuable papers
+at his death, most of which, together with many other narratives,
+were published in 1622 in the great work of the Rev. Samuel Purchas,
+entitled <i>Hakluytus Posthumus, or Purchas his Pilgrimes</i>.</p>
+
+<p>It is from these works that our knowledge of the gallant deeds of
+the English and other explorers of the Elizabethan age is mainly
+derived. The great and splendidly illustrated collections of voyages
+and travels of Theodorus de Bry and Hulsius served a similar useful
+purpose on the continent of Europe. One important object of
+English maritime adventurers of those days was to discover a route
+to Cathay by the north-west, a second was to settle Virginia, and a
+third was to raid the Spanish settlements in the West Indies. Nor
+was the trade to Muscovy and Turkey neglected; while latterly
+a resolute and successful attempt was made to establish direct
+commercial relations with India.</p>
+
+<p>The conception of the north-western route to Cathay now leads
+the story of exploration, for the first time as far as important and
+sustained efforts are concerned, towards the Arctic seas. This part
+of the story is fully told under the heading of <span class="sc"><a href="#artlinks">Polar Regions</a></span>, and
+only the names of Martin Frobisher (1576), John Davis (1585),
+Henry Hudson (1607) and William Baffin (1616) need be mentioned
+here in order to preserve the complete conspectus of the history of
+discovery. The Dutch emulated the British in the Arctic seas during
+this period, directing their efforts mainly towards the discovery of
+a north-east passage round the northern end of Novaya Zemlya;
+and William Barents or Barendsz (1594-1597) is the most famous
+name in this connexion, his boat voyage along the coast of Novaya
+Zemlya after losing his ship and wintering in a high latitude, being
+one of the most remarkable achievements in polar annals.</p>
+
+<p>Many English voyages were also made to Guinea and the West
+Indies, and twice English vessels followed in the track of Magellan,
+and circumnavigated the globe. In 1577 Francis Drake, who had
+previously served with Hawkins in the West Indies, undertook his
+celebrated voyage round the world. Reaching the Pacific through
+the Strait of Magellan, Drake proceeded northward along the west
+coast of America, resolved to attempt the discovery of a northern
+passage from the Pacific to the Atlantic. The coast from the
+southern extremity of the Californian peninsula to Cape Mendocino
+had been discovered by Juan Rodriguez Cabrillo and Francisco de
+Ulloa in 1539. Drake&rsquo;s discoveries extended from Cape Mendocino
+to 48° N., in which latitude he gave up his quest, sailed across the
+Pacific and reached the Philippine Islands, returning home round
+the Cape of Good Hope in 1580.</p>
+
+<p>Thomas Cavendish, emulous of Drake&rsquo;s example, fitted out three
+vessels for an expedition to the South sea in 1586. He took the
+same route as Drake along the west coast of America. From Cape
+San Lucas Cavendish steered across the Pacific, seeing no land until
+he reached the Ladrone Islands. He returned to England in 1588.
+The third English voyage into the Pacific was not so fortunate.
+Sir Richard Hawkins (1593) on reaching the bay of Atacames, in 1°N.
+in 1594, was attacked by a Spanish fleet, and, after a desperate
+naval engagement, was forced to surrender. Hawkins declared
+his object to be discovery and the survey of unknown lands, and
+his voyage, though terminating in disaster, bore good fruit. <i>The
+Observations of Sir Richard Hawkins in his Voyage into the South Sea</i>,
+published in 1622, are very valuable. It was long before another
+British ship entered the Pacific Ocean. Sir John Narborough took
+two ships through the Strait of Magellan in 1670 and touched on
+the coast of Chile, but it was not until 1685 that Dampier sailed over
+the part of the Pacific where Hawkins met his defeat.</p>
+
+<p>The exploring enterprise of the Spanish nation did not wane
+after the conquest of Peru and Mexico, and the acquisition of the
+vast empire of the Indies. It was spurred into renewed activity
+by the audacity of Sir John Hawkins in the West Indies, and by
+the appearance of Drake, Cavendish and Richard Hawkins in the
+Pacific.</p>
+
+<p>In the interior of South America the Spanish conquerors had
+explored the region of the Andes from the isthmus of Panama to
+Chile. Pedro de Valdivia in 1540 made an expedition into the
+country of the Araucanian Indians of Chile, and was the first to
+<span class="pagenum"><a name="page627" id="page627"></a>627</span>
+explore the eastern base of the Andes in what is now Argentine
+Patagonia. In 1541 Francisco de Orellana discovered the whole
+course of the Amazon from its source in the Andes to the Atlantic.
+A second voyage on the Amazon was made in 1561 by the mad pirate
+Lope de Aguirre; but it was not until 1639 that a full account was
+written of the great river by Father Cristoval de Acuńa, who ascended
+it from its mouth and reached the city of Quito.</p>
+
+<p>The voyage of Drake across the Pacific was preceded by that of
+Alvaro de Mendańa, who was despatched from Peru in 1567 to
+discover the great Antarctic continent which was believed
+to extend far northward into the South sea, the search
+<span class="sidenote">Spaniards in the Pacific.</span>
+for which now became one of the leading motives of
+exploration. After a voyage of eighty days across the
+Pacific, Mendańa discovered the Solomon Islands; and the expedition
+returned in safety to Callao. The appearance of Drake on
+the Peruvian coast led to an expedition being fitted out at Callao,
+to go in chase of him, under the command of Pedro Sarmiento. He
+sailed from Callao in October 1579, and made a careful survey of
+the Strait of Magellan, with the object of fortifying that entrance
+to the South sea. The colony which he afterwards took out from
+Spain was a complete failure, and is only remembered now from the
+name of &ldquo;Port Famine,&rdquo; which Cavendish gave to the site at which
+he found the starving remnant of Sarmiento&rsquo;s settlers. In June
+1595 Mendańa sailed from the coast of Peru in command of a second
+expedition to colonize the Solomon Islands. After discovering the
+Marquesas, he reached the island of Santa Cruz of evil memory,
+where he and many of the settlers died. His young widow took
+command of the survivors and brought them safely to Manila.
+The viceroys of Peru still persevered in their attempts to plant a
+colony in the hypothetical southern continent. Pedro Fernandez
+de Quiros, who was pilot under Mendańa and Luis Vaez de Torres,
+were sent in command of two ships to continue the work of exploration.
+They sailed from Callao in December 1605, and discovered
+several islands of the New Hebrides group. They anchored in a bay
+of a large island which Quiros named &ldquo;Australia del Espiritu Santo.&rdquo;
+From this place Quiros returned to America, but Torres continued
+the voyage, passed through the strait between Australia and New
+Guinea which bears his name, and explored and mapped the southern
+and eastern coasts of New Guinea.</p>
+
+<p>The Portuguese, in the early part of the 17th century (1578-1640),
+were under the dominion of Spain, and their enterprise was
+to some extent damped; but their missionaries extended geographical
+knowledge in Africa. Father Francisco Paez acquired great influence
+in Abyssinia, and explored its highlands from 1600 to 1622. Fathers
+Mendez and Lobo traversed the deserts between the coast of the
+Red sea and the mountains, became acquainted with Lake Tsana,
+and discovered the sources of the Blue Nile in 1624-1633.</p>
+
+<p>But the attention of the Portuguese was mainly devoted to vain
+attempts to maintain their monopoly of the trade of India against
+the powerful rivalry of the English and Dutch. The
+English enterprises were persevering, continuous and
+<span class="sidenote">Rivalry in the East.</span>
+successful. James Lancaster made a voyage to the Indian
+Ocean from 1591 to 1594; and in 1599 the merchants and adventurers
+of London resolved to form a company, with the object of
+establishing a trade with the East Indies. On the 31st of December
+1599 Queen Elizabeth granted the charter of incorporation to the
+East India Company, and Sir James Lancaster, one of the directors,
+was appointed general of their first fleet. He was accompanied
+by John Davis, the great Arctic navigator, as pilot-major. This
+voyage was eminently successful. The ships touched at Achin in
+Sumatra and at Java, returning with full ladings of pepper in 1603.
+The second voyage was commanded by Sir Henry Middleton; but
+it was in the third voyage, under Keelinge and Hawkins, that the
+mainland of India was first reached in 1607. Captain Hawkins
+landed at Surat and travelled overland to Agra, passing some time
+at the court of the Great Mogul. In the voyage of Sir Edward
+Michelborne in 1605, John Davis lost his life in a fight with a Japanese
+junk. The eighth voyage, led by Captain Saris, extended the
+operations of the company to Japan; and in 1613 the Japanese
+government granted privileges to the company; but the British
+retired in 1623, giving up their factory. The chief result of this
+early intercourse between Great Britain and Japan was the interesting
+series of letters written by William Adams from 1611 to 1617. From
+the tenth voyage of the East India Company, commanded by
+Captain Best, who left England in 1612, dates the establishment of
+permanent British factories on the coast of India. It was Captain
+Best who secured a regular <i>firman</i> for trade from the Great Mogul.
+From that time a fleet was despatched every year, and the company&rsquo;s
+operations greatly increased geographical knowledge of India
+and the Eastern Archipelago. British visits to Eastern countries,
+at this time, were not confined to the voyages of the company.
+Journeys were also made by land, and, among others, the entertaining
+author of the <i>Crudities</i>, Thomas Coryate, of Odcombe in
+Somersetshire, wandered on foot from France to India, and died
+(1617) in the company&rsquo;s factory at Surat. In 1561 Anthony Jenkinson
+arrived in Persia with a letter from Queen Elizabeth to the shah.
+He travelled through Russia to Bokhara, and returned by the
+Caspian and Volga. In 1579 Christopher Burroughs built a ship
+at Nizhniy Novgorod and traded across the Caspian to Baku; and
+in 1598 Sir Anthony and Robert Shirley arrived in Persia, and
+Robert was afterwards sent by the shah to Europe as his ambassador.
+He was followed by a Spanish mission under Garcia de Silva, who
+wrote an interesting account of his travels; and to Sir Dormer
+Cotton&rsquo;s mission, in 1628, we are indebted for Sir Thomas Herbert&rsquo;s
+charming narrative. In like manner Sir Thomas Roe&rsquo;s mission
+to India resulted not only in a large collection of valuable reports
+and letters of his own, but also in the detailed account of his chaplain
+Terry. But the most learned and intelligent traveller in the East,
+during the 17th century, was the German, Engelbrecht Kaempfer,
+who accompanied an embassy to Persia, in 1684, and was afterwards
+a surgeon in the service of the Dutch East India Company. He
+was in the Persian Gulf, India and Java, and resided for more than
+two years in Japan, of which he wrote a history.</p>
+
+<p>The Dutch nation, as soon as it was emancipated from Spanish
+tyranny, displayed an amount of enterprise, which, for a long time,
+was fully equal to that of the British. The Arctic voyages
+of Barents were quickly followed by the establishment of
+<span class="sidenote">Dutch exploration, 16th-17th centuries.</span>
+a Dutch East India Company; and the Dutch, ousting
+the Portuguese, not only established factories on the
+mainland of India and in Japan, but acquired a preponderating
+influence throughout the Malay Archipelago. In 1583 Jan
+Hugen van Linschoten made a voyage to India with a Portuguese
+fleet, and his full and graphic descriptions of India, Africa, China
+and the Malay Archipelago must have been of no small use to his
+countrymen in their distant voyages. The first of the Dutch Indian
+voyages was performed by ships which sailed in April 1595, and
+rounded the Cape of Good Hope. A second large Dutch fleet sailed
+in 1598; and, so eager was the republic to extend her commerce
+over the world that another fleet, consisting of five ships of Rotterdam,
+was sent in the same year by way of Magellan&rsquo;s Strait, under
+Jacob Mahu as admiral, with William Adams as pilot. Mahu died
+on the passage out, and was succeeded by Simon de Cordes, who
+was killed on the coast of Chile. In September 1599 the fleet had
+entered the Pacific. The ships were then steered direct for Japan,
+and anchored off Bungo in April 1600. In the same year, 1598, a
+third expedition was despatched under Oliver van Noort, a native
+of Utrecht, but the voyage contributed nothing to geography. The
+Dutch Company in 1614 again resolved to send a fleet to the Moluccas
+by the westward route, and Joris Spilbergen was appointed to the
+command as admiral, with a commission from the States-General.
+He was furnished with four ships of Amsterdam, two of Rotterdam
+and one from Zeeland. On the 6th of May 1615 Spilbergen entered
+the Pacific Ocean, and touched at several places on the coast of Chile
+and Peru, defeating the Spanish fleet in a naval engagement off
+Chilca. After plundering Payta and making requisitions at Acapulco,
+the Dutch fleet crossed the Pacific and reached the Moluccas in
+March 1616.</p>
+
+<p>The Dutch now resolved to discover a passage into the Pacific
+to the south of Tierra del Fuego, the insular nature of which had
+been ascertained by Sir Francis Drake. The vessels fitted out for
+this purpose were the &ldquo;Eendracht,&rdquo; of 360 tons, commanded by
+Jacob Lemaire, and the &ldquo;Hoorn,&rdquo; of 110 tons, under Willem
+Schouten. They sailed from the Texel on the 14th of June 1615,
+and by the 20th of January 1616 they were south of the entrance
+of Magellan&rsquo;s Strait. Passing through the strait of Lemaire they
+came to the southern extremity of Tierra del Fuego, which was
+named Cape Horn, in honour of the town of Hoorn in West Friesland,
+of which Schouten was a native. They passed the cape on the 31st
+of January, encountering the usual westerly winds. The great merit
+of this discovery of a second passage into the South sea lies in the
+fact that it was not accidental or unforeseen, but was due to the
+sagacity of those who designed the voyage. On the 1st of March
+the Dutch fleet sighted the island of Juan Fernandez; and, having
+crossed the Pacific, the explorers sailed along the north coast of
+New Guinea and arrived at the Moluccas on the 17th of September
+1616.</p>
+
+<p>There were several early indications of the existence of the great
+Australian continent, and the Dutch endeavoured to obtain further
+knowledge concerning the country and its extent; but only its
+northern and western coasts had been visited before the time of
+Governor van Diemen. Dirk Hartog had been on the west coast
+in latitude 26° 30&prime; S. in 1616. Pelsert struck on a reef called &ldquo;Houtman&rsquo;s
+Abrolhos&rdquo; on the 4th of June 1629. In 1697 the Dutch
+captain Vlamingh landed on the west coast of Australia, then called
+New Holland, in 31° 43&prime; S.,  and named the Swan river from the black
+swans he discovered there. In 1642 the governor and council of
+Batavia fitted out two ships to prosecute the discovery of the south
+land, then believed to be part of a vast Antarctic continent, and
+entrusted the command to Captain Abel Jansen Tasman. This
+voyage proved to be the most important to geography that had been
+undertaken since the first circumnavigation of the globe. Tasman
+sailed from Batavia in 1642, and on the 24th of November sighted
+high land in 42° 30&prime; S., which was named van Diemen&rsquo;s Land, and
+after landing there proceeded to the discovery of the western coast
+of New Zealand; at first called Staten Land, and supposed to be connected
+with the Antarctic continent from which this voyage proved
+New Holland to be separated. He then reached Tongatabu, one
+of the Friendly Islands of Cook; and returned by the north coast
+of New Guinea to Batavia. In 1644 Tasman made a second voyage
+to effect a fuller discovery of New Guinea.</p>
+
+<p><span class="pagenum"><a name="page628" id="page628"></a>628</span></p>
+
+<p>The French directed their enterprise more in the direction of
+North America than of the Indies. One of their most distinguished
+explorers was Samuel Champlain, a captain in the navy,
+<span class="sidenote">French in North America.</span>
+who, after a remarkable journey through Mexico and the
+West Indies from 1599 to 1602, established his historic
+connexion with Canada, to the geographical knowledge
+of which he made a very large addition.</p>
+
+<p>The principles and methods of surveying and position finding
+had by this time become well advanced, and the most remarkable
+example of the early application of these improvements
+is to be found in the survey of China by Jesuit missionaries.
+<span class="sidenote">Missionaries in the East.</span>
+They first prepared a map of the country round Peking,
+which was submitted to the emperor Kang-hi, and,
+being satisfied with the accuracy of the European method of surveying,
+he resolved to have a survey made of the whole empire on the
+same principles. This great work was begun in July 1708, and the
+completed maps were presented to the emperor in 1718. The
+records preserved in each city were examined, topographical information
+was diligently collected, and the Jesuit fathers checked their
+triangulation by meridian altitudes of the sun and pole star and by a
+system of remeasurements. The result was a more accurate map of
+China than existed, at that time, of any country in Europe. Kang-hi
+next ordered a similar map to be made of Tibet, the survey being
+executed by two lamas who were carefully trained as surveyors
+by the Jesuits at Peking. From these surveys were constructed
+the well-known maps which were forwarded to Duhalde, and which
+D&rsquo;Anville utilized for his atlas.</p>
+
+<p>Several European missionaries had previously found their way
+from India to Tibet. Antonio Andrada, in 1624, was the first
+European to enter Tibet since the visit of Friar Odoric
+<span class="sidenote">The 18th century.</span>
+in 1325. The next journey was that of Fathers Grueber
+and Dorville about 1660, who succeeded in passing from
+China, through Tibet, into India. In 1715 Fathers Desideri and
+Freyre made their way from Agra, across the Himalayas, to Lhasa,
+and the Capuchin Friar Orazio della Penna resided in that city
+from 1735 until 1747. But the most remarkable journey in this
+direction was performed by a Dutch traveller named Samuel van de
+Putte. He left Holland in 1718, went by land through Persia to
+India, and eventually made his way to Lhasa, where he resided for a
+long time. He went thence to China, returned to Lhasa, and was
+in India in time to be an eye-witness of the sack of Delhi by Nadir
+<span class="sidenote">Asia.</span>
+Shah in 1737. In 1743 he left India and died at Batavia
+on the 27th of September 1745. The premature death
+of this illustrious traveller is the more to be lamented because his
+vast knowledge died with him. Two English missions sent by
+Warren Hastings to Tibet, one led by George Bogle in 1774, and the
+other by Captain Turner in 1783, complete Tibetan exploration in
+the 18th century.</p>
+
+<p>From Persia much new information was supplied by Jean Chardin,
+Jean Tavernier, Charles Hamilton, Jean de Thévenot and Father
+Jude Krusinski, and by English traders on the Caspian. In 1738
+John Elton traded between Astrakhan and the Persian port of
+Enzelī on the Caspian, and undertook to build a fleet for Nadir
+Shah. Another English merchant, named Jonas Hanway, arrived
+at Astrabad from Russia, and travelled to the camp of Nadir at
+Kazvin. One lasting and valuable result of Hanway&rsquo;s wanderings
+was a charming book of travels. In 1700 Guillaume Delisle published
+his map of the continents of the Old World; and his successor
+D&rsquo;Anville produced his map of India in 1752. D&rsquo;Anville&rsquo;s map
+contained all that was then known, but ten years afterwards Major
+Rennell began his surveying labours, which extended over the
+period from 1763 to 1782. His survey covered an area 900 m. long
+by 300 wide, from the eastern confines of Bengal to Agra, and from
+the Himalayas to Calpi. Rennell was indefatigable in collecting
+geographical information; his Bengal atlas appeared in 1781, his
+famous map of India in 1788 and the memoir in 1792. Surveys
+were also made along the Indian coasts.</p>
+
+<p>Arabia received very careful attention, in the 18th century,
+from the Danish scientific mission, which included Carsten Niebuhr
+among its members. Niebuhr landed at Loheia, on the coast of
+Yemen, in December 1762, and went by land to Sana. All the other
+members of the mission died, but he proceeded from Mokha to
+Bombay. He then made a journey through Persia and Syria to
+Constantinople, returning to Copenhagen in 1767. His valuable
+work, the <i>Description of Arabia</i>, was published in 1772, and was
+followed in 1774-1778 by two volumes of travels in Asia. The great
+traveller survived until 1815, when he died at the age of eighty-two.</p>
+
+<p>James Bruce of Kinnaird, the contemporary of Niebuhr, was
+equally devoted to Eastern travel; and his principal geographical
+work was the tracing of the Blue Nile from its source to
+its junction with the White Nile. Before the death of
+<span class="sidenote">Africa.</span>
+Bruce an African Association was formed, in 1788, for collecting
+information respecting the interior of that continent, with Major
+Rennell and Sir Joseph Banks as leading members. The association
+first employed John Ledyard (who had previously made an extraordinary
+journey into Siberia) to cross Africa from east to west
+on the parallel of the Niger, and William Lucas to cross the Sahara
+to Fezzan. Lucas went from Tripoli to Mesurata, obtained some
+information respecting Fezzan and returned in 1789. One of the
+chief problems the association wished to solve was that of the existence
+and course of the river Niger, which was believed by some
+authorities to be identical with the Congo. Mungo Park, then an
+assistant surgeon of an Indiaman, volunteered his services, which
+were accepted by the association, and in 1795 he succeeded in
+reaching the town of Segu on the Niger, but was prevented from
+continuing his journey to Timbuktu. Five years later he accepted
+an offer from the government to command an expedition into the
+interior of Africa, the plan being to cross from the Gambia to the
+Niger and descend the latter river to the sea. After losing most of
+his companions he himself and the rest perished in a rapid on the
+Niger at Busa, having been attacked from the shore by order of a
+chief who thought he had not received suitable presents. His work,
+however, had established the fact that the Niger was not identical
+with the Congo.</p>
+
+<p>While the British were at work in the direction of the Niger, the
+Portuguese were not unmindful of their old exploring fame. In
+1798 Dr F.J.M. de Lacerda, an accomplished astronomer, was
+appointed to command a scientific expedition of discovery to the
+north of the Zambesi. He started in July, crossed the Muchenja
+Mountains, and reached the capital of the Cazembe, where he died
+of fever. Lacerda left a valuable record of his adventurous journey;
+but with Mungo Park and Lacerda the history of African exploration
+in the 18th century closes.</p>
+
+<p>In South America scientific exploration was active during this
+period. The great geographical event of the century, as regards
+that continent, was the measurement of an arc of the
+meridian. The undertaking was proposed by the French
+<span class="sidenote">South America.</span>
+Academy as part of an investigation with the object
+of ascertaining the length of the degree near the equator and near the
+pole respectively so as to determine the figure of the earth. A
+commission left Paris in 1735, consisting of Charles Marie de la
+Condamine, Pierre Bouguer, Louis Godin and Joseph de Jussieu
+the naturalist. Spain appointed two accomplished naval officers,
+the brothers Ulloa, as coadjutors. The operations were carried on
+during eight years on a plain to the south of Quito; and, in addition
+to his memoir on this memorable measurement, La Condamine
+collected much valuable geographical information during a voyage
+down the Amazon. The arc measured was 3° 7&prime; 3&Prime; in length;
+and the work consisted of two measured bases connected by a series
+of triangles, one north and the other south of the equator, on the
+meridian of Quito. Contemporaneously, in 1738, Pierre Louis
+Moreau de Maupertuis, Alexis Claude Clairaut, Charles Etienne
+Louis Camus, Pierre Charles Lemonnier and the Swedish physicist
+Celsius measured an arc of the meridian in Lapland.</p>
+
+<p>The British and French governments despatched several expeditions
+of discovery into the Pacific and round the world during the
+18th century. They were preceded by the wonderful
+and romantic voyages of the buccaneers. The narratives
+<span class="sidenote">The Pacific Ocean.</span>
+of such men as Woodes Rogers, Edward Davis, George
+Shelvocke, Clipperton and William Dampier, can never
+fail to interest, while they are not without geographical value.
+The works of Dampier are especially valuable, and the narratives
+of William Funnell and Lionel Wafer furnished the best accounts
+then extant of the Isthmus of Darien. Dampier&rsquo;s literary ability
+eventually secured for him a commission in the king&rsquo;s service;
+and he was sent on a voyage of discovery, during which he explored
+part of the coasts of Australia and New Guinea, and discovered the
+strait which bears his name between New Guinea and New Britain,
+returning in 1701. In 1721 Jacob Roggewein was despatched on a
+voyage of some importance across the Pacific by the Dutch West
+India Company, during which he discovered Easter Island on the
+6th of April 1722.</p>
+
+<p>The voyage of Lord Anson to the Pacific in 1740-1744 was of a
+predatory character, and he lost more than half his men from scurvy;
+while it is not pleasant to reflect that at the very time when the
+French and Spaniards were measuring an arc of the meridian at
+Quito, the British under Anson were pillaging along the coast of the
+Pacific and burning the town of Payta. But a romantic interest
+attaches to the wreck of the &ldquo;Wager,&rdquo; one of Anson&rsquo;s fleet, on a
+desert island near Chiloe, for it bore fruit in the charming narrative
+of Captain John Byron, which will endure for all time. In 1764
+Byron himself was sent on a voyage of discovery round the world,
+which led immediately after his return to the despatch of another
+to complete his work, under the command of Captain Samuel Wallis.</p>
+
+<p>The expedition, consisting of the &ldquo;Dolphin&rdquo; commanded by
+Wallis, and the &ldquo;Swallow&rdquo; under Captain Philip Carteret, sailed in
+September 1766, but the ships were separated on entering the Pacific
+from the Strait of Magellan. Wallis discovered Tahiti on the 19th
+of June 1767, and he gave a detailed account of that island. He
+returned to England in May 1768. Carteret discovered the Charlotte
+and Gloucester Islands, and Pitcairn Island on the 2nd of July 1767;
+revisited the Santa Cruz group, which was discovered by Mendańa
+and Quiros; and discovered the strait separating New Britain from
+New Ireland. He reached Spithead again in February 1769. Wallis
+and Carteret were followed very closely by the French expedition
+of Bougainville, which sailed from Nantes in November 1766.
+Bougainville had first to perform the unpleasant task of delivering
+up the Falkland Islands, where he had encouraged the formation
+of a French settlement, to the Spaniards. He then entered the
+Pacific, and reached Tahiti in April 1768. Passing through the New
+<span class="pagenum"><a name="page629" id="page629"></a>629</span>
+Hebrides group he touched at Batavia, and arrived at St Malo after
+an absence of two years and four months.</p>
+
+<p>The three voyages of Captain James Cook form an era in the history
+of geographical discovery. In 1767 he sailed for Tahiti, with the
+object of observing the transit of Venus, accompanied
+by two naturalists, Sir Joseph Banks and Dr Solander,
+<span class="sidenote">Captain Cook.</span>
+a pupil of Linnaeus, as well as by two astronomers. The
+transit was observed on the 3rd of June 1769. After exploring
+Tahiti and the Society group, Cook spent six months surveying New
+Zealand, which he discovered to be an island, and the coast of New
+South Wales from latitude 38° S. to the northern extremity. The
+belief in a vast Antarctic continent stretching far into the temperate
+zone had never been abandoned, and was vehemently asserted by
+Charles Dalrymple, a disappointed candidate nominated by the
+Royal Society for the command of the Transit expedition of 1769.
+In 1772 the French explorer Yves Kerguelen de Tremarec had discovered
+the land that bears his name in the South Indian Ocean
+without recognizing it to be an island, and naturally believed it
+to be part of the southern continent.</p>
+
+<p>Cook&rsquo;s second voyage was mainly intended to settle the question
+of the existence of such a continent once for all, and to define the
+limits of any land that might exist in navigable seas towards the
+Antarctic circle. James Cook at his first attempt reached a south
+latitude of 57° 15&prime;. On a second cruise from the Society Islands,
+in 1773, he, first of all men, crossed the Antarctic circle, and was
+stopped by ice in 71° 10&prime; S. During the second voyage Cook visited
+Easter Island, discovered several islands of the New Hebrides and
+New Caledonia; and on his way home by Cape Horn, in March 1774,
+he discovered the Sandwich Island group and described South
+Georgia. He proved conclusively that any southern continent
+that might exist lay under the polar ice. The third voyage was
+intended to attempt the passage from the Pacific to the Atlantic by
+the north-east. The &ldquo;Resolution&rdquo; and &ldquo;Discovery&rdquo; sailed in
+1776, and Cook again took the route by the Cape of Good Hope.
+On reaching the North American coast, he proceeded northward,
+fixed the position of the western extremity of America and surveyed
+Bering Strait. He was stopped by the ice in 70° 41&prime; N., and named
+the farthest visible point on the American shore Icy Cape. He then
+visited the Asiatic shore and discovered Cape North. Returning to
+Hawaii, Cook was murdered by the natives. On the 14th of February
+1779, his second, Captain Edward Clerke, took command, and
+proceeding to Petropavlovsk in the following summer, he again
+examined the edge of the ice, but only got as far as 70° 33&prime; N. The
+ships returned to England in October 1780.</p>
+
+<p>In 1785 the French government carefully fitted out an expedition
+of discovery at Brest, which was placed under the command of
+Franēois La Pérouse, an accomplished and experienced officer.
+After touching at Concepcion in Chile and at Easter Island, La
+Pérouse proceeded to Hawaii and thence to the coast of California,
+of which he has given a very interesting account. He then crossed
+the Pacific to Macao, and in July 1787 he proceeded to explore the
+Gulf of Tartary and the shores of Sakhalin, remaining some time at
+Castries Bay, so named after the French minister of marine. Thence
+he went to the Kurile Islands and Kamchatka, and sailed from the
+far north down the meridian to the Navigator and Friendly Islands.
+He was in Botany Bay in January 1788; and sailing thence, the
+explorer, his ship and crew were never seen again. Their fate was
+long uncertain. In September 1791 Captain Antoine d&rsquo;Entrecasteaux
+sailed from Brest with two vessels to seek for tidings.
+He visited the New Hebrides, Santa Cruz, New Caledonia and Solomon
+Islands, and made careful though rough surveys of the Louisiade
+Archipelago, islands north of New Britain and part of New Guinea.
+D&rsquo;Entrecasteaux died on board his ship on the 20th of July 1793,
+without ascertaining the fate of La Pérouse. Captain Peter Dillon
+at length ascertained, in 1828, that the ships of La Pérouse had been
+wrecked on the island of Vanikoro during a hurricane.</p>
+
+<p>The work of Captain Cook bore fruit in many ways. His master,
+Captain William Bligh, was sent in the &ldquo;Bounty&rdquo; to convey breadfruit
+plants from Tahiti to the West Indies. He reached Tahiti in
+October 1788, and in April 1789 a mutiny broke out, and he, with
+several officers and men, was thrust into an open boat in mid-ocean.
+During the remarkable voyage he then made to Timor, Bligh
+passed amongst the northern islands of the New Hebrides, which
+he named the Banks Group, and made several running surveys.
+He reached England in March 1790. The &ldquo;Pandora,&rdquo; under
+Captain Edwards, was sent out in search of the &ldquo;Bounty,&rdquo; and
+discovered the islands of Cherry and Mitre, east of the Santa Cruz
+group, but she was eventually lost on a reef in Torres Strait. In
+1796-1797 Captain Wilson, in the missionary ship &ldquo;Duff,&rdquo; discovered
+the Gambier and other islands, and rediscovered the islands known
+to and seen by Quiros, but since called the Duff Group. Another
+result of Captain Cook&rsquo;s work was the colonization of Australia.
+On the 18th of January 1788 Admiral Phillip and Captain Hunter
+arrived in Botany Bay in the &ldquo;Supply&rdquo; and &ldquo;Sirius,&rdquo; followed by
+six transports, and established a colony at Port Jackson. Surveys
+were then undertaken in several directions. In 1795 and 1796
+Matthew Flinders and George Bass were engaged on exploring work
+in a small boat called the &ldquo;Tom Thumb.&rdquo; In 1797 Bass, who had
+been a surgeon, made an expedition southwards, continued the work
+of Cook from Ram Head, and explored the strait which bears his
+name, and in 1798 he and Flinders were surveying on the east coast
+of Van Diemen&rsquo;s land.</p>
+
+<p>Yet another outcome of Captain Cook&rsquo;s work was the voyage of
+George Vancouver, who had served as a midshipman in Cook&rsquo;s
+second and third voyages. The Spaniards under Quadra had begun
+a survey of north-western America and occupied Nootka Sound,
+which their government eventually agreed to surrender. Captain
+Vancouver was sent out to receive the cession, and to survey the
+coast from Cape Mendocino northwards. He commanded the old
+&ldquo;Discovery,&rdquo; and was at work during the seasons of 1792, 1793 and
+1794, wintering at Hawaii. Returning home in 1795, he completed
+his narrative and a valuable series of charts.</p>
+
+<p>The 18th century saw the Arctic coast of North America reached
+at two points, as well as the first scientific attempt to reach the
+North Pole. The Hudson Bay Company had been incorporated
+in 1670, and its servants soon extended their
+<span class="sidenote">Arctic regions.</span>
+operations over a wide area to the north and west of
+Canada. In 1741 Captain Christopher Middleton was ordered to
+solve the question of a passage from Hudson Bay to the westward.
+Leaving Fort Churchill in July 1742, he discovered the Wager river
+and Repulse Bay. He was followed by Captain W. Moor in 1746,
+and Captain Coats in 1751, who examined the Wager Inlet up to the
+end. In November 1769 Samuel Hearne was sent by the Hudson
+Bay Company to discover the sea on the north side of America,
+but was obliged to return. In February 1770 he set out again from
+Fort Prince of Wales; but, after great hardships, he was again
+forced to return to the fort. He started once more in December
+1771, and at length reached the Coppermine river, which he surveyed
+to its mouth, but his observations are unreliable. With the same
+object Alexander Mackenzie, with a party of Canadians, set out from
+Fort Chippewyan on the 3rd of June 1789, and descending the great
+river which now bears the explorer&rsquo;s name reached the Arctic sea.</p>
+
+<p>In February 1773 the Royal Society submitted a proposal to the
+king for an expedition towards the North Pole. The expedition was
+fitted out under Captains Constantine Phipps and Skeffington
+Lutwidge, and the highest latitude reached was 80° 48&prime; N., but no
+opening was discovered in the heavy Polar pack. The most important
+Arctic work in the 18th century was performed by the
+Russians, for they succeeded in delineating the whole of the northern
+coast of Siberia. Some of this work was possibly done at a still
+earlier date. The Cossack Simon Dezhneff is thought to have made a
+voyage, in the summer of 1648, from the river Kolyma, through
+Bering Strait (which was rediscovered by Vitus Bering in 1728) to
+Anadyr. Between 1738 and 1750 Manin and Sterlegoff made their
+way in small sloops from the mouth of the Yenesei as far north as
+75° 15&prime; N. The land from Taimyr to Cape Chelyuskin, the most
+northern extremity of Siberia, was mapped in many years of patient
+exploration by Chelyuskin, who reached the extreme point
+(77° 34&prime; N.) in May 1742. To the east of Cape Chelyuskin the
+Russians encountered greater difficulties. They built small vessels
+at Yakutsk on the Lena, 900 m. from its mouth, whence the first
+expedition was despatched under Lieut. Prontschichev in 1735. He
+sailed from the mouth of the Lena to the mouth of the Olonek,
+where he wintered, and on the 1st of September 1736 he got as far
+as 77° 29&prime; N., within 5 m. of Cape Chelyuskin. Both he and his
+young wife died of scurvy, and the vessel returned. A second
+expedition, under Lieut. Laptyev, started from the Lena in 1739,
+but encountered masses of drift ice in Chatanga bay, and with this
+ended the voyages to the westward of the Lena. Several attempts
+were also made to navigate the sea from the Lena to the Kolyma.
+In 1736 Lieut. Laptyev sailed, but was stopped by the drift ice in
+August, and in 1739, during another trial, he reached the mouth
+of the Indigirka, where he wintered. In the season of 1740 he
+continued his voyage to beyond the Kolyma, wintering at Nizhni
+Kolymsk. In September 1740 Vitus Bering sailed from Okhotsk
+on a second Arctic voyage with George William Steller on board
+as naturalist. In June 1741 he named the magnificent peak on the
+coast of North America Mount St Elias and explored the Aleutian
+Islands. In November the ship was wrecked on Bering Island;
+and the gallant Dane, worn out with scurvy, died there on the
+8th of December 1741. In March 1770 a merchant named Liakhov
+saw a large herd of reindeer coming from the north to the Siberian
+coast, which induced him to start in a sledge in the direction whence
+they came. Thus he reached the New Siberian or Liakhov Islands,
+and for years afterwards the seekers for fossil ivory resorted to them.
+The Russian Captain Vassili Chitschakov in 1765 and 1766 made two
+persevering attempts to penetrate the ice north of Spitsbergen,
+and reached 80° 30&prime; N., while Russian parties twice wintered at Bell
+Sound.</p>
+
+<p>In reviewing the progress of geographical discovery thus far, it
+has been possible to keep fairly closely to a chronological order.
+But in the 19th century and after exploring work was so
+generally and steadily maintained in all directions, and
+<span class="sidenote">Geographical societies.</span>
+was in so many cases narrowed down from long journeys
+to detailed surveys within relatively small areas, that it
+becomes desirable to cover the whole period at one view for certain
+great divisions of the world. (See <span class="sc"><a href="#artlinks">Africa</a></span>; <span class="sc"><a href="#artlinks">Asia</a></span>; <span class="sc"><a href="#artlinks">Australia</a></span>; <span class="sc"><a href="#artlinks">Polar
+Regions</a></span>; &amp;c.) Here, however, may be noticed the development
+of geographical societies devoted to the encouragement of exploration
+and research. The first of the existing geographical societies was
+<span class="pagenum"><a name="page630" id="page630"></a>630</span>
+that of Paris, founded in 1825 under the title of La Société de
+Géographie. The Berlin Geographical Society (Gesellschaft für
+Erdkunde) is second in order of seniority, having been founded in
+1827. The Royal Geographical Society, which was founded in
+London in 1830, comes third on the list; but it may be viewed as a
+direct result of the earlier African Association founded in 1788.
+Sir John Barrow, Sir John Cam Hobhouse (Lord Broughton), Sir
+Roderick Murchison, Mr Robert Brown and Mr Bartle Frere formed
+the foundation committee of the Royal Geographical Society, and
+the first president was Lord Goderich. The action of the society in
+supplying practical instruction to intending travellers, in astronomy,
+surveying and the various branches of science useful to collectors,
+has had much to do with advancement of discovery. Since the war
+of 1870 many geographical societies have been established on the
+continent of Europe. At the close of the 19th century there were
+upwards of 100 such societies in the world, with more than 50,000
+members, and over 150 journals were devoted entirely to geographical
+subjects.<a name="fa11b" id="fa11b" href="#ft11b"><span class="sp">11</span></a> The great development of photography has been a notable
+aid to explorers, not only by placing at their disposal a faithful and
+ready means of recording the features of a country and the types
+of inhabitants, but by supplying a method of quick and accurate
+topographical surveying.</p>
+
+<p class="pt2 center sc">The Principles of Geography</p>
+
+<p>As regards the scope of geography, the order of the various
+departments and their inter-relation, there is little difference of
+opinion, and the principles of geography<a name="fa12b" id="fa12b" href="#ft12b"><span class="sp">12</span></a> are now generally accepted
+by modern geographers. The order in which the various subjects
+are treated in the following sketch is the natural succession from
+fundamental to dependent facts, which corresponds also to the
+evolution of the diversities of the earth&rsquo;s crust and of its inhabitants.</p>
+
+<p>The fundamental geographical conceptions are mathematical, the
+relations of space and form. The figure and dimensions of the
+earth are the first of these. They are ascertained by a
+combination of actual measurement of the highest
+<span class="sidenote">Mathematical geography.</span>
+precision on the surface and angular observations of the
+positions of the heavenly bodies. The science of geodesy
+is part of mathematical geography, of which the arts of surveying
+and cartography are applications. The motions of the earth
+as a planet must be taken into account, as they render possible
+the determination of position and direction by observations of the
+heavenly bodies. The diurnal rotation of the earth furnishes two
+fixed points or poles, the axis joining which is fixed or nearly so in its
+direction in space. The rotation of the earth thus fixes the directions
+of north and south and defines those of east and west. The angle
+which the earth&rsquo;s axis makes with the plane in which the planet
+revolves round the sun determines the varying seasonal distribution
+of solar radiation over the surface and the mathematical zones of
+climate. Another important consequence of rotation is the deviation
+produced in moving bodies relatively to the surface. In the form
+known as Ferrell&rsquo;s Law this runs: &ldquo;If a body moves in any direction
+on the earth&rsquo;s surface, there is a deflecting force which arises from
+the earth&rsquo;s rotation which tends to deflect it to the right in the
+northern hemisphere but to the left in the southern hemisphere.&rdquo;
+The deviation is of importance in the movement of air, of ocean
+currents, and to some extent of rivers.<a name="fa13b" id="fa13b" href="#ft13b"><span class="sp">13</span></a></p>
+
+<p>In popular usage the words &ldquo;physical geography&rdquo; have come
+to mean geography viewed from a particular standpoint rather
+than any special department of the subject. The popular
+meaning is better conveyed by the word physiography, a
+<span class="sidenote">Physical geography.</span>
+term which appears to have been introduced by Linnaeus,
+and was reinvented as a substitute for the cosmography of the middle
+ages by Professor Huxley. Although the term has since been limited
+by some writers to one particular part of the subject, it seems best
+to maintain the original and literal meaning. In the stricter sense,
+physical geography is that part of geography which involves the
+processes of contemporary change in the crust and the circulation
+of the fluid envelopes. It thus draws upon physics for the explanation
+of the phenomena with the space-relations of which it is specially
+concerned. Physical geography naturally falls into three divisions,
+dealing respectively with the surface of the lithosphere&mdash;geomorphology;
+the hydrosphere&mdash;oceanography; and the atmosphere&mdash;climatology.
+All these rest upon the facts of mathematical geography,
+and the three are so closely inter-related that they cannot
+be rigidly separated in any discussion.</p>
+
+<p>Geomorphology is the part of geography which deals with terrestrial
+relief, including the submarine as well as the subaėrial portions
+of the crust. The history of the origin of the various forms belongs
+to geology, and can be completely studied only by geological
+<span class="sidenote">Geomorphology.</span>
+methods. But the relief of the crust is not a finished piece of sculpture;
+the forms are for the most part transitional, owing
+their characteristic outlines to the process by which they
+are produced; therefore the geographer must, for strictly
+geographical purposes, take some account of the processes which are
+now in action modifying the forms of the crust. Opinion still differs
+as to the extent to which the geographer&rsquo;s work should overlap that
+of the geologist.</p>
+
+<p>The primary distinction of the forms of the crust is that between
+elevations and depressions. Granting that the geoid or mean
+surface of the ocean is a uniform spheroid, the distribution of land
+and water approximately indicates a division of the surface of the
+globe into two areas, one of elevation and one of depression. The
+increasing number of measurements of the height of land in all
+continents and islands, and the very detailed levellings in those
+countries which have been thoroughly surveyed, enable the average
+elevation of the land above sea-level to be fairly estimated, although
+many vast gaps in accurate knowledge remain, and the estimate
+is not an exact one. The only part of the sea-bed the configuration
+of which is at all well known is the zone bordering the coasts where
+the depth is less than about 100 fathoms or 200 metres, <i>i.e.</i> those
+parts which sailors speak of as &ldquo;in soundings.&rdquo; Actual or projected
+routes for telegraph cables across the deep sea have also been sounded
+with extreme accuracy in many cases; but beyond these lines of
+sounding the vast spaces of the ocean remain unplumbed save for
+the rare researches of scientific expeditions, such as those of the
+&ldquo;Challenger,&rdquo; the &ldquo;Valdivia,&rdquo; the &ldquo;Albatross&rdquo; and the &ldquo;Scotia.&rdquo;
+Thus the best approximation to the average depth of the ocean is
+little more than an expert guess; yet a fair approximation is probable
+for the features of sub-oceanic relief are so much more uniform than
+those of the land that a smaller number of fixed points is required
+to determine them.</p>
+
+<p>The chief element of uncertainty as to the largest features of the
+relief of the earth&rsquo;s crust is due to the unexplored area in the Arctic
+region and the larger regions of the Antarctic, of which
+we know nothing. We know that the earth&rsquo;s surface if
+<span class="sidenote">Crustal relief.</span>
+unveiled of water would exhibit a great region of elevation
+arranged with a certain rough radiate symmetry round the north
+pole, and extending southwards in three unequal arms which taper
+to points in the south. A depression surrounds the little-known
+south polar region in a continuous ring and extends northwards in
+three vast hollows lying between the arms of the elevated area. So
+far only is it possible to speak with certainty, but it is permissible
+to take a few steps into the twilight of dawning knowledge and
+indicate the chief subdivisions which are likely to be established
+in the great crust-hollow and the great crust-heap. The boundary
+between these should obviously be the mean surface of the
+sphere.</p>
+
+<p>Sir John Murray deduced the mean height of the land of the globe
+as about 2250 ft. above sea-level, and the mean depth of the oceans
+as 2080 fathoms or 12,480 ft. below sea-level.<a name="fa14b" id="fa14b" href="#ft14b"><span class="sp">14</span></a> Calculating the area
+of the land at 55,000,000 sq. m. (or 28.6% of the surface), and that
+of the oceans as 137,200,000 sq. m. (or 71.4% of the surface), he
+found that the volume of the land above sea-level was 23,450,000
+cub. m., the volume of water below sea-level 323,800,000, and the
+total volume of the water equal to about <span class="spp">1</span>&frasl;<span class="suu">666</span>th of the volume of the
+whole globe. From these data, as revised by A. Supan,<a name="fa15b" id="fa15b" href="#ft15b"><span class="sp">15</span></a> H.R. Mill
+calculated the position of mean sphere-level at about 10,000 ft. or
+1700 fathoms below sea-level. He showed that an imaginary
+spheroidal shell, concentric with the earth and cutting the slope
+between the elevated and depressed areas at the contour-line of 1700
+fathoms, would not only leave above it a volume of the crust equal
+to the volume of the hollow left below it, but would also divide the
+surface of the earth so that the area of the elevated region was
+equal to that of the depressed region.<a name="fa16b" id="fa16b" href="#ft16b"><span class="sp">16</span></a></p>
+
+<p>A similar observation was made almost simultaneously by
+Romieux,<a name="fa17b" id="fa17b" href="#ft17b"><span class="sp">17</span></a> who further speculated on the equilibrium between the
+weight of the elevated land mass and that of the total
+waters of the ocean, and deduced some interesting relations
+<span class="sidenote">Areas of the crust according to Murray.</span>
+between them. Murray, as the result of his study,
+divided the earth&rsquo;s surface into three zones&mdash;the <i>continental
+area</i> containing all dry land, the <i>transitional area</i> including
+the submarine slopes down to 1000 fathoms, and the <i>abysmal area</i>
+consisting of the floor of the ocean beyond that depth; and Mill
+proposed to take the line of mean-sphere level, instead of the empirical
+depth of 1000 fathoms, as the boundary between the transitional
+and abysmal areas.</p>
+
+<p>An elaborate criticism of all the existing data regarding the
+volume relations of the vertical relief of the globe was made in
+1894 by Professor Hermann Wagner, whose recalculations of volumes
+<span class="pagenum"><a name="page631" id="page631"></a>631</span>
+and mean heights&mdash;the best results which have yet been obtained&mdash;led
+to the following conclusions.<a name="fa18b" id="fa18b" href="#ft18b"><span class="sp">18</span></a></p>
+
+<p>The area of the dry land was taken as 28.3% of the surface of the
+globe, and that of the oceans as 71.7%. The mean height deduced
+for the land was 2300 ft. above sea-level, the mean depth
+of the sea 11,500 ft. below, while the position of mean-sphere
+<span class="sidenote">Areas of the crust according to Wagner.</span>
+level comes out as 7500 ft. (1250 fathoms) below
+sea-level. From this it would appear that 43% of the
+earth&rsquo;s surface was above and 57% below the mean
+level. It must be noted, however, that since 1895 the soundings
+of Nansen in the north polar area, of the &ldquo;Valdivia,&rdquo; &ldquo;Belgica,&rdquo;
+&ldquo;Gauss&rdquo; and &ldquo;Scotia&rdquo; in the Southern Ocean, and of various
+surveying ships in the North and South Pacific, have proved that
+the mean depth of the ocean is considerably greater than had been
+supposed, and mean-sphere level must therefore lie deeper than the
+calculations of 1895 show; possibly not far from the position deduced
+from the freer estimate of 1888. The whole of the available data
+were utilized by the prince of Monaco in 1905 in the preparation of a
+complete bathymetrical map of the oceans on a uniform scale,
+which must long remain the standard work for reference on ocean
+depths.</p>
+
+<p>By the device of a hypsographic curve co-ordinating the vertical
+relief and the areas of the earth&rsquo;s surface occupied by each zone of
+elevation, according to the system introduced by Supan,<a name="fa19b" id="fa19b" href="#ft19b"><span class="sp">19</span></a> Wagner
+showed his results graphically.</p>
+
+<p>This curve with the values reduced from metres to feet is reproduced
+below.</p>
+
+<p>Wagner subdivides the earth&rsquo;s surface, according to elevation,
+into the following five regions:</p>
+
+<p class="pt2 center"><i>Wagner&rsquo;s Divisions of the Earth&rsquo;s Crust:</i></p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tccm allb">Name.</td> <td class="tccm allb">Per cent of<br />Surface.</td> <td class="tccm allb">From</td> <td class="tccm allb">To</td></tr>
+
+<tr><td class="tcl lb rb">Depressed area</td> <td class="tcr rb">3</td> <td class="tcc rb">Deepest.</td> <td class="tcr rb">&minus;16,400 feet.</td></tr>
+<tr><td class="tcl lb rb">Oceanic plateau</td> <td class="tcr rb">54</td> <td class="tcr rb">&minus;16,400 feet.</td> <td class="tcr rb">&minus; 7,400 feet.</td></tr>
+<tr><td class="tcl lb rb">Continental slope</td> <td class="tcr rb">9</td> <td class="tcr rb">&minus; 7,400 feet.</td> <td class="tcr rb">&minus; &ensp; 660 feet.</td></tr>
+<tr><td class="tcl lb rb">Continental plateau</td> <td class="tcr rb">28</td> <td class="tcr rb">&minus; &ensp; 660 feet.</td> <td class="tcr rb">+ 3,000 feet.</td></tr>
+<tr><td class="tcl lb rb bb">Culminating area</td> <td class="tcr rb bb">6</td> <td class="tcr rb bb">+ 3,300 feet.</td> <td class="tcc rb bb">Highest.</td></tr>
+</table>
+
+<p>The continental plateau might for purposes of detailed study be
+divided into the <i>continental shelf</i> from -660 ft. to sea-level, and
+<i>lowlands</i> from sea-level to +660 ft. (corresponding to
+the mean level of the whole globe).<a name="fa20b" id="fa20b" href="#ft20b"><span class="sp">20</span></a> <i>Uplands</i> reaching
+from 660 ft. to 2300 (the approximate mean level of
+the land), and <i>highlands</i>, from 2300 upwards, might
+also be distinguished.</p>
+
+<div class="center ptb2"><img style="width:600px; height:364px; vertical-align: middle;" src="images/img631.jpg" alt="" /></div>
+
+<p>A striking fact in the configuration of the crust is
+that each continent, or elevated mass of the crust, is
+diametrically opposite to an ocean basin or great depression;
+the only partial exception being in the case of southern
+<span class="sidenote">Arrangement of world-ridges and hollows.</span>
+South America, which is antipodal to eastern Asia.
+Professor C. Lapworth has generalized the grand features
+of crustal relief in a scheme of attractive simplicity. He
+sees throughout all the chaos of irregular crust-forms the
+recurrence of a certain harmony, a succession of folds or
+waves which build up all the minor features.<a name="fa21b" id="fa21b" href="#ft21b"><span class="sp">21</span></a> One
+great series of crust waves from east to west is crossed by a
+second great series of crust waves from north to south, giving rise
+by their interference to six great elevated masses (the continents),
+arranged in three groups, each consisting of a northern and a
+southern member separated by a minor depression. These elevated
+masses are divided from one another by similar great depressions.</p>
+
+<p>He says: &ldquo;The surface of each of our great continental masses of
+land resembles that of a long and broad arch-like form, of
+which we see the simplest type in the New World. The
+surface of the North American arch is sagged downwards
+<span class="sidenote">Lapworth&rsquo;s fold-theory.</span>
+in the middle into a central depression which
+lies between two long marginal plateaus, and these
+plateaus are finally crowned by the wrinkled crests which form its
+two modern mountain systems. The surface of each of our ocean
+floors exactly resembles that of a continent turned upside down.
+Taking the Atlantic as our simplest type, we may say that the
+surface of an ocean basin resembles that of a mighty trough or
+syncline, buckled up more or less centrally in a medial ridge, which
+is bounded by two long and deep marginal hollows, in the cores
+of which still deeper grooves sink to the profoundest depths. This
+complementary relationship descends even to the minor features
+of the two. Where the great continental sag sinks below the ocean
+level, we have our gulfs and our Mediterraneans, seen in our type
+continent, as the Mexican Gulf and Hudson Bay. Where the
+central oceanic buckle attains the water-line we have our oceanic
+islands, seen in our type ocean, as St Helena and the Azores. Although
+the apparent crust-waves are neither equal in size nor
+symmetrical in form, this complementary relationship between
+them is always discernible. The broad Pacific depression seems to
+answer to the broad elevation of the Old World&mdash;the narrow trough
+of the Atlantic to the narrow continent of America.&rdquo;</p>
+
+<p>The most thorough discussion of the great features of terrestrial
+relief in the light of their origin is that by Professor E. Suess,<a name="fa22b" id="fa22b" href="#ft22b"><span class="sp">22</span></a> who
+points out that the plan of the earth is the result of
+two movements of the crust&mdash;one, subsidence over
+<span class="sidenote">Suess&rsquo;s theory.</span>
+wide areas, giving rise to oceanic depressions and leaving
+the continents protuberant; the other, folding along comparatively
+narrow belts, giving rise to mountain ranges. This theory of crust
+blocks dropped by subsidence is opposed to Lapworth&rsquo;s theory of
+vast crust-folds, but geology is the science which has to decide
+between them.</p>
+
+<p>Geomorphology is concerned, however, in the suggestions which
+have been made as to the cause of the distribution of heap and
+hollow in the larger features of the crust. Élie de Beaumont, in
+his speculations on the relation between the direction of mountain
+ranges and their geological age and character, was feeling towards a
+comprehensive theory of the forms of crustal relief; but his ideas
+were too geometrical, and his theory that the earth is a spheroid
+built up on a rhombic dodecahedron, the pentagonal faces of which
+determined the direction of mountain ranges, could not be proved.<a name="fa23b" id="fa23b" href="#ft23b"><span class="sp">23</span></a>
+The &ldquo;tetrahedral theory&rdquo; brought forward by Lowthian Green,<a name="fa24b" id="fa24b" href="#ft24b"><span class="sp">24</span></a>
+that the form of the earth is a spheroid based on a regular tetrahedron,
+is more serviceable, because it accounts for three very
+interesting facts of the terrestrial plan&mdash;(1) the antipodal
+position of continents and ocean basins; (2) the triangular
+outline of the continents; and (3) the excess of
+sea in the southern hemisphere. Recent investigations
+have recalled attention to the work of Lowthian Green,
+but the question is still in the controversial stage.<a name="fa25b" id="fa25b" href="#ft25b"><span class="sp">25</span></a> The
+study of tidal strain in the earth&rsquo;s crust by Sir George
+Darwin has led that physicist to indicate the possibility
+of the triangular form and southerly direction of the
+continents being a result of the differential or tidal
+attraction of the sun and moon. More recently Professor
+A.E.H. Love has shown that the great features of the
+relief of the lithosphere may be expressed by spherical
+harmonics of the first, second and third degrees, and their
+formation related to gravitational action in a sphere of
+unequal density.<a name="fa26b" id="fa26b" href="#ft26b"><span class="sp">26</span></a></p>
+
+<p>In any case it is fully recognized that the plan of the earth is so
+clear as to leave no doubt as to its being due to some general cause
+which should be capable of detection.</p>
+
+<p>If the level of the sea were to become coincident with the mean
+level of the lithosphere, there would result one tri-radiate land-mass
+of nearly uniform outline and one continuous sheet of water
+<span class="pagenum"><a name="page632" id="page632"></a>632</span>
+broken by few islands. The actual position of sea-level lies so near
+<span class="sidenote">The continents.</span>
+the summit of the crust-heap that the varied relief of the upper
+portion leads to the formation of a complicated coast-line
+and a great number of detached portions of land.
+The hydrosphere is, in fact, continuous, and the land is
+all in insular masses: the largest is the Old World of Europe,
+Asia and Africa; the next in size, America; the third, possibly,
+Antarctica; the fourth, Australia; the fifth, Greenland. After
+this there is a considerable gap before New Guinea, Borneo, Madagascar,
+Sumatra and the vast multitude of smaller islands descending
+in size by regular gradations to mere rocks. The contrast between
+island and mainland was natural enough in the days before the
+discovery of Australia, and the mainland of the Old World was
+traditionally divided into three continents. These &ldquo;continents,&rdquo;
+&ldquo;parts of the earth,&rdquo; or &ldquo;quarters of the globe,&rdquo; proved to be
+convenient divisions; America was added as a fourth, and subsequently
+divided into two, while Australia on its discovery was classed
+sometimes as a new continent, sometimes merely as an island, sometimes
+compromisingly as an island-continent, according to individual
+opinion. The discovery of the insularity of Greenland might again
+give rise to the argument as to the distinction between island and
+continent. Although the name of continent was not applied to
+large portions of land for any physical reasons, it so happens that
+there is a certain physical similarity or homology between them
+which is not shared by the smaller islands or peninsulas.</p>
+
+<p>The typical continental form is triangular as regards its sea-level
+outline. The relief of the surface typically includes a central plain,
+sometimes dipping below sea-level, bounded by lateral
+highlands or mountain ranges, loftier on one side than
+<span class="sidenote">Homology of continents.</span>
+on the other, the higher enclosing a plateau shut in by
+mountains. South America and North America follow
+this type most closely; Eurasia (the land mass of Europe and Asia)
+comes next, while Africa and Australia are farther removed from
+the type, and the structure of Antarctica and Greenland is unknown.</p>
+
+<p>If the continuous, unbroken, horizontal extent of land in a continent
+is termed its <i>trunk</i>,<a name="fa27b" id="fa27b" href="#ft27b"><span class="sp">27</span></a> and the portions cut up by inlets or
+channels of the sea into islands and peninsulas the <i>limbs</i>, it is possible
+to compare the continents in an instructive manner.</p>
+
+<p>The following table is from the statistics of Professor H. Wagner,<a name="fa28b" id="fa28b" href="#ft28b"><span class="sp">28</span></a>
+his metric measurements being transposed into British units:</p>
+
+<p class="pt2 center"><i>Comparison of the Continents.</i></p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tccm allb">&nbsp;</td> <td class="tccm allb">Area<br />total<br />mil.<br />sq. m.</td> <td class="tccm allb">Mean<br />height,<br />feet.</td> <td class="tccm allb">Area<br />trunk,<br />mil.<br />sq. m.</td> <td class="tccm allb">Area<br />penin-<br />sulas,<br />mil.<br />sq. m.</td> <td class="tccm allb">Area<br />islands,<br />mil.<br />sq. m.</td> <td class="tccm allb">Area<br />limbs,<br />mil.<br />sq. m.</td> <td class="tccm allb">Area<br />limbs,<br />per<br />cent.</td></tr>
+
+<tr><td class="tcl lb rb">Old World</td> <td class="tcr rb">35.8&ensp;</td> <td class="tcr rb">2360</td> <td class="tcr rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">New World</td> <td class="tcr rb">16.2&ensp;</td> <td class="tcr rb">2230</td> <td class="tcr rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Eurasia</td> <td class="tcr rb">20.85</td> <td class="tcr rb">2620</td> <td class="tcr rb">15.42</td> <td class="tcc rb">4.09</td> <td class="tcc rb">1.34</td> <td class="tcc rb">5.43</td> <td class="tcr rb">26&ensp;</td></tr>
+<tr><td class="tcl lb rb">Africa</td> <td class="tcr rb">11.46</td> <td class="tcr rb">2130</td> <td class="tcr rb">11.22</td> <td class="tcc rb">..</td> <td class="tcc rb">0.24</td> <td class="tcc rb">0.24</td> <td class="tcr rb">2.1</td></tr>
+<tr><td class="tcl lb rb">North America</td> <td class="tcr rb">9.26</td> <td class="tcr rb">2300</td> <td class="tcr rb">6.92</td> <td class="tcc rb">0.78</td> <td class="tcc rb">1.56</td> <td class="tcc rb">2.34</td> <td class="tcr rb">25&ensp;</td></tr>
+<tr><td class="tcl lb rb">South America</td> <td class="tcr rb">6.84</td> <td class="tcr rb">1970</td> <td class="tcr rb">6.76</td> <td class="tcc rb">0.02</td> <td class="tcc rb">0.06</td> <td class="tcc rb">0.08</td> <td class="tcr rb">1.1</td></tr>
+<tr><td class="tcl lb rb">Australia</td> <td class="tcr rb">3.43</td> <td class="tcr rb">1310</td> <td class="tcr rb">2.77</td> <td class="tcc rb">0.16</td> <td class="tcc rb">0.50</td> <td class="tcc rb">0.66</td> <td class="tcr rb">19&ensp;</td></tr>
+<tr><td class="tcl lb rb">Asia</td> <td class="tcr rb">17.02</td> <td class="tcr rb">3120</td> <td class="tcr rb">12.93</td> <td class="tcc rb">3.05</td> <td class="tcc rb">1.04</td> <td class="tcc rb">4.09</td> <td class="tcr rb">24&ensp;</td></tr>
+<tr><td class="tcl lb rb bb">Europe</td> <td class="tcr rb bb">3.83</td> <td class="tcr rb bb">980</td> <td class="tcr rb bb">2.49</td> <td class="tcc rb bb">1.04</td> <td class="tcc rb bb">0.30</td> <td class="tcc rb bb">1.34</td> <td class="tcr rb bb">35&ensp;</td></tr>
+</table>
+
+<p>The usual classification of islands is into continental and oceanic.
+The former class includes all those which rise from the continental
+shelf, or show evidence in the character of their rocks of
+having at one time been continuous with a neighbouring
+<span class="sidenote">Islands.</span>
+continent. The latter rise abruptly from the oceanic abysses.
+Oceanic islands are divided according to their geological character
+into volcanic islands and those of organic origin, including coral
+islands. More elaborate subdivisions according to structure, origin and
+position have been proposed.<a name="fa29b" id="fa29b" href="#ft29b"><span class="sp">29</span></a> In some cases a piece of land is only
+an island at high water, and by imperceptible gradation the form
+passes into a peninsula. The typical peninsula is connected with the
+mainland by a relatively narrow isthmus; the name is, however, extended
+to any limb projecting from the trunk of the mainland, even
+when, as in the Indian peninsula, it is connected by its widest part.</p>
+
+<p>Small peninsulas are known as promontories or headlands, and
+the extremity as a cape. The opposite form, an inlet of the sea, is
+known when wide as a gulf, bay or bight, according
+to size and degree of inflection, or as a fjord or ria when
+<span class="sidenote">Coasts.</span>
+long and narrow. It is convenient to employ a specific name for a
+projection of a coast-line less pronounced than a peninsula, and for
+an inlet less pronounced than a bay or bight; outcurve and incurve
+may serve the turn. The varieties of coast-lines were reduced to an
+exact classification by Richthofen, who grouped them according to
+the height and slope of the land into cliff-coasts (<i>Steilküsten</i>)&mdash;narrow
+beach coasts with cliffs, wide beach coasts with cliffs, and
+low coasts, subdividing each group according as the coast-line runs
+parallel to or crosses the line of strike of the mountains, or is not
+related to mountain structure. A further subdivision depends on
+the character of the inter-relation of land and sea along the shore
+producing such types as a fjord-coast, ria-coast or lagoon-coast.
+This extremely elaborate subdivision may be reduced, as Wagner
+points out, to three types&mdash;the continental coast where the sea comes
+up to the solid rock-material of the land; the marine coast, which is
+formed entirely of soft material sorted out by the sea; and the composite
+coast, in which both forms are combined.</p>
+
+<p>On large-scale maps it is necessary to show two coast-lines, one
+for the highest, the other for the lowest tide; but in small-scale
+maps a single line is usually wider than is required to
+represent the whole breadth of the inter-tidal zone.
+<span class="sidenote">Coast-lines.</span>
+The measurement of a coast-line is difficult, because
+the length will necessarily be greater when measured on a large-scale
+map where minute irregularities can be taken into account.
+It is usual to distinguish between the general coast-line measured
+from point to point of the headlands disregarding the smaller bays,
+and the detailed coast-line which takes account of every inflection
+shown by the map employed, and follows up river entrances to the
+point where tidal action ceases. The ratio between these two
+coast-lines represents the &ldquo;coastal development&rdquo; of any region.</p>
+
+<p>While the forms of the sea-bed are not yet sufficiently well known
+to admit of exact classification, they are recognized to be as a rule
+distinct from the forms of the land, and the importance
+of using a distinctive terminology is felt. Efforts have
+<span class="sidenote">Submarine forms.</span>
+been made to arrive at a definite international agreement
+on this subject, and certain terms suggested by a committee were
+adopted by the Eighth International Geographical Congress at New
+York in 1904.<a name="fa30b" id="fa30b" href="#ft30b"><span class="sp">30</span></a> The forms of the ocean floor include the &ldquo;shelf,&rdquo;
+or shallow sea margin, the &ldquo;depression,&rdquo; a general term applied to
+all submarine hollows, and the &ldquo;elevation.&rdquo; A depression when of
+great extent is termed a &ldquo;basin,&rdquo; when it is of a more or less round
+form with approximately equal diameters, a &ldquo;trough&rdquo; when it is
+wide and elongated with gently sloping borders, and a &ldquo;trench&rdquo;
+when narrow and elongated with steeply sloping borders, one of
+which rises higher than the other. The extension of a trough or
+basin penetrating the land or an elevation is termed an &ldquo;embayment&rdquo;
+when wide, and a &ldquo;gully&rdquo; when long and narrow; and the
+deepest part of a depression is termed a &ldquo;deep.&rdquo;
+A depression of small extent when steep-sided is
+termed a &ldquo;caldron,&rdquo; and a long narrow depression
+crossing a part of the continental border is termed
+a &ldquo;furrow.&rdquo; An elevation of great extent which
+rises at a very gentle angle from a surrounding
+depression is termed a &ldquo;rise,&rdquo; one which is relatively
+narrow and steep-sided a &ldquo;ridge,&rdquo; and one
+which is approximately equal in length and breadth
+but steep-sided a &ldquo;plateau,&rdquo; whether it springs
+direct from a depression or from a rise. An elevation
+of small extent is distinguished as a &ldquo;dome&rdquo;
+when it is more than 100 fathoms from the surface,
+a &ldquo;bank&rdquo; when it is nearer the surface than
+100 fathoms but deeper than 6 fathoms, and a
+&ldquo;shoal&rdquo; when it comes within 6 fathoms of the
+surface and so becomes a serious danger to shipping.
+The highest point of an elevation is termed
+a &ldquo;height,&rdquo; if it does not form an island or one
+of the minor forms.</p>
+
+<p>The forms of the dry land are of infinite variety, and have been
+studied in great detail.<a name="fa31b" id="fa31b" href="#ft31b"><span class="sp">31</span></a> From the descriptive or topographical
+point of view, geometrical form alone should be considered;
+<span class="sidenote">Land forms.</span>
+but the origin and geological structure of
+land forms must in many cases be taken into account
+when dealing with the function they exercise in the control of
+mobile distributions. The geographers who have hitherto given
+most attention to the forms of the land have been trained as geologists,
+and consequently there is a general tendency to make origin
+or structure the basis of classification rather than form alone.</p>
+
+<p>The fundamental form-elements may be reduced to the six
+proposed by Professor Penck as the basis of his double system of
+classification by form and origin.<a name="fa32b" id="fa32b" href="#ft32b"><span class="sp">32</span></a> These may be looked
+<span class="sidenote">The six elementary land forms.</span>
+upon as being all derived by various modifications or
+arrangements of the single form-unit, the <i>slope</i> or inclined
+plane surface. No one form occurs alone, but always
+grouped together with others in various ways to make up districts,
+regions and lands of distinctive characters. The form-elements are:</p>
+
+<p><span class="pagenum"><a name="page633" id="page633"></a>633</span></p>
+
+<p>1. The <i>plain</i> or gently inclined uniform surface.</p>
+
+<p>2. The <i>scarp</i> or steeply inclined slope; this is necessarily of
+small extent except in the direction of its length.</p>
+
+<p>3. The <i>valley</i>, composed of two lateral parallel slopes inclined
+towards a narrow strip of plain at a lower level which itself slopes
+downwards in the direction of its length. Many varieties of this
+fundamental form may be distinguished.</p>
+
+<p>4. The <i>mount</i>, composed of a surface falling away on every side
+from a particular place. This place may either be a point, as
+in a volcanic cone, or a line, as in a mountain range or ridge of
+hills.</p>
+
+<p>5. The <i>hollow</i> or form produced by a land surface sloping inwards
+from all sides to a particular lowest place, the converse of a mount.</p>
+
+<p>6. The <i>cavern</i> or space entirely surrounded by a land surface.</p>
+
+<p>These forms never occur scattered haphazard over a region,
+but always in an orderly subordination depending on their mode
+of origin. The dominant forms result from crustal
+movements, the subsidiary from secondary reactions
+<span class="sidenote">Geology and land forms.</span>
+during the action of the primitive forms on mobile distributions.
+The geological structure and the mineral composition
+of the rocks are often the chief causes determining the
+character of the land forms of a region. Thus the scenery of a limestone
+country depends on the solubility and permeability of the
+rocks, leading to the typical Karst-formations of caverns, swallow-holes
+and underground stream courses, with the contingent phenomena
+of dry valleys and natural bridges. A sandy beach or desert
+owes its character to the mobility of its constituent sand-grains,
+which are readily drifted and piled up in the form of dunes. A
+region where volcanic activity has led to the embedding of dykes or
+bosses of hard rock amongst softer strata produces a plain broken by
+abrupt and isolated eminences.<a name="fa33b" id="fa33b" href="#ft33b"><span class="sp">33</span></a></p>
+
+<p>It would be impracticable to go fully into the varieties of each
+specific form; but, partly as an example of modern geographical
+classification, partly because of the exceptional importance
+of mountains amongst the features of the land, one
+<span class="sidenote">Classification of mountains.</span>
+exception may be made. The classification of mountains
+into types has usually had regard rather to geological
+structure than to external form, so that some geologists would even
+apply the name of a mountain range to a region not distinguished
+by relief from the rest of the country if it bear geological evidence
+of having once been a true range. A mountain may be described
+(it cannot be defined) as an elevated region of irregular surface
+rising comparatively abruptly from lower ground. The actual
+elevation of a summit above sea-level does not necessarily affect its
+mountainous character; a gentle eminence, for instance, rising a
+few hundred feet above a tableland, even if at an elevation of say
+15,000 ft., could only be called a hill.<a name="fa34b" id="fa34b" href="#ft34b"><span class="sp">34</span></a> But it may be said that
+any abrupt slope of 2000 ft. or more in vertical height may justly
+be called a mountain, while abrupt slopes of lesser height may
+be called hills. Existing classifications, however, do not take
+account of any difference in kind between mountain and hills,
+although it is common in the German language to speak of <i>Hügelland</i>,
+<i>Mittelgebirge</i> and <i>Hochgebirge</i> with a definite significance.</p>
+
+<p>The simple classification employed by Professor James Geikie<a name="fa35b" id="fa35b" href="#ft35b"><span class="sp">35</span></a>
+into mountains of accumulation, mountains of elevation and mountains
+of circumdenudation, is not considered sufficiently thorough
+by German geographers, who, following Richthofen, generally
+adopt a classification dependent on six primary divisions, each of
+which is subdivided. The terms employed, especially for the subdivisions,
+cannot be easily translated into other languages, and the
+English equivalents in the following table are only put forward
+tentatively:&mdash;</p>
+
+<p class="pt2 center sc">Richthofen&rsquo;s Classification of Mountains<a name="fa36b" id="fa36b" href="#ft36b"><span class="sp">36</span></a></p>
+
+<table class="reg" summary="poem"><tr><td> <div class="poemr">
+<p>I. <i>Tektonische Gebirge</i>&mdash;Tectonic mountains.</p>
+    <p class="i2">(<i>a</i>) <i>Bruchgebirge oder Schollengebirge</i>&mdash;Block mountains.</p>
+        <p class="i3">1. <i>Einseitige Schollengebirge oder Schollenrandgebirge</i>&mdash;Scarp or tilted block mountains.</p>
+              <p class="i5">(i.) <i>Tafelscholle</i>&mdash;Table blocks.</p>
+              <p class="i5">(ii.) <i>Abrasionsscholle</i>&mdash;Abraded blocks.</p>
+              <p class="i5">(iii.) <i>Transgressionsscholle</i>&mdash;Blocks of unconformable strata.</p>
+        <p class="i3">2. <i>Flexurgebirge</i>&mdash;Flexure mountains.</p>
+        <p class="i3">3. <i>Horstgebirge</i>&mdash;Symmetrical block mountains.</p>
+    <p class="i2">(<i>b</i>) <i>Faltungsgebirge</i>&mdash;Fold mountains.</p>
+        <p class="i3">1. <i>Homöomorphe Faltungsgebirge</i>&mdash;Homomorphic fold mountains.</p>
+        <p class="i3">2. <i>Heteromorphe Faltungsgebirge</i>&mdash;Heteromorphic fold mountains.</p>
+
+<p class="s">II. <i>Rumpfgebirge oder Abrasionsgebirge</i>&mdash;Trunk or abraded mountains.</p>
+<p class="s">III. <i>Ausbruchsgebirge</i>&mdash;Eruptive mountains.</p>
+<p class="s">IV. <i>Aufschüttungsgebirge</i>&mdash;Mountains of accumulation.</p>
+<p class="s">V. <i>Flachböden</i>&mdash;Plateaux.</p>
+      <p class="i2">(<i>a</i>) <i>Abrasionsplatten</i>&mdash;Abraded plateaux.</p>
+      <p class="i2">(<i>b</i>) <i>Marines Flachland</i>&mdash;Plain of marine erosion.</p>
+      <p class="i2">(<i>c</i>) <i>Schichtungstafelland</i>&mdash;Horizontally stratified tableland.</p>
+      <p class="i2">(<i>d</i>) <i>Übergusstafelland</i>&mdash;Lava plain.</p>
+      <p class="i2">(<i>e</i>) <i>Stromflachland</i>&mdash;River plain.</p>
+      <p class="i2">(<i>f</i>) <i>Flachböden der atmosphärischen Aufschüttung</i>&mdash;Plains of aeolian formation.</p>
+<p class="s">VI. <i>Erosionsgebirge</i>&mdash;Mountains of erosion.</p>
+</div> </td></tr></table>
+
+<p>From the morphological point of view it is more important to
+distinguish the associations of forms, such as the <i>mountain mass</i>
+or group of mountains radiating from a centre, with the
+valleys furrowing their flanks spreading towards every
+<span class="sidenote">Mountain forms.</span>
+direction; the <i>mountain chain</i> or line of heights, forming a
+long narrow ridge or series of ridges separated by parallel valleys;
+the <i>dissected plateau</i> or highland, divided into mountains of circumdenudation
+by a system of deeply-cut valleys; and the <i>isolated
+peak</i>, usually a volcanic cone or a hard rock mass left projecting after
+the softer strata which embedded it have been worn away (Monadnock
+of Professor Davis).</p>
+
+<p>The geographical distribution of mountains is intimately associated
+with the great structural lines of the continents of which they form
+the culminating region. Lofty lines of fold mountains
+form the &ldquo;backbones&rdquo; of North America in the Rocky
+<span class="sidenote">Distribution of mountains.</span>
+Mountains and the west coast systems, of South America
+in the Cordillera of the Andes, of Europe in the Pyrenees,
+Alps, Carpathians and Caucasus, and of Asia in the mountains of
+Asia Minor, converging on the Pamirs and diverging thence in the
+Himalaya and the vast mountain systems of central and eastern
+Asia. The remarkable line of volcanoes around the whole coast
+of the Pacific and along the margin of the Caribbean and Mediterranean
+seas is one of the most conspicuous features of the globe.</p>
+
+<p>If land forms may be compared to organs, the part they serve in
+the economy of the earth may, without straining the term, be
+characterized as functions. The first and simplest
+<span class="sidenote">Functions of land forms.<br /><br />Land waste.</span>
+function of the land surface is that of guiding loose
+material to a lower level. The downward pull of gravity
+suffices to bring about the fall of such material, but the
+path it will follow and the distance it will travel before coming to
+rest depend upon the land form. The loose material may, and in
+an arid region does, consist only of portions of the higher
+parts of the surface detached by the expansion and
+contraction produced by heating and cooling due to
+radiation. Such broken material rolling down a uniform scarp
+would tend to reduce its steepness by the loss of material in the
+upper part and by the accumulation of a mound or scree against
+the lower part of the slope. But where the side is not a uniform
+scarp, but made up of a series of ridges and valleys, the tendency
+will be to distribute the detritus in an irregular manner, directing
+it away from one place and collecting it in great masses in another,
+so that in time the land form assumes a new appearance. Snow
+accumulating on the higher portions of the land, when compacted
+into ice and caused to flow downwards by gravity, gives rise, on
+<span class="sidenote">Glaciers.</span>
+account of its more coherent character, to continuous
+glaciers, which mould themselves to the slopes down
+which they are guided, different ice-streams converging to send
+forward a greater volume. Gradually coming to occupy definite
+beds, which are deepened and polished by the friction, they impress
+a characteristic appearance on the land, which guides them as they
+traverse it, and, although the ice melts at lower levels, vast quantities
+of clay and broken stones are brought down and deposited in terminal
+moraines where the glacier ends.</p>
+
+<p>Rain is by far the most important of the inorganic mobile distributions
+upon which land forms exercise their function of guidance
+and control. The precipitation of rain from the aqueous
+<span class="sidenote">Rain.</span>
+vapour of the atmosphere is caused in part by vertical
+movements of the atmosphere involving heat changes and apparently
+independent of the surface upon which precipitation occurs; but in
+greater part it is dictated by the form and altitude of the land surface
+and the direction of the prevailing winds, which itself is largely
+influenced by the land. It is on the windward faces of the highest
+ground, or just beyond the summit of less dominant heights upon the
+leeward side, that most rain falls, and all that does not evaporate
+or percolate into the ground is conducted back to the sea by a route
+which depends only on the form of the land. More mobile and more
+searching than ice or rock rubbish, the trickling drops are guided by
+the deepest lines of the hillside in their incipient flow, and as these
+<span class="sidenote">River systems.</span>
+lines converge, the stream, gaining strength, proceeds in
+its torrential course to carve its channel deeper and entrench
+itself in permanent occupation. Thus the stream-bed,
+from which at first the water might be blown away into a new
+channel by a gale of wind, ultimately grows to be the strongest line
+of the landscape. As the main valley deepens, the tributary stream-beds
+are deepened also, and gradually cut their way headwards,
+enlarging the area whence they draw their supplies. Thus new
+land forms are created&mdash;valleys of curious complexity, for example&mdash;by
+<span class="pagenum"><a name="page634" id="page634"></a>634</span>
+the &ldquo;capture&rdquo; and diversion of the water of one river by another,
+leading to a change of watershed.<a name="fa37b" id="fa37b" href="#ft37b"><span class="sp">37</span></a> The minor tributaries become
+more numerous and more constant, until the system of torrents
+has impressed its own individuality on the mountain side. As
+the river leaves the mountain, ever growing by the accession of
+tributaries, it ceases, save in flood time, to be a formidable instrument
+of destruction; the gentler slope of the land surface gives to
+it only power sufficient to transport small stones, gravel, sand and
+ultimately mud. Its valley banks are cut back by the erosion of
+minor tributaries, or by rain-wash if the climate be moist, or left
+steep and sharp while the river deepens its bed if the climate be
+arid. The outline of the curve of a valley&rsquo;s sides ultimately depends
+on the angle of repose of the detritus which covers them, if there
+has been no subsequent change, such as the passage of a glacier
+along the valley, which tends to destroy the regularity of the cross-section.
+The slope of the river bed diminishes until the plain compels
+the river to move slowly, swinging in <i>meanders</i> proportioned to its
+size, and gradually, controlled by the flattening land, ceasing to
+transport material, but raising its banks and silting up its bed by
+the dropped sediment, until, split up and shoaled, its distributaries
+struggle across its delta to the sea. This is the typical river of which
+there are infinite varieties, yet every variety would, if time were
+given, and the land remained unchanged in level relatively to the sea,
+ultimately approach to the type. Movements of the land
+<span class="sidenote">Adjustment of rivers to land.</span>
+either of subsidence or elevation, changes in the land by
+the action of erosion in cutting back an escarpment or
+cutting through a col, changes in climate by affecting the
+rainfall and the volume of water, all tend to throw the
+river valley out of harmony with the actual condition of
+its stream. There is nothing more striking in geography than the
+perfection of the adjustment of a great river system to its valleys
+when the land has remained stable for a very lengthened period.
+Before full adjustment has been attained the river bed may be
+broken in places by waterfalls or interrupted by lakes; after adjustment
+the bed assumes a permanent outline, the slope diminishing
+more and more gradually, without a break in its symmetrical descent.
+Excellent examples of the indecisive drainage of a new land surface,
+on which the river system has not had time to impress itself, are to be
+seen in northern Canada and in Finland, where rivers are separated
+by scarcely perceptible divides, and the numerous lakes frequently
+belong to more than one river system.</p>
+
+<p>The action of rivers on the land is so important that it has been
+made the basis of a system of physical geography by Professor
+W.M. Davis, who classifies land surfaces in terms of
+the three factors&mdash;structure, process and time.<a name="fa38b" id="fa38b" href="#ft38b"><span class="sp">38</span></a> Of
+<span class="sidenote">The geographical cycle.</span>
+these time, during which the process is acting on the
+structure, is the most important. A land may thus be
+characterized by its position in the &ldquo;geographical cycle&rdquo;, or cycle
+of erosion, as young, mature or old, the last term being reached
+when the base-level of erosion is attained, and the land, however
+varied its relief may have been in youth or maturity, is reduced to
+a nearly uniform surface or peneplain. By a re-elevation of a
+peneplain the rivers of an old land surface may be restored to
+youthful activity, and resume their shaping action, deepening the
+old valleys and initiating new ones, starting afresh the whole course
+of the geographical cycle. It is, however, not the action of the
+running water on the land, but the function exercised by the land
+on the running water, that is considered here to be the special
+province of geography. At every stage of the geographical cycle
+the land forms, as they exist at that stage, are concerned in guiding
+the condensation and flow of water in certain definite ways. Thus,
+for example, in a mountain range at right angles to a prevailing
+sea-wind, it is the land forms which determine that one side of the
+range shall be richly watered and deeply dissected by a complete
+system of valleys, while the other side is dry, indefinite in its valley
+systems, and sends none of its scanty drainage to the sea. The
+action of rain, ice and rivers conspires with the movement of land
+waste to strip the layer of soil from steep slopes as rapidly as it
+forms, and to cause it to accumulate on the flat valley bottoms, on
+the graceful flattened cones of alluvial fans at the outlet of the gorges
+of tributaries, or in the smoothly-spread surface of alluvial plains.</p>
+
+<p>The whole question of the régime of rivers and lakes is sometimes
+treated under the name hydrography, a name used by some writers
+in the sense of marine surveying, and by others as synonymous with
+oceanography. For the study of rivers alone the name potamology<a name="fa39b" id="fa39b" href="#ft39b"><span class="sp">39</span></a>
+has been suggested by Penck, and the subject being of much practical
+importance has received a good deal of attention.<a name="fa40b" id="fa40b" href="#ft40b"><span class="sp">40</span></a></p>
+
+<p>The study of lakes has also been specialized under the name of
+limnology (see <span class="sc"><a href="#artlinks">Lake</a></span>).<a name="fa41b" id="fa41b" href="#ft41b"><span class="sp">41</span></a> The existence of lakes in hollows of the land
+depends upon the balance between precipitation and evaporation.
+A stream flowing into a hollow will tend to fill it up, and
+<span class="sidenote">Lakes and internal drainage.</span>
+the water will begin to escape as soon as its level rises high
+enough to reach the lowest part of the rim. In the case
+of a large hollow in a very dry climate the rate of
+evaporation may be sufficient to prevent the water from ever rising
+to the lip, so that there is no outflow to the sea, and a basin of internal
+drainage is the result. This is the case, for instance, in the Caspian
+sea, the Aral and Balkhash lakes, the Tarim basin, the Sahara, inner
+Australia, the great basin of the United States and the Titicaca
+basin. These basins of internal drainage are calculated to amount
+to 22% of the land surface. The percentages of the land surface
+draining to the different oceans are approximately&mdash;Atlantic, 34.3%;
+Arctic sea, 16.5%; Pacific, 14.4%; Indian Ocean, 12.8%.<a name="fa42b" id="fa42b" href="#ft42b"><span class="sp">42</span></a></p>
+
+<p>The parts of a river system have not been so clearly defined as is
+desirable, hence the exaggerated importance popularly attached to
+&ldquo;the source&rdquo; of a river. A well-developed river system
+has in fact many equally important and widely-separated
+<span class="sidenote">Terminology of river systems.</span>
+sources, the most distant from the mouth, the highest,
+or even that of largest initial volume not being necessarily
+of greater geographical interest than the rest.
+The whole of the land which directs drainage towards one river is
+known as its basin, catchment area or drainage area&mdash;sometimes,
+by an incorrect expression, as its valley or even its watershed.
+The boundary line between one drainage area and others is rightly
+termed the watershed, but on account of the ambiguity which has
+been tolerated it is better to call it water-parting or, as in America,
+divide. The only other important term which requires to be noted
+here is <i>talweg</i>, a word introduced from the German into French
+and English, and meaning the deepest line along the valley, which
+is necessarily occupied by a stream unless the valley is dry.</p>
+
+<p>The functions of land forms extend beyond the control of the
+circulation of the atmosphere, the hydrosphere and the water which
+is continually being interchanged between them; they are exercised
+with increased effect in the higher departments of biogeography and
+anthropogeography.</p>
+
+<p>The sum of the organic life on the globe is termed by some geographers
+the biosphere, and it has been estimated that the whole
+mass of living substance in existence at one time would
+cover the surface of the earth to a depth of one-fifth of
+<span class="sidenote">Biogeography.</span>
+an inch.<a name="fa43b" id="fa43b" href="#ft43b"><span class="sp">43</span></a> The distribution of living organisms is a
+complex problem, a function of many factors, several of which
+are yet but little known. They include the biological nature of
+the organism and its physical environment, the latter involving
+conditions in which geographical elements, direct or indirect, preponderate.
+The direct geographical elements are the arrangement
+of land and sea (continents and islands standing in sharp contrast)
+and the vertical relief of the globe, which interposes barriers of a
+less absolute kind between portions of the same land area or oceanic
+depression. The indirect geographical elements, which, as a rule,
+act with and intensify the direct, are mainly climatic; the prevailing
+winds, rainfall, mean and extreme temperatures of every
+locality depending on the arrangement of land and sea and of land
+forms. Climate thus guided affects the weathering of rocks, and
+so determines the kind and arrangement of soil. Different species
+of organisms come to perfection in different climates; and it may
+be stated as a general rule that a species, whether of plant or animal,
+once established at one point, would spread over the whole zone
+of the climate congenial to it unless some barrier were interposed
+to its progress. In the case of land and fresh-water organisms
+the sea is the chief barrier; in the case of marine organisms, the
+land. Differences in land forms do not exert great influence on the
+distribution of living creatures directly, but indirectly such land
+forms as mountain ranges and internal drainage basins are very
+potent through their action on soil and climate. A snow-capped
+mountain ridge or an arid desert forms a barrier between different
+forms of life which is often more effective than an equal breadth of
+sea. In this way the surface of the land is divided into numerous
+natural regions, the flora and fauna of each of which include some
+distinctive species not shared by the others. The distribution of
+life is discussed in the various articles in this <i>Encyclopaedia</i> dealing
+with biological, botanical and zoological subjects.<a name="fa44b" id="fa44b" href="#ft44b"><span class="sp">44</span></a></p>
+
+<p><span class="pagenum"><a name="page635" id="page635"></a>635</span></p>
+
+<p>The classification of the land surface into areas inhabited by
+distinctive groups of plants has been attempted by many phyto-geographers,
+but without resulting in any scheme of
+general acceptance. The simplest classification is perhaps
+<span class="sidenote">Floral zones.</span>
+that of Drude according to climatic zones, subdivided
+according to continents. This takes account of&mdash;(1) the <i>Arctic-Alpine</i>
+zone, including all the vegetation of the region bordering
+on perpetual snow; (2) the <i>Boreal</i> zone, including the temperate
+lands of North America, Europe and Asia, all of which are substantially
+alike in botanical character; (3) the <i>Tropical</i> zone, divided
+sharply into (<i>a</i>) the tropical zone of the New World, and (<i>b</i>) the
+tropical zone of the Old World, the forms of which differ in a significant
+degree; (4) the <i>Austral</i> zone, comprising all continental
+land south of the equator, and sharply divided into three regions
+the floras of which are strikingly distinct&mdash;(<i>a</i>) South American,
+(<i>b</i>) South African and (<i>c</i>) Australian; (5) the <i>Oceanic</i>, comprising
+all oceanic islands, the flora of which consists exclusively of forms
+whose seeds could be drifted undestroyed by ocean currents or
+carried by birds. To these might be added the antarctic, which is
+still very imperfectly known. Many subdivisions and transitional
+zones have been suggested by different authors.</p>
+
+<p>From the point of view of the economy of the globe this classification
+by species is perhaps less important than that by mode
+of life and physiological character in accordance with
+environment. The following are the chief areas of
+<span class="sidenote">Vegetation areas.</span>
+vegetational activity usually recognized: (1) The ice-deserts
+of the arctic and antarctic and the highest mountain regions,
+where there is no vegetation except the lowest forms, like that
+which causes &ldquo;red snow.&rdquo; (2) The tundra or region of intensely
+cold winters, forbidding tree-growth, where mosses and lichens
+cover most of the ground when unfrozen, and shrubs occur of
+species which in other conditions are trees, here stunted to the
+height of a few inches. A similar zone surrounds the permanent
+snow on lofty mountains in all latitudes. The tundra passes by
+imperceptible gradations into the moor, bog and heath of warmer
+climates. (3) The temperate forests of evergreen or deciduous trees,
+according to circumstances, which occupy those parts of both
+temperate zones where rainfall and sunlight are both abundant.
+(4) The grassy steppes or prairies where the rainfall is diminished
+and temperatures are extreme, and grass is the prevailing form of
+vegetation. These pass imperceptibly into&mdash;(5) the arid desert,
+where rainfall is at a minimum, and the only plants are those modified
+to subsist with the smallest supply of water. (6) The tropical forest,
+which represents the maximum of plant luxuriance, stimulated by
+the heaviest rainfall, greatest heat and strongest light. These
+divisions merge one into the other, and admit of almost indefinite
+subdivision, while they are subject to great modifications by human
+interference in clearing and cultivating. Plants exhibit the controlling
+power of environment to a high degree, and thus vegetation is
+usually in close adjustment to the bolder geographical features of
+a region.</p>
+
+<p>The divisions of the earth into faunal regions by Dr P.L. Sclater
+have been found to hold good for a large number of groups of animals
+as different in their mode of life as birds and mammals,
+and they may thus be accepted as based on nature.
+<span class="sidenote">Faunal realms.</span>
+They are six in number: (1) <i>Palaearctic</i>, including
+Europe, Asia north of the Himalaya, and Africa north of the Sahara;
+(2) <i>Ethiopian</i>, consisting of Africa south of the Atlas range, and
+Madagascar; (3) <i>Oriental</i>, including India, Indo-China and the
+Malay Archipelago north of Wallace&rsquo;s line, which runs between
+Bali and Lombok; (4) <i>Australian</i>, including Australia, New Zealand,
+New Guinea and Polynesia; (5) <i>Nearctic</i> or North America, north
+of Mexico; and (6) <i>Neotropical</i> or South America. Each of these
+divisions is the home of a special fauna, many species of which
+are confined to it alone; in the Australian region, indeed, practically
+the whole fauna is peculiar and distinctive, suggesting a prolonged
+period of complete biological isolation. In some cases, such as the
+Ethiopian and Neotropical and the Palaearctic and Nearctic regions,
+the faunas, although distinct, are related, several forms on opposite
+sides of the Atlantic being analogous, <i>e.g.</i> the lion and puma, ostrich
+and rhea. Where two of the faunal realms meet there is usually,
+though not always, a mixing of faunas. These facts have led some
+naturalists to include the Palaearctic and Nearctic regions in one,
+termed <i>Holarctic</i>, and to suggest transitional regions, such as the
+<i>Sonoran</i>, between North and South America, and the <i>Mediterranean</i>,
+between Europe and Africa, or to create sub-regions, such as Madagascar
+and New Zealand. Oceanic islands have, as a rule, distinctive
+faunas and floras which resemble, but are not identical with, those of
+other islands in similar positions.</p>
+
+<p>The study of the evolution of faunas and the comparison of the
+faunas of distant regions have furnished a trustworthy
+instrument of pre-historic geographical research, which
+<span class="sidenote">Biological distribution as a means of geographical research.</span>
+enables earlier geographical relations of land and sea to
+be traced out, and the approximate period, or at least the
+chronological order of the larger changes, to be estimated.
+In this way, for example, it has been suggested that a
+land, &ldquo;Lemuria,&rdquo; once connected Madagascar with the
+Malay Archipelago, and that a northern extension of
+the antarctic land once united the three southern continents.</p>
+
+<p>The distribution of fossils frequently makes it possible to map out
+approximately the general features of land and sea in long-past
+geological periods, and so to enable the history of crustal relief to be
+traced.<a name="fa45b" id="fa45b" href="#ft45b"><span class="sp">45</span></a></p>
+
+<p>While the tendency is for the living forms to come into harmony
+with their environment and to approach the state of equilibrium
+by successive adjustments if the environment should
+happen to change, it is to be observed that the action
+<span class="sidenote">Reaction of organisms on environment.</span>
+of organisms themselves often tends to change their
+environment. Corals and other quick-growing calcareous
+marine organisms are the most powerful in this
+respect by creating new land in the ocean. Vegetation of all sorts
+acts in a similar way, either in forming soil and assisting in breaking
+up rocks, in filling up shallow lakes, and even, like the mangrove,
+in reclaiming wide stretches of land from the sea. Plant life,
+utilizing solar light to combine the inorganic elements of water,
+soil and air into living substance, is the basis of all animal life.
+This is not by the supply of food alone, but also by the withdrawal
+of carbonic acid from the atmosphere, by which vegetation maintains
+the composition of the air in a state fit for the support of animal
+life. Man in the primitive stages of culture is scarcely to be distinguished
+from other animals as regards his subjection to environment,
+but in the higher grades of culture the conditions of control
+and reaction become much more complicated, and the department
+of anthropogeography is devoted to their consideration.</p>
+
+<p>The first requisites of all human beings are food and protection,
+in their search for which men are brought into intimate relations
+with the forms and productions of the earth&rsquo;s surface.
+The degree of dependence of any people upon environment
+<span class="sidenote">Anthropogeography.</span>
+varies inversely as the degree of culture or civilization,
+which for this purpose may perhaps be defined as the power
+of an individual to exercise control over the individual and over
+the environment for the benefit of the community. The development
+of culture is to a certain extent a question of race, and although
+forming one species, the varieties of man differ in almost imperceptible
+gradations with a complexity defying classification (see <span class="sc"><a href="#artlinks">Anthropology</a></span>).
+Professor Keane groups man round four leading types,
+which may be named the black, yellow, red and white, or the Ethiopic,
+Mongolic, American and Caucasic. Each may be subdivided,
+though not with great exactness, into smaller groups, either according
+to physical characteristics, of which the form of the head is most
+important, or according to language.</p>
+
+<p>The black type is found only in tropical or sub-tropical countries,
+and is usually in a primitive condition of culture, unless educated
+by contact with people of the white type. They follow
+the most primitive forms of religion (mainly fetishism),
+<span class="sidenote">Types of man.</span>
+live on products of the woods or of the chase, with the
+minimum of work, and have only a loose political organization.
+The red type is peculiar to America, inhabiting every climate from
+polar to equatorial, and containing representatives of many stages
+of culture which had apparently developed without the aid or
+interference of people of any other race until the close of the 15th
+century. The yellow type is capable of a higher culture, cherishes
+higher religious beliefs, and inhabits as a rule the temperate zone,
+although extending to the tropics on one side and to the arctic
+regions on the other. The white type, originating in the north
+temperate zone, has spread over the whole world. They have
+attained the highest culture, profess the purest forms of monotheistic
+religion, and have brought all the people of the black type
+and many of those of the yellow under their domination.</p>
+
+<p>The contrast between the yellow and white types has been softened
+by the remarkable development of the Japanese following the
+assimilation of western methods.</p>
+
+<p>The actual number of human inhabitants in the world has been
+calculated as follows:</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcl">&nbsp;</td> <td class="tcc f80">By Continents.<a name="fa46b" id="fa46b" href="#ft46b"><span class="sp">46</span></a></td> <td class="tcc f80">&nbsp;</td> <td class="tcc f80">By Race.<a name="fa47b" id="fa47b" href="#ft47b"><span class="sp">47</span></a></td></tr>
+
+<tr><td class="tcl">Asia</td> <td class="tcr rb">875,000,000</td> <td class="tcl">White (Caucasic)</td> <td class="tcr">770,000,000</td></tr>
+<tr><td class="tcl">Europe</td> <td class="tcr rb">392,000,000</td> <td class="tcl">Yellow (Mong.)</td> <td class="tcr">540,000,000</td></tr>
+<tr><td class="tcl">Africa</td> <td class="tcr rb">170,000,000</td> <td class="tcl">Black (Ethiopic)</td> <td class="tcr">175,000,000</td></tr>
+<tr><td class="tcl">America</td> <td class="tcr rb">143,000,000</td> <td class="tcl">Red (American)</td> <td class="tcr">22,000,000</td></tr>
+<tr><td class="tcl">Australia and Polynesia</td> <td class="tcr rb">7,000,000</td> <td class="tcl">&nbsp;</td> <td class="tcr">&mdash;&mdash;&mdash;&mdash;&mdash;</td></tr>
+<tr><td class="tcl">&nbsp;</td> <td class="tcr rb">&mdash;&mdash;&mdash;&mdash;&mdash;</td> <td class="tcc">Total</td> <td class="tcr">1,507,000,000</td></tr>
+<tr><td class="tcc">Total</td> <td class="tcr rb">1,587,000,000</td> <td class="tcl">&nbsp;</td> <td class="tcl">&nbsp;</td></tr>
+</table>
+
+<p>In round numbers the population of the world is about
+1,600,000,000, and, according to an estimate by Ravenstein,<a name="fa48b" id="fa48b" href="#ft48b"><span class="sp">48</span></a> the
+maximum population which it will be possible for the earth to
+maintain is 6000 millions, a number which, if the average rate of
+increase in 1891 continued, would be reached within 200 years.</p>
+
+<p>While highly civilized communities are able to evade many of
+the restrictions of environment, to overcome the barriers to intercommunication
+interposed by land or sea, to counteract the adverse
+<span class="pagenum"><a name="page636" id="page636"></a>636</span>
+influence of climate, and by the development of trade even to
+inhabit countries which cannot yield a food-supply, the mass of
+mankind is still completely under the control of those conditions
+which in the past determined the distribution and the mode of life
+of the whole human race.</p>
+
+<p>In tropical forests primitive tribes depend on the collection of
+wild fruits, and in a minor degree on the chase of wild animals, for
+their food. Clothing is unnecessary; hence there is
+little occasion for exercising the mental faculties beyond
+<span class="sidenote">Influence of environment on man.</span>
+the sense of perception to avoid enemies, or the inventive
+arts beyond what is required for the simplest
+weapons and the most primitive fortifications. When
+the pursuit of game becomes the chief occupation of a people there
+is of necessity a higher development of courage, skill, powers of
+observation and invention; and these qualities are still further
+enhanced in predatory tribes who take by force the food, clothing
+and other property prepared or collected by a feebler people. The
+fruit-eating savage cannot stray beyond his woods which bound
+his life as the water bounds that of a fish; the hunter is free to
+live on the margin of forests or in open country, while the robber
+or warrior from some natural stronghold of the mountains sweeps
+over the adjacent plains and carries his raids into distant lands.
+Wide grassy steppes lead to the organization of the people as nomads
+whose wealth consists in flocks and herds, and their dwellings
+are tents. The nomad not only domesticates and turns to his
+own use the gentler and more powerful animals, such as sheep,
+cattle, horses, camels, but even turns some predatory creatures,
+like the dog, into a means of defending their natural prey. They
+hunt the beasts of prey destructive to their flocks, and form armed
+bands for protection against marauders or for purposes of aggression
+on weaker sedentary neighbours. On the fertile low grounds along
+the margins of rivers or in clearings of forests, agricultural communities
+naturally take their rise, dwelling in villages and cultivating
+the wild grains, which by careful nurture and selection have been
+turned into rich cereals. The agriculturist as a rule is rooted to
+the soil. The land he tills he holds, and acquires a closer connexion
+with a particular patch of ground than either the hunter or the herdsman.
+In the temperate zone, where the seasons are sharply contrasted,
+but follow each other with regularity, foresight and self-denial
+were fostered, because if men did not exercise these qualities seed-time
+or harvest might pass into lost opportunities and the tribes would
+suffer. The more extreme climates of arid regions on the margins of
+the tropics, by the unpredictable succession of droughts and floods,
+confound the prevision of uninstructed people, and make prudence
+and industry qualities too uncertain in their results to be worth
+cultivating. Thus the civilization of agricultural peoples of the
+temperate zone grew rapidly, yet in each community a special type
+arose adapted to the soil, the crop and the climate. On the seashore
+fishing naturally became a means of livelihood, and dwellers
+by the sea, in virtue of the dangers to which they are exposed from
+storm and unseaworthy craft, are stimulated to a higher degree of
+foresight, quicker observation, prompter decision and more energetic
+action in emergencies than those who live inland. The building
+and handling of vessels also, and the utilization of such uncontrollable
+powers of nature as wind and tide, helped forward mechanical
+invention. To every type of coast there may be related a special
+type of occupation and even of character; the deep and gloomy
+fjord, backed by almost impassable mountains, bred bold mariners
+whose only outlet for enterprise was seawards towards other lands&mdash;the
+<i>viks</i> created the vikings. On the gently sloping margin of the
+estuary of a great river a view of tranquil inland life was equally
+presented to the shore-dweller, and the ocean did not present the
+only prospect of a career. Finally the mountain valley, with its
+patches of cultivable soil on the alluvial fans of tributary torrents,
+its narrow pastures on the uplands only left clear of snow in summer,
+its intensified extremes of climates and its isolation, almost equal to
+that of an island, has in all countries produced a special type of
+brave and hardy people, whose utmost effort may bring them comfort,
+but not wealth, by honest toil, who know little of the outer
+world, and to whom the natural outlet for ambition is marauding
+on the fertile plains. The highlander and viking, products of the
+valleys raised high amid the mountains or half-drowned in the sea,
+are everywhere of kindred spirit.</p>
+
+<p>It is in some such manner as these that the natural conditions
+of regions, which must be conformed to by prudence and utilized
+by labour to yield shelter and food, have led to the growth of peoples
+differing in their ways of life, thought and speech. The initial
+differences so produced are confirmed and perpetuated by the
+same barriers which divide the faunal or floral regions, the sea,
+mountains, deserts and the like, and much of the course of past
+history and present politics becomes clear when the combined
+results of differing race and differing environment are taken into
+account.<a name="fa49b" id="fa49b" href="#ft49b"><span class="sp">49</span></a></p>
+
+<p>The specialization which accompanies the division of labour has
+important geographical consequences, for it necessitates communication
+between communities and the interchange of their products.
+<span class="sidenote">Density of population.</span>
+Trade makes it possible to work mineral resources
+in localities where food can only be grown with great
+difficulty and expense, or which are even totally barren
+and waterless, entirely dependent on supplies from distant sources.</p>
+
+<p>The population which can be permanently supported by a given
+area of land differs greatly according to the nature of the resources
+and the requirements of the people. Pastoral communities are
+always scattered very thinly over large areas; agricultural populations
+may be almost equally sparse where advanced methods of
+agriculture and labour-saving machinery are employed; but where
+a frugal people are situated on a fertile and inexhaustible soil, such
+as the deltas and river plains of Egypt, India and China, an enormous
+population may be supported on a small area. In most cases,
+however, a very dense population can only be maintained in regions
+where mineral resources have fixed the site of great manufacturing
+industries. The maximum density of population which a given
+region can support is very difficult to determine; it depends partly
+on the race and standard of culture of the people, partly on the
+nature and origin of the resources on which they depend, partly
+on the artificial burdens imposed and very largely on the climate.
+Density of population is measured by the average number of people
+residing on a unit of area; but in order to compare one part of the
+world with another the average should, strictly speaking, be taken
+for regions of equal size or of equal population; and the portions
+of the country which are permanently uninhabitable ought to be
+excluded from the calculation.<a name="fa50b" id="fa50b" href="#ft50b"><span class="sp">50</span></a> Considering the average density
+of population within the political limits of countries, the following
+list is of some value; the figures for a few smaller divisions of
+large countries are added (in brackets) for comparison:</p>
+
+<p class="pt2 center"><i>Average Population on 1 sq. m.</i> (<i>For 1900 or 1901.</i>)</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tccm allb">Country.</td> <td class="tccm allb">Density<br />of pop.</td> <td class="tccm allb">Country.</td> <td class="tccm allb">Density<br />of pop.</td></tr>
+
+<tr><td class="tcl lb rb">(Saxony)</td> <td class="tcl rb">743*</td> <td class="tcl rb">Ceylon</td> <td class="tcl rb">141**</td></tr>
+<tr><td class="tcl lb rb">Belgium</td> <td class="tcl rb">589*</td> <td class="tcl rb">Greece</td> <td class="tcl rb">&ensp;97</td></tr>
+<tr><td class="tcl lb rb">Java</td> <td class="tcl rb">568**</td> <td class="tcl rb">European Turkey</td> <td class="tcl rb">&ensp;90</td></tr>
+<tr><td class="tcl lb rb">(England and Wales)</td> <td class="tcl rb">558</td> <td class="tcl rb">Spain</td> <td class="tcl rb">&ensp;97</td></tr>
+<tr><td class="tcl lb rb">(Bengal)</td> <td class="tcl rb">495**</td> <td class="tcl rb">European Russia</td> <td class="tcl rb">&ensp;55**</td></tr>
+<tr><td class="tcl lb rb">Holland</td> <td class="tcl rb">436</td> <td class="tcl rb">Sweden</td> <td class="tcl rb">&ensp;30</td></tr>
+<tr><td class="tcl lb rb">United Kingdom</td> <td class="tcl rb">344</td> <td class="tcl rb">United States</td> <td class="tcl rb">&ensp;25</td></tr>
+<tr><td class="tcl lb rb">Japan</td> <td class="tcl rb">317</td> <td class="tcl rb">Mexico</td> <td class="tcl rb">&ensp;18</td></tr>
+<tr><td class="tcl lb rb">Italy</td> <td class="tcl rb">293</td> <td class="tcl rb">Norway</td> <td class="tcl rb">&ensp;18</td></tr>
+<tr><td class="tcl lb rb">China proper</td> <td class="tcl rb">270**</td> <td class="tcl rb">Persia</td> <td class="tcl rb">&ensp;15</td></tr>
+<tr><td class="tcl lb rb">German Empire</td> <td class="tcl rb">270</td> <td class="tcl rb">New Zealand</td> <td class="tcl rb">&emsp;7</td></tr>
+<tr><td class="tcl lb rb">Austria</td> <td class="tcl rb">226</td> <td class="tcl rb">Argentina</td> <td class="tcl rb">&emsp;5</td></tr>
+<tr><td class="tcl lb rb">Switzerland</td> <td class="tcl rb">207</td> <td class="tcl rb">Brazil</td> <td class="tcl rb">&emsp;4.5</td></tr>
+<tr><td class="tcl lb rb">France</td> <td class="tcl rb">188</td> <td class="tcl rb">Eastern States of</td> <td class="tcl rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Indian Empire</td> <td class="tcl rb">167**</td> <td class="tcl rb">&emsp;Australia</td> <td class="tcl rb">&emsp;3</td></tr>
+<tr><td class="tcl lb rb">Denmark</td> <td class="tcl rb">160**</td> <td class="tcl rb">Dominion of Canada</td> <td class="tcl rb">&emsp;1.5</td></tr>
+<tr><td class="tcl lb rb">Hungary</td> <td class="tcl rb">154**</td> <td class="tcl rb">Siberia</td> <td class="tcl rb">&emsp;1</td></tr>
+<tr><td class="tcl lb rb bb">Portugal</td> <td class="tcl rb bb">146</td> <td class="tcl rb bb">West Australia</td> <td class="tcl rb bb">&emsp;0.2</td></tr>
+
+<tr><td class="tcl" colspan="4">&emsp;* Almost exclusively industrial.</td></tr>
+
+<tr><td class="tcl" colspan="4">&emsp;** Almost exclusively agricultural.</td></tr>
+</table>
+
+<p>The movement of people from one place to another without the
+immediate intention of returning is known as migration, and according
+to its origin it may be classed as centrifugal (directed
+<i>from</i> a particular area) and centripetal (directed <i>towards</i>
+<span class="sidenote">Migration.</span>
+a particular area). Centrifugal migration is usually a matter of
+compulsion; it may be necessitated by natural causes, such as a
+change of climate leading to the withering of pastures or destruction
+of agricultural land, to inundation, earthquake, pestilence or to an
+excess of population over means of support; or to artificial causes,
+such as the wholesale deportation of a conquered people; or to
+political or religious persecution. In any case the people are driven
+out by some adverse change; and when the urgency is great they
+may require to drive out in turn weaker people who occupy a desirable
+territory, thus propagating the wave of migration, the direction of
+which is guided by the forms of the land into inevitable channels.
+Many of the great historic movements of peoples were doubtless due
+to the gradual change of geographical or climatic conditions; and the
+slow desiccation of Central Asia has been plausibly suggested as the
+real cause of the peopling of modern Europe and of the medieval
+wars of the Old World, the theatres of which were critical points on
+the great natural lines of communication between east and west.</p>
+
+<p>In the case of centripetal migrations people flock to some particular
+place where exceptionally favourable conditions have been found to
+exist. The rushes to gold-fields and diamond-fields are typical instances;
+the growth of towns on coal-fields and near other sources
+of power, and the rapid settlement of such rich agricultural districts
+as the wheat-lands of the American prairies and great plains are
+other examples.</p>
+
+<p>There is, however, a tendency for people to remain rooted to the
+<span class="pagenum"><a name="page637" id="page637"></a>637</span>
+land of their birth, when not compelled or induced by powerful
+external causes to seek a new home.</p>
+
+<p>Thus arises the spirit of patriotism, a product of purely geographical
+conditions, thereby differing from the sentiment of loyalty,
+which is of racial origin. Where race and soil conspire to
+evoke both loyalty and patriotism in a people, the moral
+<span class="sidenote">Political geography.</span>
+qualities of a great and permanent nation are secured.
+It is noticeable that the patriotic spirit is strongest in those places
+where people are brought most intimately into relation with the land;
+dwellers in the mountain or by the sea, and, above all, the people of
+rugged coasts and mountainous archipelagoes, have always been
+renowned for love of country, while the inhabitants of fertile plains
+and trading communities are frequently less strongly attached to
+their own land.</p>
+
+<p>Amongst nomads the tribe is the unit of government, the political
+bond is personal, and there is no definite territorial association
+of the people, who may be loyal but cannot be patriotic. The idea
+of a country arises only when a nation, either homogeneous or
+composed of several races, establishes itself in a region the boundaries
+of which may be defined and defended against aggression from
+without. Political geography takes account of the partition of the
+earth amongst organized communities, dealing with the relation of
+races to regions, and of nations to countries, and considering the
+conditions of territorial equilibrium and instability.</p>
+
+<p>The definition of boundaries and their delimitation is one of the
+most important parts of political geography. Natural boundaries
+are always the most definite and the strongest, lending
+themselves most readily to defence against aggression.
+<span class="sidenote">Boundaries.</span>
+The sea is the most effective of all, and an island state is
+recognized as the most stable. Next in importance comes a mountain
+range, but here there is often difficulty as to the definition of
+the actual crest-line, and mountain ranges being broad regions, it
+may happen that a small independent state, like Switzerland or
+Andorra, occupies the mountain valleys between two or more great
+countries. Rivers do not form effective international boundaries,
+although between dependent self-governing communities they are
+convenient lines of demarcation. A desert, or a belt of country
+left purposely without inhabitants, like the mark, marches or
+debatable lands of the middle ages, was once a common means
+of separating nations which nourished hereditary grievances. The
+&ldquo;buffer-state&rdquo; of modern diplomacy is of the same ineffectual
+type. A less definite though very practical boundary is that formed
+by the meeting-line of two languages, or the districts inhabited
+by two races. The line of fortresses protecting Austria from Italy
+lies in some places well back from the political boundary, but
+just inside the linguistic frontier, so as to separate the German
+and Italian races occupying Austrian territory. Arbitrary lines,
+either traced from point to point and marked by posts on the ground,
+or defined as portions of meridians and parallels, are now the most
+common type of boundaries fixed by treaty. In Europe and Asia
+frontiers are usually strongly fortified and strictly watched in times
+of peace as well as during war. In South America strictly defined
+boundaries are still the exception, and the claims of neighbouring
+nations have very frequently given rise to war, though now more
+commonly to arbitration.<a name="fa51b" id="fa51b" href="#ft51b"><span class="sp">51</span></a></p>
+
+<p>The modes of government amongst civilized peoples have little
+influence on political geography; some republics are as arbitrary
+and exacting in their frontier regulations as some absolute
+monarchies. It is, however, to be noticed that absolute
+<span class="sidenote">Forms of government.</span>
+monarchies are confined to the east of Europe and to
+Asia, Japan being the only established constitutional
+monarchy east of the Carpathians. Limited monarchies are (with
+the exception of Japan) peculiar to Europe, and in these the degree
+of democratic control may be said to diminish as one passes eastwards
+from the United Kingdom. Republics, although represented
+in Europe, are the peculiar form of government of America and
+are unknown in Asia.</p>
+
+<p>The forms of government of colonies present a series of transitional
+types from the autocratic administration of a governor
+appointed by the home government to complete democratic
+self-government. The latter occurs only in the temperate possessions
+of the British empire, in which there is no great preponderance
+of a coloured native population. New colonial forms have been
+developed during the partition of Africa amongst European powers,
+the sphere of influence being especially worthy of notice. This
+is a vaguer form of control than a protectorate, and frequently
+amounts merely to an agreement amongst civilized powers to respect
+the right of one of their number to exercise government within
+a certain area, if it should decide to do so at any future time.</p>
+
+<p>The central governments of all civilized countries concerned with
+external relations are closely similar in their modes of action, but
+the internal administration may be very varied. In this respect a
+country is either centralized, like the United Kingdom or France,
+or federated of distinct self-governing units like Germany (where
+the units include kingdoms, at least three minor types of monarchies,
+municipalities and a crown land under a nominated governor), or the
+United States, where the units are democratic republics. The ultimate
+cause of the predominant form of federal government may be
+the geographical diversity of the country, as in the cantons occupying
+the once isolated mountain valleys of Switzerland, the racial diversity
+of the people, as in Austria-Hungary, or merely political expediency,
+as in republics of the American type.</p>
+
+<p>The minor subdivisions into provinces, counties and parishes, or
+analogous areas, may also be related in many cases to natural
+features or racial differences perpetuated by historical causes. The
+territorial divisions and subdivisions often survive the conditions
+which led to their origin; hence the study of political geography is
+allied to history as closely as the study of physical geography is allied
+to geology, and for the same reason.</p>
+
+<p>The aggregation of population in towns was at one time mainly
+brought about by the necessity for defence, a fact indicated by the
+defensive sites of many old towns. In later times,
+towns have been more often founded in proximity to
+<span class="sidenote">Towns.</span>
+valuable mineral resources, and at critical points or nodes on lines
+of communication. These are places where the mode of travelling
+or of transport is changed, such as seaports, river ports and railway
+termini, or natural resting-places, such as a ford, the foot of a
+steep ascent on a road, the entrance of a valley leading up from a
+plain into the mountains, or a crossing-place of roads or railways.<a name="fa52b" id="fa52b" href="#ft52b"><span class="sp">52</span></a>
+The existence of a good natural harbour is often sufficient to
+give origin to a town and to fix one end of a line of land communication.</p>
+
+<p>In countries of uniform surface or faint relief, roads and railways
+may be constructed in any direction without regard to the configuration.
+In places where the low ground is marshy,
+roads and railways often follow the ridge-lines of hills,
+<span class="sidenote">Lines of communication.</span>
+or, as in Finland, the old glacial eskers, which run parallel
+to the shore. Wherever the relief of the land is pronounced,
+roads and railways are obliged to occupy the lowest ground
+winding along the valleys of rivers and through passes in the mountains.
+In exceptional cases obstructions which it would be impossible
+or too costly to turn are overcome by a bridge or tunnel, the magnitude
+of such works increasing with the growth of engineering skill
+and financial enterprise. Similarly the obstructions offered to
+water communication by interruption through land or shallows are
+overcome by cutting canals or dredging out channels. The economy
+and success of most lines of communication depend on following
+as far as possible existing natural lines and utilizing existing natural
+sources of power.<a name="fa53b" id="fa53b" href="#ft53b"><span class="sp">53</span></a></p>
+
+<p>Commercial geography may be defined as the description of the
+earth&rsquo;s surface with special reference to the discovery, production,
+transport and exchange of commodities. The transport
+concerns land routes and sea routes, the latter being
+<span class="sidenote">Commercial geography.</span>
+the more important. While steam has been said to
+make a ship independent of wind and tide, it is still
+true that a long voyage even by steam must be planned so as to
+encounter the least resistance possible from prevailing winds and
+permanent currents, and this involves the application of oceanographical
+and meteorological knowledge. The older navigation by
+utilizing the power of the wind demands a very intimate knowledge
+of these conditions, and it is probable that a revival of sailing
+ships may in the present century vastly increase the importance of
+the study of maritime meteorology.</p>
+
+<p>The discovery and production of commodities require a knowledge
+of the distribution of geological formations for mineral products,
+of the natural distribution, life-conditions and cultivation
+or breeding of plants and animals and of the labour market. Attention
+must also be paid to the artificial restrictions of political geography,
+to the legislative restrictions bearing on labour and trade
+as imposed in different countries, and, above all, to the incessant
+fluctuations of the economic conditions of supply and demand and
+the combinations of capitalists or workers which affect the market.<a name="fa54b" id="fa54b" href="#ft54b"><span class="sp">54</span></a>
+The term &ldquo;applied geography&rdquo; has been employed to designate
+commercial geography, the fact being that every aspect of scientific
+geography may be applied to practical purposes, including the
+purposes of trade. But apart from the applied science, there is an
+aspect of pure geography which concerns the theory of the relation
+of economics to the surface of the earth.</p>
+
+<p>It will be seen that as each successive aspect of geographical
+science is considered in its natural sequence the conditions become
+<span class="pagenum"><a name="page638" id="page638"></a>638</span>
+<span class="sidenote">Conclusion.</span>
+more numerous, complex, variable and practically important.
+From the underlying abstract mathematical considerations all
+through the superimposed physical, biological, anthropological,
+political and commercial development of the
+subject runs the determining control exercised by crust-forms
+acting directly or indirectly on mobile distributions; and this
+is the essential principle of geography.</p>
+</div>
+<div class="author">(H. R. M.)</div>
+
+<hr class="foot" style="clear: both;" /> <div class="note">
+
+<p><a name="ft1b" id="ft1b" href="#fa1b"><span class="fn">1</span></a> A concise sketch of the whole history of geographical method or
+theory as distinguished from the history of geographical discovery
+(see later section of this article) is only to be found in the introduction
+to H. Wagner&rsquo;s <i>Lehrbuch der Geographie</i>, vol. i. (Leipzig, 1900),
+which is in every way the most complete treatise on the principles of
+geography.</p>
+
+<p><a name="ft2b" id="ft2b" href="#fa2b"><span class="fn">2</span></a> <i>History of Ancient Geography</i> (Cambridge, 1897), p. 70.</p>
+
+<p><a name="ft3b" id="ft3b" href="#fa3b"><span class="fn">3</span></a> See J.L. Myres, &ldquo;An Attempt to reconstruct the Maps used by
+Herodotus,&rdquo; <i>Geographical Journal</i>, viii. (1896), p. 605.</p>
+
+<p><a name="ft4b" id="ft4b" href="#fa4b"><span class="fn">4</span></a> <i>Geschichte der wissenschaftlichen Erdkunde der Griechen</i> (Leipzig,
+1891), Abt. 3, p. 60.</p>
+
+<p><a name="ft5b" id="ft5b" href="#fa5b"><span class="fn">5</span></a> Bunbury&rsquo;s <i>History of Ancient Geography</i> (2 vols., London, 1879),
+Müller&rsquo;s <i>Geographi Graeci minores</i> (2 vols., Paris, 1855, 1861) and
+Berger&rsquo;s <i>Geschichte der wissenschaftlichen Erdkunde der Griechen</i>
+(4 vols., Leipzig, 1887-1893) are standard authorities on the Greek
+geographers.</p>
+
+<p><a name="ft6b" id="ft6b" href="#fa6b"><span class="fn">6</span></a> The period of the early middle ages is dealt with in Beazley&rsquo;s
+<i>Dawn of Modern Geography</i> (London; part i., 1897; part ii., 1901;
+part iii., 1906); see also Winstedt, <i>Cosmos Indicopleustes</i> (1910).</p>
+
+<p><a name="ft7b" id="ft7b" href="#fa7b"><span class="fn">7</span></a> From translator&rsquo;s preface to the English version by Mr Dugdale
+(1733), entitled <i>A Complete System of General Geography</i>, revised
+by Dr Peter Shaw (London, 1756).</p>
+
+<p><a name="ft8b" id="ft8b" href="#fa8b"><span class="fn">8</span></a> Printed in <i>Schriften zur physischen Geographie</i>, vol. vi. of
+Schubert&rsquo;s edition of the collected works of Kant (Leipzig, 1839).
+First published with notes by Rink in 1802.</p>
+
+<p><a name="ft9b" id="ft9b" href="#fa9b"><span class="fn">9</span></a> <i>History of Civilization</i>, vol. i. (1857).</p>
+
+<p><a name="ft10b" id="ft10b" href="#fa10b"><span class="fn">10</span></a> See H.J. Mackinder in <i>British Association Report</i> (Ipswich),
+1895, p. 738, for a summary of German opinion, which has been
+expressed by many writers in a somewhat voluminous literature.</p>
+
+<p><a name="ft11b" id="ft11b" href="#fa11b"><span class="fn">11</span></a> H. Wagner&rsquo;s year-book, <i>Geographische Jahrbuch</i>, published at
+Gotha, is the best systematic record of the progress of geography
+in all departments; and Haack&rsquo;s <i>Geographen Kalender</i>, also published
+annually at Gotha, gives complete lists of the geographical societies
+and geographers of the world.</p>
+
+<p><a name="ft12b" id="ft12b" href="#fa12b"><span class="fn">12</span></a> This phrase is old, appearing in one of the earliest English works
+on geography, William Cuningham&rsquo;s <i>Cosmographical Glasse conteinyng
+the pleasant Principles of Cosmographie, Geographie, Hydrographie
+or Navigation</i> (London, 1559).</p>
+
+<p><a name="ft13b" id="ft13b" href="#fa13b"><span class="fn">13</span></a> See also S. Günther, <i>Handbuch der mathematischen Geographie</i>
+(Stuttgart, 1890).</p>
+
+<p><a name="ft14b" id="ft14b" href="#fa14b"><span class="fn">14</span></a> &ldquo;On the Height of the Land and the Depth of the Ocean,&rdquo; <i>Scot.
+Geog. Mag.</i> iv. (1888), p. 1. Estimates had been made previously by
+Humboldt, De Lapparent, H. Wagner, and subsequently by Penck
+and Heiderich, and for the oceans by Karstens.</p>
+
+<p><a name="ft15b" id="ft15b" href="#fa15b"><span class="fn">15</span></a> <i>Petermanns Mitteilungen</i>, xxv. (1889), p. 17.</p>
+
+<p><a name="ft16b" id="ft16b" href="#fa16b"><span class="fn">16</span></a> <i>Proc. Roy. Soc. Edin.</i> xvii. (1890) p. 185.</p>
+
+<p><a name="ft17b" id="ft17b" href="#fa17b"><span class="fn">17</span></a> <i>Comptes rendus Acad. Sci.</i> (Paris, 1890), vol. iii. p. 994.</p>
+
+<p><a name="ft18b" id="ft18b" href="#fa18b"><span class="fn">18</span></a> &ldquo;Areal und mittlere Erhebung der Landflächen sowie der Erdkruste&rdquo;
+in Gerland&rsquo;s <i>Beiträge zur Geophysik</i>, ii. (1895) p. 667. See
+also <i>Nature</i>, 54 (1896), p. 112.</p>
+
+<p><a name="ft19b" id="ft19b" href="#fa19b"><span class="fn">19</span></a> <i>Petermanns Mitteilungen</i>, xxxv. (1889) p. 19.</p>
+
+<p><a name="ft20b" id="ft20b" href="#fa20b"><span class="fn">20</span></a> The areas of the continental shelf and lowlands are approximately
+equal, and it is an interesting circumstance that, taken as a
+whole, the actual coast-line comes just midway on the most nearly
+level belt of the earth&rsquo;s surface, excepting the ocean floor. The configuration
+of the continental slope has been treated in detail by
+Nansen in <i>Scientific Results of Norwegian North Polar Expedition</i>,
+vol. iv. (1904), where full references to the literature of the subject
+will be found.</p>
+
+<p><a name="ft21b" id="ft21b" href="#fa21b"><span class="fn">21</span></a> <i>British Association Report</i> (Edinburgh, 1892), p. 699.</p>
+
+<p><a name="ft22b" id="ft22b" href="#fa22b"><span class="fn">22</span></a> <i>Das Antlitz der Erde</i> (4 vols., Leipzig, 1885, 1888, 1901). Translated
+under the editorship of E. de Margerie, with much additional
+matter, as <i>La Face de la terre</i>, vols. i. and ii. (Paris, 1897, 1900), and
+into English by Dr Hertha Sollas as <i>The Face of the Earth</i>, vols. i.
+and ii. (Oxford, 1904, 1906).</p>
+
+<p><a name="ft23b" id="ft23b" href="#fa23b"><span class="fn">23</span></a> Élie de Beaumont, <i>Notice sur les systčmes de montagnes</i> (3 vols.,
+Paris, 1852).</p>
+
+<p><a name="ft24b" id="ft24b" href="#fa24b"><span class="fn">24</span></a> <i>Vestiges of the Molten Globe</i> (London, 1875).</p>
+
+<p><a name="ft25b" id="ft25b" href="#fa25b"><span class="fn">25</span></a> See J.W. Gregory, &ldquo;The Plan of the Earth and its Causes,&rdquo;
+<i>Geog. Journal</i>, xiii. (1899) p. 225; Lord Avebury, <i>ibid.</i> xv. (1900)
+p. 46; Marcel Bertrand, &ldquo;Déformation tétraédrique de la terre et
+déplacement du pōle,&rdquo; <i>Comptes rendus Acad. Sci.</i> (Paris, 1900),
+vol. cxxx. p. 449; and A. de Lapparent, <i>ibid.</i> p. 614.</p>
+
+<p><a name="ft26b" id="ft26b" href="#fa26b"><span class="fn">26</span></a> See A.E.H. Love, &ldquo;Gravitational Stability of the Earth,&rdquo; <i>Phil.
+Trans.</i> ser. A. vol. ccvii. (1907) p. 171.</p>
+
+<p><a name="ft27b" id="ft27b" href="#fa27b"><span class="fn">27</span></a> <i>Rumpf</i>, in German, the language in which this distinction was
+first made.</p>
+
+<p><a name="ft28b" id="ft28b" href="#fa28b"><span class="fn">28</span></a> <i>Lehrbuch der Geographie</i> (Hanover and Leipzig, 1900), Bd. i. S.
+245, 249.</p>
+
+<p><a name="ft29b" id="ft29b" href="#fa29b"><span class="fn">29</span></a> See, for example, F.G. Hahn&rsquo;s <i>Insel-Studien</i> (Leipzig, 1883).</p>
+
+<p><a name="ft30b" id="ft30b" href="#fa30b"><span class="fn">30</span></a> See <i>Geographical Journal</i>, xxii. (1903) pp. 191-194.</p>
+
+<p><a name="ft31b" id="ft31b" href="#fa31b"><span class="fn">31</span></a> The most important works on the classification of land forms are
+F. von Richthofen, <i>Führer für Forschungsreisende</i> (Berlin, 1886);
+G. de la Noė and E. de Margerie, <i>Les Formes du terrain</i> (Paris, 1888);
+and above all A. Penck, <i>Morphologie der Erdoberfläche</i> (2 vols.,
+Stuttgart, 1894). Compare also A. de Lapparent, <i>Leēons de géographie
+physique</i> (2nd ed., Paris, 1898), and W.M. Davis, <i>Physical
+Geography</i> (Boston, 1899).</p>
+
+<p><a name="ft32b" id="ft32b" href="#fa32b"><span class="fn">32</span></a> &ldquo;Geomorphologie als genetische Wissenschaft,&rdquo; in <i>Report of
+Sixth International Geog. Congress</i> (London, 1895), p. 735 (English
+Abstract, p. 748).</p>
+
+<p><a name="ft33b" id="ft33b" href="#fa33b"><span class="fn">33</span></a> On this subject see J. Geikie, <i>Earth Sculpture</i> (London, 1898);
+J.E. Marr, <i>The Scientific Study of Scenery</i> (London, 1900); Sir A.
+Geikie, <i>The Scenery and Geology of Scotland</i> (London, 2nd ed., 1887);
+Lord Avebury (Sir J. Lubbock), <i>The Scenery of Switzerland</i> (London,
+1896) and <i>The Scenery of England</i> (London, 1902).</p>
+
+<p><a name="ft34b" id="ft34b" href="#fa34b"><span class="fn">34</span></a> Some geographers distinguish a mountain from a hill by origin;
+thus Professor Seeley says &ldquo;a mountain implies elevation and a hill
+implies denudation, but the external forms of both are often identical.&rdquo;
+<i>Report VI. Int. Geog. Congress</i> (London, 1895), p. 751.</p>
+
+<p><a name="ft35b" id="ft35b" href="#fa35b"><span class="fn">35</span></a> &ldquo;Mountains,&rdquo; in <i>Scot. Geog. Mag.</i> ii. (1896) p. 145.</p>
+
+<p><a name="ft36b" id="ft36b" href="#fa36b"><span class="fn">36</span></a> <i>Führer für Forschungsreisende</i>, pp. 652-685.</p>
+
+<p><a name="ft37b" id="ft37b" href="#fa37b"><span class="fn">37</span></a> See, for a summary of river-action, A. Phillipson, <i>Studien über
+Wasserscheiden</i> (Leipzig, 1886); also I.C. Russell, <i>River Development</i>,
+(London, 1898) (published as <i>The Rivers of North America</i>, New York,
+1898).</p>
+
+<p><a name="ft38b" id="ft38b" href="#fa38b"><span class="fn">38</span></a> W.M. Davis, &ldquo;The Geographical Cycle,&rdquo; <i>Geog. Journ.</i> xiv.
+(1899) p. 484.</p>
+
+<p><a name="ft39b" id="ft39b" href="#fa39b"><span class="fn">39</span></a> A. Penck, &ldquo;Potamology as a Branch of Physical Geography,&rdquo;
+<i>Geog. Journ.</i> x. (1897) p. 619.</p>
+
+<p><a name="ft40b" id="ft40b" href="#fa40b"><span class="fn">40</span></a> See, for instance, E. Wisotzki, <i>Hauptfluss und Nebenfluss</i>
+(Stettin, 1889). For practical studies see official reports on the
+Mississippi, Rhine, Seine, Elbe and other great rivers.</p>
+
+<p><a name="ft41b" id="ft41b" href="#fa41b"><span class="fn">41</span></a> F.A. Forel, <i>Handbuch der Seenkunde: allgemeine Limnologie</i>
+(Stuttgart, 1901); F.A. Forel, &ldquo;La Limnologie, branche de la géographie,&rdquo;
+<i>Report VI. Int. Geog. Congress</i> (London, 1895), p. 593;
+also <i>Le Léman</i> (2 vols., Lausanne, 1892, 1894); H. Lullies, &ldquo;Studien
+über Seen,&rdquo; <i>Jubiläumsschrift der Albertus-Universität</i> (Königsberg,
+1894); and G.R. Credner, &ldquo;Die Reliktenseen,&rdquo; <i>Petermanns Mitteilungen</i>,
+Ergänzungshefte 86 and 89 (Gotha., 1887, 1888).</p>
+
+<p><a name="ft42b" id="ft42b" href="#fa42b"><span class="fn">42</span></a> J. Murray, &ldquo;Drainage Areas of the Continents,&rdquo; <i>Scot. Geog. Mag.</i>
+ii. (1886) p. 548.</p>
+
+<p><a name="ft43b" id="ft43b" href="#fa43b"><span class="fn">43</span></a> Wagner, <i>Lehrbuch der Geographie</i> (1900), i. 586.</p>
+
+<p><a name="ft44b" id="ft44b" href="#fa44b"><span class="fn">44</span></a> For details, see A.R. Wallace, <i>Geographical Distribution of
+Animals and Island Life</i>; A. Heilprin, <i>Geographical and Geological
+Distribution of Animals</i> (1887); O. Drude, <i>Handbuch der Pflanzengeographie</i>;
+A. Engler, <i>Entwickelungsgeschichte der Pflanzenwelt</i>;
+also Beddard, <i>Zoogeography</i> (Cambridge, 1895); and Sclater, <i>The
+Geography of Mammals</i> (London, 1899).</p>
+
+<p><a name="ft45b" id="ft45b" href="#fa45b"><span class="fn">45</span></a> See particularly A. de Lapparent, <i>Traité de géologie</i> (4th ed.,
+Paris, 1900).</p>
+
+<p><a name="ft46b" id="ft46b" href="#fa46b"><span class="fn">46</span></a> Estimate for 1900. H. Wagner, <i>Lehrbuch der Geographie</i>, i.
+P. 658.</p>
+
+<p><a name="ft47b" id="ft47b" href="#fa47b"><span class="fn">47</span></a> Estimate for year not stated. A.H. Keane in <i>International
+Geography</i>, p. 108.</p>
+
+<p><a name="ft48b" id="ft48b" href="#fa48b"><span class="fn">48</span></a> In <i>Proc. R. G. S.</i> xiii. (1891) p. 27.</p>
+
+<p><a name="ft49b" id="ft49b" href="#fa49b"><span class="fn">49</span></a> On the influence of land on people see Shaler, <i>Nature and
+Man in America</i> (New York and London, 1892); and Ellen C.
+Semple&rsquo;s <i>American History and its Geographic Conditions</i> (Boston,
+1903).</p>
+
+<p><a name="ft50b" id="ft50b" href="#fa50b"><span class="fn">50</span></a> See maps of density of population in Bartholomew&rsquo;s great large-scale
+atlases, <i>Atlas of Scotland</i> and <i>Atlas of England</i>.</p>
+
+<p><a name="ft51b" id="ft51b" href="#fa51b"><span class="fn">51</span></a> For the history of territorial changes in Europe, see Freeman,
+<i>Historical Geography of Europe</i>, edited by Bury (Oxford), 1903;
+and for the official definition of existing boundaries, see Hertslet,
+<i>The Map of Europe by Treaty</i> (4 vols., London, 1875, 1891); <i>The
+Map of Africa by Treaty</i> (3 vols., London, 1896). Also Lord Curzon&rsquo;s
+Oxford address on <i>Frontiers</i> (1907).</p>
+
+<p><a name="ft52b" id="ft52b" href="#fa52b"><span class="fn">52</span></a> For numerous special instances of the determining causes of
+town sites, see G.G. Chisholm, &ldquo;On the Distribution of Towns
+and Villages in England,&rdquo; <i>Geographical Journal</i> (1897), ix. 76,
+x. 511.</p>
+
+<p><a name="ft53b" id="ft53b" href="#fa53b"><span class="fn">53</span></a> The whole subject of anthropogeography is treated in a masterly
+way by F. Ratzel in his <i>Anthropogeographie</i> (Stuttgart, vol. i. 2nd
+ed., 1899, vol. ii. 1891), and in his <i>Politische Geographie</i> (Leipzig,
+1897). The special question of the reaction of man on his environment
+is handled by G.P. Marsh in <i>Man and Nature, or Physical
+Geography as modified by Human Action</i> (London, 1864).</p>
+
+<p><a name="ft54b" id="ft54b" href="#fa54b"><span class="fn">54</span></a> For commercial geography see G.G. Chisholm, <i>Manual of Commercial
+Geography</i> (1890).</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">GEOID<a name="ar16" id="ar16"></a></span> (from Gr. <span class="grk" title="gź">&#947;&#8134;</span>, the earth), an imaginary surface employed
+by geodesists which has the property that every element
+of it is perpendicular to the plumb-line where that line cuts it.
+Compared with the &ldquo;spheroid of reference&rdquo; the surface of the
+geoid is in general depressed over the oceans and raised over
+the great land masses. (See <span class="sc"><a href="#artlinks">Earth, Figure of the</a></span>.)</p>
+
+
+<hr class="art" />
+<p><span class="bold">GEOK-TEPE<a name="ar17" id="ar17"></a></span>, a former fortress of the Turkomans, in Russian
+Transcaspia, in the oasis of Akhal-tekke, on the Transcaspian
+railway, 28 m. N.W. of Askabad. It consisted of a walled
+enclosure 1¾ m. in circuit, the wall being 18 ft. high and 20 to
+30 ft. thick. In December 1880 the place was attacked by
+6000 Russians under General Skobelev, and after a siege of
+twenty-three days was carried by storm, although the defenders
+numbered 25,000. A monument and a small museum commemorate
+the event.</p>
+
+
+<hr class="art" />
+<p><span class="bold">GEOLOGY<a name="ar18" id="ar18"></a></span> (from Gr. <span class="grk" title="gź">&#947;&#8134;</span>, the earth, and <span class="grk" title="logos">&#955;&#972;&#947;&#959;&#962;</span>, science), the
+science which investigates the physical history of the earth.
+Its object is to trace the structural progress of our planet from
+the earliest beginnings of its separate existence, through its
+various stages of growth, down to the present condition of
+things. It seeks to determine the manner in which the evolution
+of the earth&rsquo;s great surface features has been effected. It unravels
+the complicated processes by which each continent has
+been built up. It follows, even into detail, the varied sculpture
+of mountain and valley, crag and ravine. Nor does it confine
+itself merely to changes in the inorganic world. Geology shows
+that the present races of plants and animals are the descendants
+of other and very different races which once peopled the earth.
+It teaches that there has been a progressive development of the
+inhabitants, as well as one of the globe on which they have
+dwelt; that each successive period in the earth&rsquo;s history, since
+the introduction of living things, has been marked by characteristic
+types of the animal and vegetable kingdoms; and that,
+however imperfectly the remains of these organisms have been
+preserved or may be deciphered, materials exist for a history
+of life upon the planet. The geographical distribution of existing
+faunas and floras is often made clear and intelligible by geological
+evidence; and in the same way light is thrown upon some of
+the remoter phases in the history of man himself. A subject
+so comprehensive as this must require a wide and varied basis
+of evidence. It is one of the characteristics of geology to gather
+evidence from sources which at first sight seem far removed
+from its scope, and to seek aid from almost every other leading
+branch of science. Thus, in dealing with the earliest conditions
+of the planet, the geologist must fully avail himself of the
+labours of the astronomer. Whatever is ascertainable by
+telescope, spectroscope or chemical analysis, regarding the constitution
+of other heavenly bodies, has a geological bearing.
+The experiments of the physicist, undertaken to determine
+conditions of matter and of energy, may sometimes be taken
+as the starting-points of geological investigation. The work
+of the chemical laboratory forms the foundation of a vast and
+increasing mass of geological inquiry. To the botanist, the
+zoologist, even to the unscientific, if observant, traveller by land
+or sea, the geologist turns for information and assistance.</p>
+
+<p>But while thus culling freely from the dominions of other
+sciences, geology claims as its peculiar territory the rocky
+framework of the globe. In the materials composing that
+framework, their composition and arrangement, the processes
+of their formation, the changes which they have undergone,
+and the terrestrial revolutions to which they bear witness, lie
+the main data of geological history. It is the task of the geologist
+to group these elements in such a way that they may be made
+to yield up their evidence as to the march of events in the
+evolution of the planet. He finds that they have in large
+measure arranged themselves in chronological sequence,&mdash;the
+oldest lying at the bottom and the newest at the top. Relics
+of an ancient sea-floor are overlain by traces of a vanished
+land-surface; these are in turn covered by the deposits of a
+former lake, above which once more appear proofs of the return
+of the sea. Among these rocky records lie the lavas and ashes
+of long-extinct volcanoes. The ripple left upon the shore, the
+cracks formed by the sun&rsquo;s heat upon the muddy bottom of a
+dried-up pool, the very imprint of the drops of a passing rainshower,
+have all been accurately preserved, and yield their
+evidence as to geographical conditions often widely different
+from those which exist where such markings are now found.</p>
+
+<p>But it is mainly by the remains of plants and animals imbedded
+in the rocks that the geologist is guided in unravelling the
+chronological succession of geological changes. He has found
+that a certain order of appearance characterizes these organic
+remains, that each great group of rocks is marked by its own
+special types of life, and that these types can be recognized,
+and the rocks in which they occur can be correlated even in
+distant countries, and where no other means of comparison
+would be possible. At one moment he has to deal with the bones
+of some large mammal scattered through a deposit of superficial
+gravel, at another time with the minute foraminifers and ostracods
+of an upraised sea-bottom. Corals and crinoids crowded and
+crushed into a massive limestone where they lived and died,
+ferns and terrestrial plants matted together into a bed of coal
+where they originally grew, the scattered shells of a submarine
+sand-bank, the snails and lizards which lived and died within
+a hollow-tree, the insects which have been imprisoned within
+the exuding resin of old forests, the footprints of birds and
+quadrupeds, the trails of worms left upon former shores&mdash;these,
+and innumerable other pieces of evidence, enable the geologist
+to realize in some measure what the faunas and floras of successive
+periods have been, and what geographical changes the site of
+every land has undergone.</p>
+
+<p>It is evident that to deal successfully with these varied
+materials, a considerable acquaintance with different branches
+of science is needful. Especially necessary is a tolerably wide
+knowledge of the processes now at work in changing the surface
+of the earth, and of at least those forms of plant and animal
+life whose remains are apt to be preserved in geological deposits,
+or which in their structure and habitat enable us to realize what
+their forerunners were. It has often been insisted that the
+present is the key to the past; and in a wide sense this assertion
+is eminently true. Only in proportion as we understand the
+present, where everything is open on all sides to the fullest investigation,
+can we expect to decipher the past, where so much is
+obscure, imperfectly preserved or not preserved at all. A
+study of the existing economy of nature ought thus to be the
+foundation of the geologist&rsquo;s training.</p>
+
+<p>While, however, the present condition of things is thus employed,
+we must obviously be on our guard against the danger
+of unconsciously assuming that the phase of nature&rsquo;s operations
+which we now witness has been the same in all past time, that
+geological changes have always or generally taken place in former
+ages in the manner and on the scale which we behold to-day,
+and that at the present time all the great geological processes,
+which have produced changes in the past eras of the earth&rsquo;s
+history, are still existent and active. As a working hypothesis
+we may suppose that the nature of geological processes has
+remained constant from the beginning; but we cannot postulate
+that the action of these processes has never varied in energy.
+The few centuries wherein man has been observing nature
+obviously form much too brief an interval by which to measure
+the intensity of geological action in all past time. For aught
+we can tell the present is an era of quietude and slow change,
+compared with some of the eras which have preceded it. Nor
+perhaps can we be quite sure that, when we have explored
+every geological process now in progress, we have exhausted
+all the causes of change which, even in comparatively recent
+times, have been at work.</p>
+
+<p>In dealing with the geological record, as the accessible solid
+part of the globe is called, we cannot too vividly realize that at
+<span class="pagenum"><a name="page639" id="page639"></a>639</span>
+the best it forms but an imperfect chronicle. Geological history
+cannot be compiled from a full and continuous series of documents.
+From the very nature of its origin the record is necessarily
+fragmentary, and it has been further mutilated and obscured
+by the revolutions of successive ages. And even where the
+chronicle of events is continuous, it is of very unequal value in
+different places. In one case, for example, it may present us
+with an unbroken succession of deposits many thousands of
+feet in thickness, from which, however, only a few meagre facts
+as to geological history can be gleaned. In another instance
+it brings before us, within the compass of a few yards, the
+evidence of a most varied and complicated series of changes
+in physical geography, as well as an abundant and interesting
+suite of organic remains. These and other characteristics of
+the geological record become more apparent and intelligible as
+we proceed in the study of the science.</p>
+
+<p><i>Classification.</i>&mdash;For systematic treatment the subject may be
+conveniently arranged in the following parts:&mdash;</p>
+
+<p>1. <i>The Historical Development of Geological Science.</i>&mdash;Here
+a brief outline will be given of the gradual growth of geological
+conceptions from the days of the Greeks and Romans down to
+modern times, tracing the separate progress of the more important
+branches of inquiry and noting some of the stages which in each
+case have led up to the present condition of the science.</p>
+
+<p>2. <i>The Cosmical Aspects of Geology.</i>&mdash;This section embraces
+the evidence supplied by astronomy and physics regarding the
+form and motions of the earth, the composition of the planets
+and sun, and the probable history of the solar system. The
+subjects dealt with under this head are chiefly treated in separate
+articles.</p>
+
+<p>3. <i>Geognosy.</i>&mdash;An inquiry into the materials of the earth&rsquo;s
+substance. This division, which deals with the parts of the
+earth, its envelopes of air and water, its solid crust and the
+probable condition of its interior, especially treats of the more
+important minerals of the crust, and the chief rocks of which
+that crust is built up. Geognosy thus lays a foundation of knowledge
+regarding the nature of the materials constituting the mass
+of the globe, and prepares the way for an investigation of the
+processes by which these materials are produced and altered.</p>
+
+<p>4. <i>Dynamical Geology</i> studies the nature and working of the
+various geological processes whereby the rocks of the earth&rsquo;s
+crust are formed and metamorphosed, and by which changes
+are effected upon the distribution of sea and land, and upon
+the forms of terrestrial surfaces. Such an inquiry necessitates
+a careful examination of the existing geological economy of
+nature, and forms a fitting introduction to an inquiry into the
+geological changes of former periods.</p>
+
+<p>5. <i>Geotectonic or Structural Geology</i> has for its object the
+architecture of the earth&rsquo;s crust. It embraces an inquiry into the
+manner in which the various materials composing this crust
+have been arranged. It shows that some have been formed
+in beds or strata of sediment on the floor of the sea, that others
+have been built up by the slow aggregation of organic forms,
+that others have been poured out in a molten condition or in
+showers of loose dust from subterranean sources. It further
+reveals that, though originally laid down in almost horizontal
+beds, the rocks have subsequently been crumpled, contorted
+and dislocated, that they have been incessantly worn down,
+and have often been depressed and buried beneath later
+accumulations.</p>
+
+<p>6. <i>Palaeontological Geology.</i>&mdash;This branch of the subject,
+starting from the evidence supplied by the organic forms which
+are found preserved in the crust of the earth, includes such
+questions as the relations between extinct and living types,
+the laws which appear to have governed the distribution of life
+in time and in space, the relative importance of different genera
+of animals in geological inquiry, the nature and use of the
+evidence from organic remains regarding former conditions
+of physical geography. Some of these problems belong also to
+zoology and botany, and are more fully discussed in the articles
+<span class="sc"><a href="#artlinks">Palaeontology</a></span> and <span class="sc"><a href="#artlinks">Palaeobotany</a></span>.</p>
+
+<p>7. <i>Stratigraphical Geology.</i>&mdash;This section might be called
+geological history. It works out the chronological succession
+of the great formations of the earth&rsquo;s crust, and endeavours to
+trace the sequence of events of which they contain the record.
+More particularly, it determines the order of succession of the
+various plants and animals which in past time have peopled
+the earth, and thus ascertains what has been the grand march
+of life upon this planet.</p>
+
+<p>8. <i>Physiographical Geology</i>, proceeding from the basis of
+fact laid down by stratigraphical geology regarding former
+geographical changes, embraces an inquiry into the origin and
+history of the features of the earth&rsquo;s surface&mdash;continental ridges
+and ocean basins, plains, valleys and mountains. It explains
+the causes on which local differences of scenery depend, and
+shows under what very different circumstances, and at what
+widely separated intervals, the hills and mountains, even of a
+single country, have been produced.</p>
+
+<p>Most of the detail embraced in these several sections is
+relegated to separate articles, to which references are here
+inserted. The following pages thus deal mainly with the general
+principles and historical development of the science:&mdash;</p>
+
+<p class="pt2 center sc">Part I.&mdash;Historical Development</p>
+
+<div class="condensed">
+<p><i>Geological Ideas among the Greeks and Romans.</i>&mdash;Many geological
+phenomena present themselves in so striking a form that they could
+hardly fail to impress the imagination of the earliest and rudest
+races of mankind. Such incidents as earthquakes and volcanic
+eruptions, destructive storms on land and sea, disastrous floods and
+landslips suddenly strewing valleys with ruin, must have awakened
+the terror of those who witnessed them. Prominent features of
+landscape, such as mountain-chains with their snows, clouds and
+thunderstorms, dark river-chasms that seem purposely cleft open in
+order to give passage to the torrents that rush through them, crags
+with their impressive array of pinnacles and recesses must have
+appealed of old, as they still do, to the awe and wonder of those
+who for the first time behold them. Again, banks of sea-shells in
+far inland districts would, in course of time, arrest the attention of
+the more intelligent and reflective observers, and raise in their minds
+some kind of surmise as to how such shells could ever have come
+there. These and other conspicuous geological problems found
+their earliest solution in legends and myths, wherein the more
+striking terrestrial features and the elemental forces of nature were
+represented to be the manifestation of the power of unseen supernatural
+beings.</p>
+
+<p>The basin of the Mediterranean Sea was especially well adapted,
+from its physical conditions, to be the birth-place of such fables.
+It is a region frequently shaken by earthquakes, and contains two
+distinct centres of volcanic activity, one in the Aegean Sea and one
+in Italy. It is bounded on the north by a long succession of lofty
+snow-capped mountain-ranges, whence copious rivers, often swollen
+by heavy rains or melted snows, carry the drainage into the sea.
+On the south it boasts the Nile, once so full of mystery; likewise
+wide tracts of arid desert with their dreaded dust storms. The
+Mediterranean itself, though an inland sea, is subject to gales,
+which, on exposed coasts, raise breakers quite large enough to give a
+vivid impression of the power of ocean waves. The countries that
+surround this great sheet of water display in many places widely-spread
+deposits full of sea shells, like those that still live in the
+neighbouring bays and gulfs. Such a region was not only well fitted
+to supply subjects for mythology, but also to furnish, on every side,
+materials which, in their interest and suggestiveness, would appeal
+to the reason of observant men.</p>
+
+<p>It was natural, therefore, that the early philosophers of Greece
+should have noted some of these geological features, and should have
+sought for other explanations of them than those to be found in the
+popular myths. The opinions entertained in antiquity on these
+subjects may be conveniently grouped under two heads: (1) Geological
+processes now in operation, and (2) geological changes in
+the past.</p>
+
+<p>1. <i>Contemporary Processes.</i>&mdash;The geological processes of the present
+time are partly at work underground and partly on the surface of the
+earth. The former, from their frequently disastrous
+character, received much attention from Greek and
+<span class="sidenote">Earthquakes and volcanoes.</span>
+Roman authors. Aristotle, in his <i>Meteorics</i>, cites the
+speculations of several of his predecessors which he rejects
+in favour of his own opinion to the effect that earthquakes are due
+to the generation of wind within the earth, under the influence of the
+warmth of the sun and the internal heat. Wind, being the lightest
+and most rapidly moving body, is the cause of motion in other
+bodies, and fire, united with wind, becomes flame, which is endowed
+with great rapidity of motion. Aristotle looked upon earthquakes
+and volcanic eruptions as closely connected with each other, the
+discharge of hot materials to the surface being the result of a severe
+earthquake, when finally the wind rushes out with violence, and
+sometimes buries the surrounding country under sparks and cinders,
+<span class="pagenum"><a name="page640" id="page640"></a>640</span>
+as had happened at Lipari. These crude conceptions of the nature
+of volcanic action, and the cause of earthquakes, continued to prevail
+for many centuries. They are repeated by Lucretius, who, however,
+following Anaximenes, includes as one of the causes of earthquakes
+the fall of mountainous masses of rock undermined by time, and the
+consequent propagation of gigantic tremors far and wide through
+the earth. Strabo, having travelled through the volcanic districts
+of Italy, was able to recognize that Vesuvius had once been an
+active volcano, although no eruption had taken place from it within
+human memory. He continued to hold the belief that volcanic
+energy arose from the movement of subterranean wind. He believed
+that the district around the Strait of Messina, which had formerly
+suffered from destructive earthquakes, was seldom visited by them
+after the volcanic vents of that region had been opened, so as to
+provide an escape for the subterranean fire, wind, water and burning
+masses. He cites in his <i>Geography</i> a number of examples of widespread
+as well as local sinkings of land, and alludes also to the uprise
+of the sea-bottom. He likewise regards some islands as having been
+thrown up by volcanic agency, and others as torn from the mainland
+by such convulsions as earthquakes.</p>
+
+<p>The most detailed account of earthquake phenomena which has
+come down to us from antiquity is that of Seneca in his <i>Quaestiones
+Naturales</i>. This philosopher had been much interested in the
+accounts given him by survivors and witnesses of the earthquake
+which convulsed the district of Naples in February <span class="scs">A.D.</span> 63. He
+distinguished several distinct movements of the ground: 1st, the
+up and down motion (<i>succussio</i>); 2nd, the oscillatory motion (<i>inclinatio</i>);
+and probably a third, that of trembling or vibration.
+While admitting that some earthquakes may arise from the collapse
+of the walls of subterranean cavities, he adhered to the old idea,
+held by the most numerous and important previous writers, that
+these commotions are caused mainly by the movements of wind
+imprisoned within the earth. As to the origin of volcanic outbursts
+he supposed that the subterranean wind in struggling for an outlet,
+and whirling through the chasms and passages, meets with great
+store of sulphur and other combustible substances, which by mere
+friction are set on fire. The elder Pliny reiterates the commonly
+accepted opinion as to the efficacy of wind underground. In
+discussing the phenomena of earthquakes he remarks that towns
+with many culverts and houses with cellars suffer less than others,
+and that at Naples those houses are most shaken which stand on
+hard ground. It thus appears that with regard to subterranean
+geological operations, no advance was made during the time of the
+Greeks and Romans as to the theoretical explanation of these phenomena;
+but a considerable body of facts was collected, especially
+as to the effects of earthquakes and the occurrence of volcanic
+eruptions.</p>
+
+<p>The superficial processes of geology, being much less striking than
+those of subterranean energy, naturally attracted less attention in
+antiquity. The operations of rivers, however, which so
+intimately affect a human population, were watched with
+<span class="sidenote">Action of rivers.</span>
+more or less care. Herodotus, struck by the amount of
+alluvial silt brought down annually by the Nile and spread over the
+flat inundated land, inferred that &ldquo;Egypt is the gift of the river.&rdquo;
+Aristotle, in discussing some of the features of rivers, displays considerable
+acquaintance with the various drainage-systems on the
+north side of the Mediterranean basin. He refers to the mountains
+as condensers of the atmospheric moisture, and shows that the largest
+rivers rise among the loftiest high grounds. He shows how sensibly
+the alluvial deposits carried down to the sea increase the breadth
+of the land, and cites some parts of the shores of the Black Sea,
+where, in sixty years, the rivers had brought down such a quantity
+of material that the vessels then in use required to be of much
+smaller draught than previously, the water shallowing so much that
+the marshy ground would, in course of time, become dry land.
+Strabo supplies further interesting information as to the work of
+rivers in making their alluvial plains and in pushing their deltas
+seaward. He remarks that these deltas are prevented from advancing
+farther outward by the ebb and flow of the tides.</p>
+
+<p>2. <i>Past Processes.</i>&mdash;The abundant well-preserved marine shells
+exposed among the upraised Tertiary and post-Tertiary deposits in
+the countries bordering the Mediterranean are not infrequently
+alluded to in Greek and Latin literature.
+<span class="sidenote">Occurrences of fossils.</span>
+Xenophanes of Colophon (614 <span class="scs">B.C.</span>) noticed the occurrence
+of shells and other marine productions inland among the
+mountains, and inferred from them that the land had risen out of
+the sea. A similar conclusion was drawn by Xanthus the Lydian
+(464 <span class="scs">B.C.</span>) from shells like scallops and cockles, which were found far
+from the sea in Armenia and Lower Phrygia. Herodotus, Eratosthenes,
+Strato and Strabo noted the vast quantities of fossil shells in
+different parts of Egypt, together with beds of salt, as evidence that
+the sea had once spread over the country. But by far the most
+philosophical opinions on the past mutations of the earth&rsquo;s surface
+are those expressed by Aristotle in the treatise already cited. Reviewing
+the evidence of these changes, he recognized that the sea
+now covers tracts that were once dry land, and that land will one
+day reappear where there is now sea. These alternations are to be
+regarded as following each other in a certain order and periodicity.
+But they are apt to escape our notice because they require successive
+periods of time, which, compared with our brief existence, are of
+enormous duration, and because they are brought about so imperceptibly
+that we fail to detect them in progress. In a celebrated
+passage in his <i>Metamorphoses</i>, Ovid puts into the mouth of the
+philosopher Pythagoras an account of what was probably regarded
+as the Pythagorean view of the subject in the Augustan age. It
+affirms the interchange of land and sea, the erosion of valleys by
+descending rivers, the washing down of mountains into the sea, the
+disappearance of the rivers and the submergence of land by earthquake
+movements, the separation of some islands from, and the union
+of others with, the mainland, the uprise of hills by volcanic action,
+the rise and extinction of burning mountains. There was a time
+before Etna began to glow, and the time is coming when the mountain
+will cease to burn.</p>
+
+<p>From this brief sketch it will be seen that while the ancients had
+accumulated a good deal of information regarding the occurrence of
+geological changes, their interpretations of the phenomena were to
+a considerable extent mere fanciful speculation. They had acquired
+only a most imperfect conception of the nature and operation of the
+geological processes; and though many writers realized that the
+surface of the earth has not always been, and will not always remain,
+as it is now, they had no glimpse of the vast succession of changes
+of that surface which have been revealed by geology. They built
+hypotheses on the slenderest basis of fact, and did not realize the
+necessity of testing or verifying them.</p>
+
+<p><i>Progress of Geological Conceptions in the Middle Ages.</i>&mdash;During the
+centuries that succeeded the fall of the Western empire little progress
+was made in natural science. The schoolmen in the monasteries
+and other seminaries were content to take their science from the
+literature of Greece and Rome. The Arabs, however, not only
+collected and translated that literature, but in some departments
+made original observations themselves. To one of the most illustrious
+of their number, Avicenna, the translator of Aristotle, a treatise has
+been ascribed, in which singularly modern ideas are expressed
+regarding mountains, some of which are there stated to have been
+produced by an uplifting of the ground, while others have been left
+prominent, owing to the wearing away of the softer rocks around
+them. In either case, it is confessed that the process would demand
+long tracts of time for its completion.</p>
+
+<p>After the revival of learning the ancient problem presented by
+fossil shells imbedded in the rocks of the interior of many countries
+received renewed attention. But the conditions for its solution
+were no longer what they had been in the days of the philosophers
+of antiquity. Men were not now free to adopt and teach any doctrine
+they pleased on the subject. The Christian church had meanwhile
+arisen to power all over Europe, and adjudged as heretics all
+who ventured to impugn any of her dogmas. She taught that the
+land and the sea had been separated on the third day of creation,
+before the appearance of any animal life, which was not created until
+the fifth day. To assert that the dry land is made up in great part
+of rocks that were formed in the sea, and are crowded with the
+remains of animals, was plainly to impugn the veracity of the Bible.
+Again, it had come to be the orthodox belief that only somewhere
+about 6000 years had elapsed since the time of Adam and Eve.
+If any thoughtful observer, impressed with the overwhelming force
+of the evidence that the fossiliferous formations of the earth&rsquo;s crust
+must have taken long periods of time for their accumulation, ventured
+to give public expression to his conviction, he ran considerable
+risk of being proceeded against as a heretic. It was needful, therefore,
+to find some explanation of the facts of nature, which would not
+run counter to the ecclesiastical system of the day. Various such
+interpretations were proposed, doubtless in an honest endeavour at
+reconciliation. Three of these deserve special notice: (1) Many
+able observers and diligent collectors of fossils persuaded themselves
+that these objects never belonged to organisms of any kind, but
+should be regarded as mere &ldquo;freaks of nature,&rdquo; having no more
+connexion with any once living creature than the frost patterns
+on a window. They were styled &ldquo;formed&rdquo; or &ldquo;figured&rdquo; stones,
+&ldquo;lapides sui generis,&rdquo; and were asserted to be due to some inorganic
+imitative process within the earth or to the influence of the stars.
+(2) Observers who could not resist the evidence of their senses that
+the fossil shells once belonged to living animals, and who, at the
+same time, felt the necessity of accounting for the presence of marine
+organisms in the rocks of which the dry land is largely built up,
+sought a way out of the difficulty by invoking the Deluge of Noah.
+Here was a catastrophe which, they said, extended over the whole
+globe, and by which the entire dry land was submerged even up to
+the tops of the high hills. True, it only lasted one hundred and fifty
+days, but so little were the facts then appreciated that no difficulty
+seems to have been generally felt in crowding the accumulation of
+the thousands of feet of fossiliferous formations into that brief space
+of time. (3) Some more intelligent men in Italy, recognizing that
+these interpretations could not be upheld, fell back upon the idea
+that the rocks in which fossil shells are imbedded might have been
+heaped up by repeated and vigorous eruptions from volcanic centres.
+Certain modern eruptions in the Aegean Sea and in the Bay of Naples
+had drawn attention to the rapidity with which hills of considerable
+size could be piled around an active crater. It was argued that if
+Monte Nuovo near Naples could have been accumulated to a height
+of nearly 500 ft. in two days, there seemed to be no reason against
+believing that, during the time of the Flood, and in the course of the
+<span class="pagenum"><a name="page641" id="page641"></a>641</span>
+centuries that have elapsed since that event, the whole of the fossiliferous
+rocks might have been deposited. Unfortunately for this
+hypothesis it ignored the fact that these rocks do not consist of
+volcanic materials.</p>
+
+<p>So long as the fundamental question remained in dispute as to
+the true character and history of the stratified portion of the earth&rsquo;s
+crust containing organic remains, geology as a science could not
+begin its existence. The diluvialists (those who relied on the hypothesis
+of the Flood) held the field during the 16th, 17th and a great
+part of the 18th century. They were looked on as the champions of
+orthodoxy; and, on that account, they doubtless wielded much
+more influence than would have been gained by them from the
+force of their arguments. Yet during those ages there were not
+wanting occasional observers who did good service in combating the
+prevalent misconceptions, and in preparing the way for the ultimate
+triumph of truth. It was more especially in Italy, where many of
+the more striking phenomena of geology are conspicuously displayed,
+that the early pioneers of the science arose, and that for several
+generations the most marked progress was made towards placing
+the investigations of the past history of the earth upon a basis of
+careful observation and scientific deduction. One of the first of
+<span class="sidenote">Leonardo da Vinci; Fracastorio; Falloppio.</span>
+these leaders was Leonardo da Vinci (1452-1519), who,
+besides his achievements in painting, sculpture, architecture
+and engineering, contributed some notable observations
+regarding the great problem of the origin of fossil
+shells. He ridiculed the notion that these objects could
+have been formed by the influence of the stars, and maintained
+that they had once belonged to living organisms, and therefore
+that what is now land was formerly covered by the sea.
+Girolamo Fracastorio (1483-1553) claimed that the shells could
+never have been left by the Flood, which was a mere temporary
+inundation, but that they proved the mountains, in which they
+occur, to have been successively uplifted out of the sea. On the
+other hand, even an accomplished anatomist like Gabriello Falloppio
+(1523-1562) found it easier to believe that the bones of elephants,
+teeth of sharks, shells and other fossils were mere earthy inorganic
+concretions, than that the waters of Noah&rsquo;s Flood could ever nave
+reached as far as Italy.</p>
+
+<p>By much the most important member of this early band of Italian
+writers was undoubtedly Nicolas Steno (1631-1687), who, though
+born in Copenhagen, ultimately settled in Florence.
+Having made a European reputation as an anatomist,
+<span class="sidenote">Nicolas Steno.</span>
+his attention was drawn to geological problems by finding
+that the rocks of the north of Italy contained what appeared to be
+sharks&rsquo; teeth closely resembling those of a dog-fish, of which he had
+published the anatomy. Cautiously at first, for fear of offending
+orthodox opinions, but afterwards more boldly, he proclaimed his
+conviction that those objects had once been part of living animals,
+and that they threw light on some of the past history of the earth.
+He published in 1669 a small tract, <i>De solido intra solidum naturaliter
+contento</i>, in which he developed the ideas he had formed of this
+history from an attentive study of the rocks. He showed that the
+stratified formations of the hills and valleys consist of such materials
+as would be laid down in the form of sediment in turbid water;
+that where they contain marine productions this water is proved
+to have been the sea; that diversities in their composition point to
+commingling of currents, carrying different kinds of sediment of
+which the heaviest would first sink to the bottom. He made original
+and important observations on stratification, and laid down some
+of the fundamental axioms in stratigraphy. He reasoned that as
+the original position of strata was approximately horizontal, when
+they are found to be steeply inclined or vertical, or bent into arches,
+they have been disrupted by subterranean exhalations, or by the
+falling in of the roofs of underground cavernous spaces. It is to
+this alteration of the original position of the strata that the inequalities
+of the earth&rsquo;s surface, such as mountains, are to be ascribed,
+though some have been formed by the outburst of fire, ashes and
+stones from inside the earth. Another effect of the dislocation has
+been to provide fissures, which serve as outlets for springs. Steno&rsquo;s
+anatomical training peculiarly fitted him for dealing authoritatively
+with the question of the nature and origin of the fossils contained
+in the rocks. He had no hesitation in affirming that, even if no shells
+had ever been found living in the sea, the internal structure of these
+fossils would demonstrate that they once formed parts of living
+animals. And not only shells, but teeth, bones and skeletons of
+many kinds of fishes had been quarried out of the rocks, while some
+of the strata had skulls, horns and teeth of land-animals. Illustrating
+his general principles by a sketch of what he supposed to have been
+the past history of Tuscany, he added a series of diagrams which
+show how clearly he had conceived the essential elements of stratigraphy.
+He thought he could perceive the records of six successive
+phases in the evolution of the framework of that country, and was
+inclined to believe that a similar chronological sequence would be
+found all over the world. He anticipated the objections that would
+be brought against his views on account of the insuperable difficulty
+in granting the length of time that would be required for all the
+geographical vicissitudes which his interpretation required. He
+thought that many of the fossils must be as old as the time of the
+general deluge, but he was careful not to indulge in any speculation
+as to the antiquity of the earth.</p>
+
+<p>To the Italian school, as especially typified in Steno, must be
+assigned the honour of having thus begun to lay firmly and truly
+the first foundation stones of the modern science of
+geology. The same school included Antonio Vallisneri
+<span class="sidenote">Lazzaro Moro.</span>
+(1661-1730), who surpassed his predecessors in his wider
+and more exact knowledge of the fossiliferous rocks that form the
+backbone of the Italian peninsula, which he contended were formed
+during a wide and prolonged submergence of the region, altogether
+different from the brief deluge of Noah. There was likewise Lazzaro
+Moro (1687-1740), who did good service against the diluvialists,
+but the fundamental feature of his system of nature lay in the
+preponderant part which, unaware of the great difference between
+volcanic materials and ordinary sediment, he assigned to volcanic
+action in the production of the sedimentary rocks of the earth&rsquo;s
+crust. He supposed that in the beginning the globe was completely
+surrounded with water, beneath which the solid earth lay as a smooth
+ball. On the third day of creation, however, vast fires were kindled
+inside the globe, whereby the smooth surface of stone was broken
+up, and portions of it, appearing above the water, formed the earliest
+land. From that time onward, volcanic eruptions succeeded each
+other, not only on the emerged land, but on the sea-floor, over which
+the ejected material spread in an ever augmenting thickness of
+sedimentary strata. In this way Moro carried the history of the
+stratified rocks beyond the time of the Flood back to the Creation,
+which was supposed to have been some 1600 years earlier; and he
+brought it down to the present day, when fresh sedimentary deposits
+are continually accumulating. He thus incurred no censure from
+the ecclesiastical guardians of the faith, and he succeeded in attracting
+increased public attention to the problems of geology. The
+influence of his teaching, however, was subsequently in great part
+due to the Carmelite friar Generelli, who published an eloquent
+exposition of Moro&rsquo;s views.</p>
+
+<p><i>The Cosmogonists and Theories of the Earth.</i>&mdash;While in Italy
+substantial progress was made in collecting information regarding
+the fossiliferous formations of that country, and in forming conclusions
+concerning them based upon more or less accurate observations,
+the tendency to mere fanciful speculation, which could not be
+wholly repressed in any country, reached a remarkable extravagance
+in England. In proportion as materials were yet lacking from
+which to construct a history of the evolution of our planet in accordance
+with the teaching of the church, imagination supplied the place
+of ascertained fact, and there appeared during the last twenty years
+of the 18th century a group of English cosmogonists, who, by the
+sensational character of their speculations, aroused general attention
+both in Britain and on the continent. It may be doubted, however,
+whether the effect of their writings was not to hinder the advance
+of true science by diverting men from the observation of nature into
+barren controversy over unrealities. It is not needful here to do
+more than mention the names of Thomas Burnet, whose <i>Sacred
+Theory of the Earth</i> appeared in 1681, and William Whiston, whose
+New Theory of the Earth was published in 1696. Hardly less fanciful
+than these writers, though his practical acquaintance with rocks
+and fossils was infinitely greater, was John Woodward, whose
+<i>Essay towards a Natural History of the Earth</i> dates from 1695. More
+important as a contribution to science was the catalogue of the large
+collection of fossils, which he had made from the rocks of England
+and which he bequeathed to the university of Cambridge. This
+catalogue appeared in 1728-1729 with the title of <i>An attempt towards
+a Natural History of the Fossils of England</i>.</p>
+
+<p>A striking contrast to these cosmogonists is furnished by another
+group, which arose in France and Germany, and gave to the world
+the first rational ideas concerning the probable primeval
+evolution of our globe. The earliest of these pioneers was
+<span class="sidenote">Descartes.</span>
+the illustrious philosopher René Descartes (1596-1650). He propounded
+a scheme of cosmical development in which he represented
+the earth, like the other planets, to have been originally a mass of
+glowing material like the sun, and to have gradually cooled on the
+outside, while still retaining an incandescent, self-luminous nucleus.
+Yet with this noble conception, which modern science has accepted,
+Descartes could not shake himself free from the time-honoured
+error in regard to the origin of volcanic action. He thought that
+certain exhalations within the earth condense into oil, which, when
+in violent motion, enters into the subterranean cavities, where it
+passes into a kind of smoke. This smoke is from time to time ignited
+by a spark of fire and, pressing violently against its containing
+walls, gives rise to earthquakes. If the flame breaks through to the
+surface at the top of a mountain, it may escape with enormous
+energy, hurling forth much earth mingled with sulphur or bitumen,
+and thus producing a volcano. The mountain might burn for a
+long time until at last its store of fuel in the shape of sulphur or
+bitumen would be exhausted. Not only did the philosopher refrain
+from availing himself of the high internal temperature of the globe
+as the source of volcanic energy, he even did not make use of it as
+the cause of the ignition of his supposed internal fuel, but speculated
+on the kindling of the subterranean fires by the spirits or gases
+setting fire to the exhalations, or by the fall of masses of rock and
+the sparks produced by their friction or percussion.</p>
+
+<p>The ideas of Descartes regarding planetary evolution were enlarged
+and made more definite by Wilhelm Gottfried Leibnitz (1646-1716),
+whose teaching has largely influenced all subsequent speculation
+<span class="pagenum"><a name="page642" id="page642"></a>642</span>
+on the subject. In his great tract, the <i>Protogaea</i> (published in 1749,
+<span class="sidenote">Leibnitz.</span>
+thirty-three years after his death), he traced the probable passage
+of our earth from an original condition of incandescent
+vapour into that of a smooth molten globe, which, by
+continuous cooling, acquired an external solid crust and rugose
+surface. He thought that the more ancient rocks, such as granite
+and gneiss, might be portions of the earliest outer crust; and that as
+the external solidification advanced, immense subterranean cavities
+were left which were filled with air and water. By the collapse of
+the roofs of these caverns, valleys might be originated at the surface,
+while the solid intervening walls would remain in place and form
+mountains. By the disruption of the crust, enormous bodies of
+water were launched over the surface of the earth, which swept vast
+quantities of sediment together, and thus gave rise to sedimentary
+deposits. After many vicissitudes of this kind, the terrestrial forces
+calmed down, and a more stable condition of things was established.</p>
+
+<p>An important feature in the cosmogony of Leibnitz is the
+prominent place which he assigned to organic remains in the stratified
+rocks of the crust. Ridiculing the foolish attempts to account for
+the presence of these objects by calling them &ldquo;sports of nature,&rdquo;
+he showed that they are to be regarded as historical monuments;
+and he adduced a number of instances wherein successive platforms
+of strata, containing organic remains, bear witness to a series of
+advances and retreats of the sea. He recognized that some of the
+fossils appeared to have nothing like them in the living world of
+to-day, but some analogous forms might yet be found, he thought,
+in still unexplored parts of the earth; and even if no living representatives
+should ever be discovered, many types of animals might
+have undergone transformation during the great changes which had
+affected the surface of the earth. In spite of his clear realization
+of the vast store of potential energy residing within the highly heated
+interior of the earth, Leibnitz continued to regard volcanic action
+as due to the combustion of inflammable substances enclosed within
+the terrestrial crust, such as stone-coal, naphtha and sulphur.</p>
+
+<p>Appealing to a much wider public than Descartes or Leibnitz, and
+basing his speculations on a wider acquaintance with the organic
+and inorganic realms of nature, G.L.L. de Buffon (1707-1788)
+was undoubtedly one of the most influential forces
+<span class="sidenote">Buffon.</span>
+that in Europe guided the growth of geological ideas during the
+18th century. He published in 1749 a <i>Theory of the Earth</i>, in which
+he adopted views similar to those of Descartes and Leibnitz as to
+planetary evolution; but though he realized the importance of
+fossils as records of former conditions of the earth&rsquo;s surface, he
+accounted for them by supposing that they had been deposited from
+a universal ocean, a large part of which had subsequently been
+engulfed into caverns in the interior of the globe. Thirty years
+later, after having laboured with skill and enthusiasm in all branches
+of natural history, he published another work, his famous <i>Époques
+de la nature</i> (1778), which is specially remarkable as the first attempt
+to deal with the history of the earth in a chronological manner, and
+to compute, on a basis of experiment, the antiquity of the several
+stages of this history. His experiments were made with globes of
+cast iron, and could not have yielded results of any value for his
+purpose; but in so far as his calculations were not mere random
+guesses but had some kind of foundation on experiment, they
+deserve respectful recognition. He divided the history of our earth
+into six periods of unequal duration, the whole comprising a period
+of some 70,000 or 75,000 years. He supposed that the stage of
+incandescence, before the globe had consolidated to the centre,
+lasted 2936 years, and that about 35,000 years elapsed before the
+surface had cooled sufficiently to be touched, and therefore to be
+capable of supporting living things. Terrestrial animal life, however,
+was not introduced until 55,000 or 60,000 years after the beginning
+of the world or about 15,000 years before our time. Looking into
+the future, he foresaw that, by continued refrigeration, our globe
+will eventually become colder than ice, and this fair face of nature,
+with its manifold varieties of plant and animal life, will perish after
+having existed for 132,000 years.</p>
+
+<p>Buffon&rsquo;s conception of the operation of the geological agents did
+not become broader or more accurate in the interval between the
+appearance of his two treatises. He still continued to believe in
+the lowering of the ocean by subsidence into vast subterranean
+cavities, with a consequent emergence of land. He still looked on
+volcanoes as due to the burning of &ldquo;pyritous and combustible
+stones,&rdquo; though he now called in the co-operation of electricity.
+He calculated that the first volcanoes could not arise until some
+50,000 years after the beginning of the world, by which time a
+sufficient extent of dense vegetation had been buried in the earth
+to supply them with fuel. He appears to have had but an imperfect
+acquaintance with the literature of his own time. At least there
+can be little doubt that had he availed himself of the labours of his
+own countryman, Jean Etienne Guettard (1715-1786), of Giovanni
+Arduķno (1714-1795) in Italy, and of Johann Gottlob Lehmann
+(d. 1767) and George Christian Füchsel (1722-1773) in Germany, he
+would have been able to give to his &ldquo;epochs&rdquo; a more definite succession
+of events and a greater correspondence with the facts of nature.</p>
+
+<p>Among the writers of the 18th century, who formed philosophical
+conceptions of the system of processes by which the life of our earth
+as a habitable globe is carried on, a foremost place must be assigned
+to James Hutton (1726-1797). Educated for the medical profession,
+<span class="sidenote">James Hutton.</span>
+he studied at Edinburgh and at Paris, and took his doctor&rsquo;s degree
+at Leiden. But having inherited a small landed property in
+Berwickshire, he took to agriculture, and after putting
+his land into excellent order, let his farm and betook
+himself to Edinburgh, there to gratify the scientific
+tastes which he had developed early in life. He had been more
+especially led to study minerals and rocks, and to meditate on the
+problems which they suggest as to the constitution and history of
+the earth. His journeys in Britain and on the continent of Europe
+had furnished him with material for reflection; and he had gradually
+evolved a system or theory in which all the scattered facts
+could be arranged so as to show their mutual dependence and their
+place in the orderly mechanism of the world. He used to discuss
+his views with one or two of his friends, but refrained from publishing
+them to the world until, on the foundation of the Royal Society of
+Edinburgh, he communicated an outline of his doctrine to that
+learned body in 1785. Some years later he expanded this first essay
+into a larger work in two volumes, which were published in 1795
+with the title of <i>Theory of the Earth, with Proofs and Illustrations</i>.</p>
+
+<p>Hutton&rsquo;s teaching has exercised a profound influence on modern
+geology. This influence, however, has arisen less from his own
+writings than from the account of his doctrines given by
+his friend John Playfair in the classic work entitled
+<span class="sidenote">John Playfair.</span>
+<i>Illustrations of the Huttonian Theory</i>, published in 1802.
+Hutton wrote in so prolix and obscure a style as rather to repel than
+attract readers. Playfair, on the other hand, expressed himself in
+such clear and graceful language as to command general attention,
+and to gain wide acceptance for his master&rsquo;s views. Unlike the
+older cosmogonists, Hutton refrained from trying to explain the
+origin of things, and from speculations as to what might possibly
+have been the early history of our globe. He determined from the
+outset to interpret the past by what can be seen to be the present
+order of nature; and he refused to admit the operation of causes
+which cannot be shown to be part of the actual terrestrial system.
+Like other observers who had preceded him, he recognized in the
+various rocks composing the dry land evidence of former geographical
+conditions very different from those which now prevail. He saw
+that the vast majority of rocks consist of hardened sediments and
+must have been deposited in the sea. He could distinguish among
+them an older or Primary series, and a younger or Secondary series;
+and did not dispute the existence of a Tertiary series claimed by
+Peter Simon Pallas (1741-1811). He believed that these various
+aqueous accumulations had been consolidated by subterranean heat,
+that the oldest and lowest rocks had suffered most from this action,
+that into these more deep-seated masses subsequent veins and
+larger bodies of molten matter were injected from below, and thus
+that what was originally loose detritus eventually became changed
+in such crystalline schists as are now found in mountain-chains.
+In the course of these terrestrial revolutions sedimentary strata,
+originally more or less nearly horizontal, have been pushed upward,
+dislocated, crumpled, placed on end, and even elevated to form
+ranges of lofty mountains. Hutton looked upon these disturbances
+as due to the expansive power of subterranean heat; but he did not
+attempt to sketch the mechanism of the process, and he expressly
+declined to offer any conjecture as to how the land so elevated
+remains in that position. He thought that the interior of our
+planet may &ldquo;be a fluid mass, melted, but unchanged by the action
+of heat&rdquo;; and, far from connecting volcanoes with the combustion of
+inflammable substances, as had been the prevalent belief for so many
+centuries, he looked upon them as a beneficent provision of &ldquo;spiracles
+to the subterranean furnace, in order to prevent the unnecessary
+elevation of land and fatal effects of earthquakes.&rdquo;</p>
+
+<p>A distinguishing feature of the Huttonian philosophy is to be
+seen in the breadth of its conceptions regarding the geological
+operations continually in progress on the surface of the globe.
+Hutton saw that the land is undergoing a ceaseless process of degradation,
+through the influence of the air, frost, rain, rivers and the sea,
+and that in course of time, if no countervailing agency should intervene,
+the whole of the dry land will be washed away into the sea.
+But he also perceived that this universal erosion is not everywhere
+carried on at the same rate; that it is specially active along the
+channels of torrents and rivers, and that, owing to this difference
+these channels are gradually deepened and widened, until the
+complicated valley-system of a country is carved out. He recognized
+that the detritus worn away from the land must be spread out over
+the floor of the sea, so as to form there strata similar to those that
+compose most of the dry land. As he could detect in the structure
+of land convincing evidence that former sea floors had been elevated
+to form the continents and islands of to-day, he could look forward
+to future ages, when the same subterranean agency which had raised
+up the present land would again be employed to uplift the bed of
+the existing ocean, thus to renew the surface of our earth as a
+habitable globe, and to start a fresh cycle of erosion and deposition.</p>
+
+<p>Though Hutton was not unaware that organic remains abound in
+many of the stratified rocks, he left them out of consideration in
+the elaboration of his theory. It was otherwise with
+one of his French contemporaries, the illustrious J.B.
+<span class="sidenote">Lamarck.</span>
+Lamarck (1744-1829), who, after having attained great eminence as
+a botanist, turned to zoology when he was nearly fifty years of age,
+and before long rose to even greater distinction in that department
+<span class="pagenum"><a name="page643" id="page643"></a>643</span>
+of science. His share in the classification and description of the
+mollusca and in founding invertebrate palaeontology, his theory
+of organic evolution and his philosophical treatment of many
+biological questions have been tardily recognized, but his contributions
+to geology have been less generally acknowledged. When he
+accepted the &ldquo;professorship of zoology; of insects, of worms and of
+microscopic animals&rdquo; at the Museum of Natural History, Paris,
+in 1793, he at once entered with characteristic ardour and capacity
+into the new field of research then opened to him. In dealing with
+the mollusca he considered not merely the living but also the extinct
+forms, especially the abundant, varied and well-preserved genera
+and species furnished by the Tertiary deposits of the Paris basin,
+of which he published descriptions and plates that proved of essential
+service in the stratigraphical work of Cuvier and Alexandre
+Brongniart (1770-1847). His labours among these relics of ancient
+seas and lakes led him to ponder over the past history of the globe,
+and as he was seldom dilatory in making known the opinions he had
+formed, he communicated some of his conclusions to the National
+Institute in 1799. These, including a further elaboration of his
+views, he published in 1802 in a small volume entitled Hydrogéologie.</p>
+
+<p>This treatise, though it did not reach a second edition and has
+never been reprinted, deserves an honourable place in geological
+literature. Its object, the author states, was to present some important
+and novel considerations, which he thought should form
+the basis of a true theory of the earth. He entirely agreed with the
+doctrine of the subaerial degradation of the land and the erosion of
+valleys by running water. Not even Playfair could have stated this
+doctrine more emphatically, and it is worthy of notice that Playfair&rsquo;s
+<i>Illustrations of the Huttonian Theory</i> appeared in the same year
+with Lamarck&rsquo;s book. The French naturalist, however, carried his
+conclusions so far as to take no account of any great movements of
+the terrestrial crust, which might have produced or modified the
+main physical features of the surface of the globe. He thought that
+all mountains, except such as were thrown up by volcanic agency or
+local accidents, have been cut out of plains, the original surfaces of
+which are indicated by the crests and summits of these elevations.</p>
+
+<p>Lamarck, in reflecting upon the wide diffusion of fossil shells and
+the great height above the sea at which they are found, conceived
+the extraordinary idea that the ocean basin has been scoured out
+by the sea, and that, by an impulse communicated to the waters
+through the influence chiefly of the moon, the sea is slowly eating
+away the eastern margins of the continents, and throwing up detritus
+on their western coasts, and is thus gradually shifting its basin
+round the globe. He would not admit the operation of cataclysms;
+but insisted as strongly as Hutton on the continuity of natural
+processes, and on the necessity of explaining former changes of the
+earth&rsquo;s surface by causes which can still be seen to be in operation.
+As might be anticipated from his previous studies, he brought living
+things and their remains into the forefront of his theory of the earth.
+He looked upon fossils as one of the chief means of comprehending
+the revolutions which the surface of the earth has undergone;
+and in his little volume he again and again dwells on the vast
+antiquity to which these revolutions bear witness. He acutely
+argues, from the condition of fossil shells, that they must have lived
+and died where their remains are now found.</p>
+
+<p>In the last part of his treatise Lamarck advances some peculiar
+opinions in physics and chemistry, which he had broached eighteen
+years before, but which had met with no acceptance among the
+scientific men of his time. He believed that the tendency of all
+compound substances is to decay, and thereby to be resolved into
+their component constituents. Yet he saw that the visible crust
+of the earth consists almost wholly of compound bodies. He therefore
+set himself to solve the problem thus presented. Perceiving
+that the biological action of living organisms is constantly forming
+combinations of matter, which would never have otherwise come
+into existence, he proceeded to draw the extraordinary conclusion
+that the action of plant and animal life (the <i>Pouvoir de la vie</i>) upon
+the inorganic world is so universal and so potent, that the rocks and
+minerals which form the outer part of the earth&rsquo;s crust are all,
+without exception, the result of the operations of once living bodies.
+Though this sweeping deduction must be allowed to detract from
+the value of Lamarck&rsquo;s work, there can be no doubt that he realized,
+more fully than any one had done before him, the efficacy of plants
+and animals as agents of geological change.</p>
+
+<p>The last notable contributor to the cosmological literature of
+geology was another illustrious Frenchman, the comparative anatomist
+Cuvier (1769-1832). He was contemporary with
+Lamarck, but of a very different type of mind. The
+<span class="sidenote">Cuvier.</span>
+brilliance of his speculations, and the charm with which he expounded
+them, early gained for him a prominent place in the society of Paris.
+He too was drawn by his zoological studies to investigate fossil
+organic remains, and to consider the former conditions of the earth&rsquo;s
+surface, of which they are memorials. It was among the vertebrate
+organisms of the Paris basin that he found his chief material, and
+from them that he prepared the memoirs which led to him being
+regarded as the founder of vertebrate palaeontology. But beyond
+their biological interest, they awakened in him a keen desire to
+ascertain the character and sequence of the geographical revolutions
+to which they bear witness. He approached the subject from an
+opposite and less philosophical point of view than that of Lamarck,
+coming to it with certain preconceived notions, which affected all
+his subsequent writings. While Lamarck was by instinct an evolutionist,
+who sought to trace in the history of the past the operation
+of the same natural processes as are still at work, Cuvier, on the
+other hand, was a catastrophist, who invoked a succession of vast
+cataclysms to account for the interruptions in the continuity of the
+geological record.</p>
+
+<p>In a preliminary <i>Discourse</i> prefixed to his <i>Recherches sur les
+ossemens fossiles</i> (1821) Cuvier gave an outline of what he conceived
+to have been the past history of our globe, so far as he had been able
+to comprehend it from his investigations of the Tertiary formations
+of France. He believed that in that history evidence can be
+recognized of the occurrence of many sudden and disastrous revolutions,
+which, to judge from their effects on the animal life of the
+time, must have exceeded in violence anything we can conceive at
+the present day, and must have been brought about by other agencies
+than those which are now in operation. Yet, in spite of these
+catastrophes, he saw that there has been an upward progress in the
+animal forms inhabiting the globe, until the series ended in the
+advent of man. He could not, however, find any evidence that one
+species has been developed from another, for in that case there should
+have been traces of intermediate forms among the stratified formations,
+where he affirmed that they had never been found. A
+prominent position in the <i>Discourse</i> is given to a strenuous argument
+to disprove the alleged antiquity of some nations, and to show that
+the last great catastrophe occurred not more than some 5000 or
+6000 years ago. Cuvier thus linked himself with those who in
+previous generations had contended for the efficacy of the Deluge.
+But his researches among fossil animals had given him a far wider
+outlook into the geological past, and had opened up to him a succession
+of deeply interesting problems in the history of life upon the
+earth, which, though he had not himself material for their solution,
+he could foresee would be cleared up in the future.</p>
+
+<p><i>Gradual Shaping of Geology into a Distinct Branch of Science.</i>&mdash;It
+will be seen from the foregoing historical sketch that it was only
+after the lapse of long centuries, and from the labours of many
+successive generations of observers and writers, that what we now
+know as the science of geology came to be recognized as a distinct
+department of natural knowledge, founded upon careful and extended
+study of the structure of the earth, and upon observation of
+the natural processes, which are now at work in changing the earth&rsquo;s
+surface. The term &ldquo;geology,&rdquo;<a name="fa1c" id="fa1c" href="#ft1c"><span class="sp">1</span></a> descriptive of this branch of the
+investigation of nature, was not proposed until the last quarter of
+the 18th century by Jean André De Luc (1727-1817) and Horace
+Benedict De Saussure (1740-1749). But the science was then in a
+markedly half-formed condition, theoretical speculation still in large
+part supplying the place of deductions from a detailed examination
+of actual fact. In 1807 a few enterprising spirits founded the
+Geological Society of London for the special purpose of counteracting
+the prevalent tendency and confining their intention &ldquo;to
+investigate the mineral structure of the earth.&rdquo; The cosmogonists
+and framers of Theories of the Earth were succeeded by other schools
+of thought. The Catastrophists saw in the composition of the crust
+of the earth distinct evidence that the forces of nature were once
+much more stupendous in their operation than they now are, and
+that they had from time to time devastated the earth&rsquo;s surface;
+extirpating the races of plants and animals, and preparing the ground
+for new creations of organized life. Then came the Uniformitarians,
+who, pushing the doctrines of Hutton to an extreme which he did
+not propose, saw no evidence that the activity of the various geological
+causes has ever seriously differed from what it is at present.
+They were inclined to disbelieve that the stratified formations of
+the earth&rsquo;s crust furnish conclusive evidence of a gradual progression,
+from simple types of life in the oldest strata to the most
+highly developed forms in the youngest; and saw no reason why
+remains of the higher vertebrates should not be met with among
+the Palaeozoic formations. Sir Charles Lyell (1797-1875) was the
+great leader of this school. His admirably clear and philosophical
+presentations of geological facts which, with unwearied industry,
+he collected from the writings of observers in all parts of the world,
+impressed his views upon the whole English-speaking world, and
+gave to geological science a coherence and interest which largely
+accelerated its progress. In his later years, however, he frankly
+accepted the views of Darwin in regard to the progressive character
+of the geological record.</p>
+
+<p>The youngest of the schools of geological thought is that of the
+Evolutionists. Pointing to the whole body of evidence from inorganic
+and organic nature, they maintain that the history of our
+planet has been one of continual and unbroken development from
+the earliest cosmical beginnings down to the present time, and that
+the crust of the earth contains an abundant, though incomplete,
+record of the successive stages through which the plant and animal
+<span class="pagenum"><a name="page644" id="page644"></a>644</span>
+kingdoms have reached their existing organization. The publication
+of Darwin&rsquo;s <i>Origin of Species</i> in 1859, in which evolution was made
+the key to the history of the animal and vegetable kingdoms, produced
+an extraordinary revolution in geological opinion. The older
+schools of thought rapidly died out, and evolution became the
+recognized creed of geologists all over the world.</p>
+
+<p><i>Development of Opinion regarding Igneous Rocks.</i>&mdash;So long as the
+idea prevailed that volcanoes are caused by the combustion of
+inflammable substances underground, there could be no rational
+conception of volcanic action and its products. Even so late as
+the middle of the 18th century, as above remarked, such a good
+observer as Lazzaro Moro drew so little distinction between volcanic
+and other rocks that he could believe the fossiliferous formations
+to have been mainly formed of materials ejected from eruptive vents.
+After his time the notion continued to prevail that all the rocks which
+form the dry land were laid down under water. Even streams of
+lava, which were seen to flow from an active crater, were regarded
+only as portions of sedimentary or other rocks, which had been
+melted by the fervent heat of the burning inflammable materials
+that had been kindled underground. In spite of the speculations
+of Descartes and Leibnitz, it was not yet generally comprehended
+that there exists beneath the terrestrial crust a molten magma,
+which, from time to time, has been injected into that crust, and has
+pierced through it, so as to escape at the surface with all the energy
+of an active volcano. What we now recognize to be memorials of
+these former injections and propulsions were all confounded with the
+rocks of unquestionably aqueous origin. The last great teacher by
+whom these antiquated doctrines were formulated into a system
+<span class="sidenote">Werner.</span>
+and promulgated to the world was Abraham Gottlob
+Werner (1749-1815), the most illustrious German mineralogist
+and geognost of the second half of the 18th century. While
+still under twenty-six years of age, he was appointed teacher of
+mining and mineralogy at the Mining Academy of Freiberg in Saxony&mdash;a
+post which he continued to fill up to the end of his life. Possessed
+of great enthusiasm for his subject, clear, methodical and eloquent
+in his exposition of it, he soon drew around him men from all parts
+of the world, who repaired to study under the great oracle of what
+he called geognosy (Gr. <span class="grk" title="gź">&#947;&#8134;</span>, the earth, <span class="grk" title="gnōsis">&#947;&#957;&#8182;&#963;&#953;&#962;</span>, knowledge) or earth-knowledge.
+Reviving doctrines that had been current long before
+his time, he taught that the globe was once completely surrounded
+with an ocean, from which the rocks of the earth&rsquo;s crust were
+deposited as chemical precipitates, in a certain definite order over
+the whole planet. Among these &ldquo;universal formations&rdquo; of aqueous
+origin were included many rocks, which have long been recognized
+to have been once molten, and to have risen from below into the
+upper parts of the terrestrial crust. Werner, following the old
+tradition, looked upon volcanoes as modern features in the history
+of the planet, which could not have come into existence until a
+sufficient amount of vegetation had been buried to furnish fuel for
+their maintenance. Hence he attached but little importance to
+them, and did not include in his system of rocks any division of
+volcanic or igneous materials. From the predominant part assigned
+by him to the sea in the accumulation of the materials of the visible
+part of the earth, Werner and his school were known as &ldquo;Neptunists.&rdquo;</p>
+
+<p>But many years before the Saxon professor began to teach, clear
+evidence had been produced from central France that basalt, one
+of the rocks claimed by him as a chemical precipitate and
+a universal formation, is a lava which has been poured
+<span class="sidenote">Origin of basalt.</span>
+out in a molten state at various widely separated periods
+of time and at many different places. So far back as 1752 J.E.
+Guettard (1715-1786) had shown that the basaltic rocks of Auvergne
+are true lavas, which have flowed out in streams from groups of
+once active cones. Eleven years later the observation was confirmed
+and greatly extended by Nicholas Desmarest (1725-1815), who,
+during a long course of years, worked out and mapped the complicated
+volcanic records of that interesting region, and demonstrated
+to all who were willing impartially to examine the evidence the true
+volcanic nature of basalt. These views found acceptance from some
+observers, but they were vehemently opposed by the followers of
+Werner, who, by the force of his genius, made his theoretical conceptions
+predominate all over Europe. The controversy as to the
+origin of basalt was waged with great vigour during the later decades
+of the 18th century. Desmarest took no part in it. He had accumulated
+such conclusive proof of the correctness of his deductions,
+and had so fully expounded the clearness of the evidence in their
+favour furnished by the region of Auvergne, that, when any one
+came to consult him on the subject, he contented himself with giving
+the advice to &ldquo;go and see.&rdquo; While the debate was in progress
+on the continent, the subject was approached from a new and
+independent point of view by Hutton in Scotland. This illustrious
+philosopher, as already stated, realized the importance of the internal
+heat of the globe in consolidating the sedimentary rocks, and believed
+that molten material from the earth&rsquo;s interior has been protruded
+from below into the overlying crust. Some of the material thus
+injected could be recognized, he thought, in granite and in the
+various dark massive rocks which, known in Scotland under the
+name of &ldquo;whinstone,&rdquo; were afterwards called &ldquo;Trap,&rdquo; and are now
+grouped under various names, such as basalt, dolerite and diorite.
+So important a share did Hutton thus assign to the internal heat in
+the geological evolution of the planet, that he and those who adopted
+the same opinions were styled &ldquo;Plutonists,&rdquo; or, especially where
+they concerned themselves with the volcanic origin of basalt, &ldquo;Vulcanists.&rdquo;
+The geological world was thus divided into two hostile
+camps, that of the Neptunists or Wernerians, and that of the
+Plutonists, Vulcanists or Huttonians.</p>
+
+<p>After many years of futile controversy the first serious weakening
+of the position of the dominant Neptunist school arose from the
+defection of some of the most prominent of Werner&rsquo;s pupils. In
+particular Jean Franēois D&rsquo;Aubuisson de Voisins (1769-1819), who
+had written a treatise on the aqueous origin of the basalts of Saxony,
+went afterwards to Auvergne, where he was speedily a convert to
+the views expounded by Desmarest as to the volcanic nature of
+basalt. Having thus to relinquish one of the fundamental articles
+of the Freiberg faith, he was subsequently led to modify his adherence
+to others until, as he himself confessed, his views came almost wholly
+to agree with those of Hutton. Not less complete, and even more
+important, was the conversion of the great Leopold von Buch (1774-1853).
+He, too, was trained by Werner himself, and proved to be
+the most illustrious pupil of the Saxon professor. Full of admiration
+for the Neptunism in which he had been reared, he, in his earliest
+separate work, maintained the aqueous origin of basalt, and contrasted
+the wide field opened up to the spirit of observation by his
+master&rsquo;s teaching with the narrower outlook offered by &ldquo;the volcanic
+theory.&rdquo; But a little further acquaintance with the facts of nature
+led Von Buch also to abandon his earlier prepossessions. It was a
+personal visit to the volcanic region of Auvergne that first opened
+his eyes, and led him to recant what he had believed and written
+about basalt. But the abandonment of so essential a portion of the
+Wernerian creed prepared the way for further relinquishments.
+When a few years later he went to Norway and found to his astonishment
+that granite, which he had been taught to regard as the oldest
+chemical precipitate from the universal ocean, could there be seen
+to have broken through and metamorphosed fossiliferous limestones,
+and to have sent veins into them, his faith in Werner&rsquo;s order of the
+succession of the rocks in the earth&rsquo;s crust received a further momentous
+shock. While one after another of the Freiberg doctrines
+crumbled away before him, he was now able to interrogate nature
+on a wider field than the narrow limits of Saxony, and he was thus
+gradually led to embrace the tenets of the opposite school. His
+commanding position, as the most accomplished geologist on the
+continent, gave great importance to his recantation of the Neptunist
+creed. His defection indeed was the severest blow that this creed
+had yet sustained. It may be said to have rung the knell of
+Wernerianism, which thereafter rapidly declined in influence, while
+Plutonism came steadily to the front, where it has ever since remained.</p>
+
+<p>Although Desmarest had traced in Auvergne a long succession
+of volcanic eruptions, of which the oldest went back to a remote
+period of time, and although he had shown that this succession,
+coupled with the records of contemporaneous denudation, might
+be used in defining epochs of geological history, it was not until
+many years after his day that volcanic action came to be recognized
+as a normal part of the mechanism of our globe, which had been in
+operation from the remotest past, and which had left numerous
+records among the rocks of the terrestrial crust. During the progress
+of the controversy between the two great opposing factions in the
+later portion of the 18th and the first three decades of the 19th
+century, those who espoused the Vulcanist cause were intent on
+proving that certain rocks, which are intercalated among the
+stratified formations and which were claimed by the Neptunists as
+obviously formed by water, are nevertheless of truly igneous origin.
+These observers fixed their eyes on the evidence that the material of
+such rocks, instead of having been deposited from aqueous solution,
+had once been actually molten, and had in that condition been thrust
+between the strata, had enveloped portions of them, and had indurated
+or otherwise altered them. They spoke of these masses
+as &ldquo;unerupted lavas&rdquo;; and undoubtedly in innumerable instances
+they were right. But their zeal to establish an intrusive origin led
+them to overlook the proofs that some intercalated sheets of igneous
+material had not been injected into the strata, but had been poured
+out at the surface as truly volcanic discharges, and therefore belonged
+to the ancient periods represented by the strata between which they
+are interposed. It may readily be supposed that any proofs of the
+contemporaneous intercalation of such sheets would be eagerly
+seized upon by the Neptunists in favour of their aqueous theory.
+The influence of the ancient belief that &ldquo;burning mountains&rdquo;
+could only rise from the combustion of subterranean inflammable
+materials extended even into the ranks of the Vulcanists, so far at
+least as to lead to a general acquiescence in the assumption that
+volcanoes appeared to belong to a late phase in the history of the
+planet. It was not until after considerable progress had been made
+in determining the palaeontological distinctions and order of succession
+of the stratified formations of the earth&rsquo;s crust that it became
+possible to trace among these formations a succession of volcanic
+episodes which were contemporaneous with them. In no part of
+the world has an ampler record of such episodes been preserved than
+in the British Isles. It was natural, therefore, that the subject
+should there receive most attention. As far back as 1820 Ami Boué
+(1794-1881) showed that the Old Red Sandstone of Scotland includes
+a great series of volcanic rocks, and that other rocks of volcanic
+origin are associated with the Carboniferous formations. H.T.
+<span class="pagenum"><a name="page645" id="page645"></a>645</span>
+de la Beche (1796-1855) afterwards traced proofs of contemporaneous
+eruptions among the Devonian rocks of the south-west of England.
+Adam Sedgwick (1785-1873) showed, first in the Lake District,
+and afterwards in North Wales, the presence of abundant volcanic
+sheets among the oldest divisions of the Palaeozoic series; while
+Roderick Impey Murchison (1792-1871) made similar discoveries
+among the Lower Silurian rocks. From the time of these pioneers
+the volcanic history of the country has been worked out by many
+observers until it is now known with a fulness as yet unattained
+in any other region.</p>
+
+<p><i>Growth of Opinion regarding Earthquakes.</i>&mdash;We have seen how
+crude were the conceptions of the ancients regarding the causes of
+volcanic action, and that they connected volcanoes and earthquakes
+as results of the commotion of wind imprisoned within subterranean
+caverns and passages. One of the earliest treatises, in which the
+phenomena of terrestrial movements were discussed in the spirit
+of modern science, was the posthumous collection of papers by
+Robert Hooke (1635-1703), entitled <i>Lectures and Discourses of
+Earthquakes and Subterranean Eruptions</i>, where the probable agency
+of earthquakes in upheaving and depressing land is fully considered,
+but without any definite pronouncement as to the author&rsquo;s conception
+of its origin. Hooke still associated earthquakes with volcanic
+action, and connected both with what he called &ldquo;the general congregation
+of sulphurous subterraneous vapours.&rdquo; He conceived
+that some kind of &ldquo;fermentation&rdquo; takes place within the earth,
+and that the materials which catch fire and give rise to eruptions
+or earthquakes are analogous to those that constitute gunpowder.
+The first essay wherein earthquakes are treated from the modern
+point of view as the results of a shock that sends waves through the
+crust of the earth was written by the Rev. John Michell, and communicated
+to the Royal Society in the year 1760. Still under the
+old misconception that volcanoes are due to the combustion of
+inflammable materials, which he thought might be set on fire by the
+spontaneous combustion of pyritous strata, he supposed that, by the
+sudden access of large bodies of water to these subterranean fires,
+vapour is produced in such quantity and with such force as to give
+rise to the shock. From the centre of origin of this shock waves,
+he thought, are propagated through the earth, which are largest
+at the start and gradually diminish as they travel outwards. By
+drawing lines at different places in the direction of the track of these
+waves, he believed that the place of common intersection of these
+lines would be nearly the centre of the disturbance. In this way he
+showed that the great Lisbon earthquake of 1755 had its focus under
+the Atlantic, somewhere between the latitudes of Lisbon and Oporto,
+and he estimated that the depth at which it originated could not
+be much less than 1 m., and probably did not exceed 3 m. Michell,
+however, misconceived the character of the waves which he described,
+seeing that he believed them to be due to the actual propagation of
+the vapour itself underneath the surface of the earth. A century
+had almost passed after the date of his essay before modern scientific
+methods of observation and the use of recording instruments began
+to be applied to the study of earthquake phenomena. In 1846 Robert
+Mallet (1810-1881) published an important paper &ldquo;On the Dynamics
+of Earthquakes&rdquo; in the <i>Transactions of the Royal Irish Academy</i>.
+From that time onward he continued to devote his energies to the
+investigation, studying the effects of the Calabrian earthquake of
+1857, experimenting on the transmission of waves of shock through
+various materials, caused by exploding charges of gunpowder, and
+collecting all the information to be obtained on the subject. His
+writings, and especially his work in two volumes on <i>The First
+Principles of Observational Seismology</i>, must be regarded as having
+laid the foundations of this branch of modern geology (see <span class="sc"><a href="#artlinks">Earthquake</a></span>;
+<span class="sc"><a href="#artlinks">Seismometer</a></span>).</p>
+
+<p><i>History of the Evolution of Stratigraphical Geology.</i>&mdash;Men had long
+been familiar with the evidence that the present dry land once lay
+under the sea, before they began to realize that the rocks, of which
+the land consists, contain a record of many alternations of land and
+sea, and relics of a long succession of plants and animals from early
+and simple types up to the manifold and complex forms of to-day.
+In countries where coal-mining had been prosecuted for generations,
+it had been recognized that the rocks consist of strata superposed
+on each other in a definite order, which was found to extend over
+the whole of a district. As far back as 1719 John Strachey drew
+attention to this fact in a communication published in the <i>Philosophical
+Transactions</i>. John Michell (1760), in the paper on earthquakes
+already cited, showed that he had acquired a clear understanding
+of the order of succession among stratified formations, and
+perceived that to disturbances of the terrestrial crust must be ascribed
+the fact that the lower or older and more inclined strata form the
+mountains, while the younger and more horizontal strata are spread
+over the plains.</p>
+
+<p>In Italy G. Arduķno (1713-1795) classified the rocks in the north
+of the peninsula as Primitive, Secondary, Tertiary and Volcanic.
+A similar threefold order was announced for the Harz and Erzgebirge
+by J.G. Lehmann in 1756. He recognized in that region an ancient
+series of rocks in inclined or vertical strata, which rise to the tops
+of the hills and descend to an unknown depth into the interior.
+These masses, he thought, were contemporaneous with the making
+of the world. Next came the Flötzgebirge, consisting of younger
+sediments, disposed in flat or gently inclined sheets which overlie
+the first and more disturbed series, and are full of petrified remains
+of plants and animals. Lastly he included the mountains which
+have from time to time been formed by local accidents. Still more
+advanced were the conceptions of G.C. Füchsel, who in the year
+1762 published in Latin <i>A History of the Earth and the Sea, based on
+a History of the Mountains of Thuringia</i>; and in 1773, in German,
+a <i>Sketch of the most Ancient History of the Earth and Man</i>. In these
+works he described the stratigraphical relations and general characters
+of the various geological formations in his little principality;
+and taking them as indicative of a general order of succession, he
+traced what he believed to have been a series of revolutions through
+which the earth has passed. In interpreting this geological history,
+he laid great stress on the evidence of the fossils contained in the
+rocks. He recognized that the various formations differ from each
+other in their enclosed organic remains, and that from these differences
+the existence of former sea-bottoms and land surfaces can
+be determined.</p>
+
+<p>The labours of these pioneers paved the way for the advent
+of Werner. Though the system evolved by this teacher claimed to
+discard theory and to be established on a basis of observed facts,
+it rested on a succession of hypotheses, for which no better foundation
+could be shown than the belief of their author in their validity.
+Starting from the extremely limited stratigraphical range displayed
+in the geological structure of Saxony, he took it as a type for the rest
+of the globe, persuading himself and impressing upon his followers
+that the rocks of that small kingdom were to be taken as examples
+of his &ldquo;universal formations.&rdquo; The oldest portion of the series,
+classed by him as &ldquo;Primitive,&rdquo; consisted of rocks which he maintained
+had been deposited from chemical solution. Yet they
+included granite, gneiss, basalt, porphyry and serpentine, which,
+even in his own day, were by many observers correctly regarded
+as of igneous origin. A later group of rocks, to which he gave the
+name of &ldquo;Transition,&rdquo; comprised, in his belief, partly chemical,
+partly mechanical sediments, and contained the earliest fossil
+organic remains. A third group, for which he reserved Lehmann&rsquo;s
+name &ldquo;Flötz,&rdquo; was made up chiefly of mechanical detritus, while
+youngest of all came the &ldquo;Alluvial&rdquo; series of loams, clays, sands,
+gravels and peat. It was by the gradual subsidence of the ocean
+that, as he believed, the general mass of the dry land emerged, the
+first-formed rocks being left standing up, sometimes on end, to form
+the mountains, while those of later date, less steeply inclined,
+occupied successively lower levels down to the flat alluvial accumulations
+of the plains. Neither Werner, nor any of his followers,
+ventured to account for what became of the water as the sea-level
+subsided, though, in despite of their antipathy to anything like
+speculation, they could not help suggesting, as an answer to the
+cogent arguments of their opponents, that &ldquo;one of the celestial
+bodies which sometimes approach near to the earth may have been
+able to withdraw a portion of our atmosphere and of our ocean.&rdquo;
+Nor was any attempt made to explain the extraordinary nature of
+the supposed chemical precipitates of the universal ocean. The
+progress of inquiry even in Werner&rsquo;s lifetime disproved some of
+the fundamental portions of his system. Many of the chemical
+precipitates were shown to be masses that had been erupted in a
+molten state from below. His order of succession was found not
+to hold good; and though he tried to readjust his sequence and to
+introduce into it modifications to suit new facts, its inherent artificiality
+led to its speedy decline after his death. It must be conceded,
+however, that the stress which he laid upon the fact that the
+rocks of the earth&rsquo;s crust were deposited in a definite order had an
+important influence in directing attention to this subject, and in
+preparing the way for a more natural system, based not on mere
+mineralogical characters, but having regard to the organic remains,
+which were now being gathered in ever-increasing numbers and
+variety from stratified formations of many different ages and from
+all parts of the globe.</p>
+
+<p>It was in France and in England that the foundations of stratigraphy,
+based upon a knowledge of organic remains, were first
+successfully laid. Abbé J.L. Giraud-Soulavie (1752-1813), in his
+<i>Histoire naturelle de la France méridionale</i>, which appeared in seven
+volumes, subdivided the limestones of Vivarais into five ages, each
+marked by a distinct assemblage of shells. In the lowest strata,
+representing the first age, none of the fossils were believed by him
+to have any living representatives, and he called these rocks &ldquo;Primordial.&rdquo;
+In the next group a mingling of living with extinct forms
+was observable. The third age was marked by the presence of
+shells of still existing species. The strata of the fourth series were
+characterized by carbonaceous shales or slates, containing remains
+of primordial vegetation, and perhaps equivalents of the first three
+calcareous series. The fifth age was marked by recent deposits
+containing remains of terrestrial vegetation and of land animals.
+It is remarkable that these sagacious conclusions should have been
+formed and published at a time when the geologists of the Continent
+were engaged in the controversy about the origin of basalt, or in
+disputes about the character and stratigraphical position of the
+supposed universal formations, and when the interest and importance
+of fossil organic remains still remained unrecognized by the vast
+majority of the combatants.</p>
+
+<p>The rocks of the Paris basin display so clearly an orderly
+arrangement, and are so distinguished for the variety and perfect
+<span class="pagenum"><a name="page646" id="page646"></a>646</span>
+preservation of their enclosed organic remains, that they could not
+fail to attract the early notice of observers. J. É. Guettard, G.F.
+Rouelle (1703-1770), N. Desmarest, A.L. Lavoisier (1743-1794)
+and others made observations in this interesting district. But it
+was reserved for Cuvier (1769-1832) and A. Brongniart (1770-1847)
+to work out the detailed succession of the Tertiary formations, and
+to show how each of these is characterized by its own peculiar
+assemblage of organic remains. The later progress of investigation
+has slightly corrected and greatly amplified the tabular arrangement
+established by these authors in 1808, but the broad outlines of the
+Tertiary stratigraphy of the Paris basin remain still as Cuvier and
+Brongniart left them. The most important subsequent change
+in the classification of the Tertiary formations was made by Sir
+Charles Lyell, who, conceiving in 1828 the idea of a classification
+of these rocks by reference to their relative proportions of living
+and extinct species of shells, established, in collaboration with
+G.P. Deshayes, the now universally accepted divisions Eocene,
+Miocene and Pliocene.</p>
+
+<p>Long before Cuvier and Brongniart published an account of their
+researches, another observer had been at work among the Secondary
+formations of the west of England, and had independently discovered
+that the component members of these formations were each
+distinguished by a peculiar group of organic remains; and that this
+distinction could be used to discriminate them over all the region
+through which he had traced them. The remarkable man who
+arrived at this far-reaching generalization was William Smith (1769-1839),
+a land surveyor who, in the prosecution of his professional
+business, found opportunities of traversing a great part of England,
+and of putting his deductions to the test. As the result of these
+journeys he accumulated materials enough to enable him to produce
+a geological map of the country, on which the distribution and
+succession of the rocks were for the first time delineated. Smith&rsquo;s
+labours laid the foundation of stratigraphical geology in England
+and he was styled even in his lifetime the &ldquo;Father of English
+geology.&rdquo; From his day onward the significance of fossil organic
+remains gained rapidly increasing recognition. Thus in England
+the outlines traced by him among the Secondary and Tertiary
+formations were admirably filled in by Thomas Webster (1773-1844);
+while the Cretaceous series was worked out in still greater detail
+in the classic memoirs of William Henry Fitton (1780-1861).</p>
+
+<p>There was one stratigraphical domain, however, into which William
+Smith did not enter. He traced his sequence of rocks down into the
+Coal Measures, but contented himself with only a vague reference
+to what lay underneath that formation. Though some of these
+underlying rocks had in various countries yielded abundant fossils,
+they had generally suffered so much from terrestrial disturbances,
+and their order of succession was consequently often so much
+obscured throughout western Europe, that they remained but little
+known for many years after the stratigraphy of the Secondary and
+Tertiary series had been established. At last in 1831 Murchison
+began to attack this <i>terra incognita</i> on the borders of South Wales,
+working into it from the Old Red Sandstone, the stratigraphical
+position of which was well known. In a few years he succeeded in
+demonstrating the existence of a succession of formations, each
+distinguished by its own peculiar assemblage of organic remains
+which were distinct from those in any of the overlying strata. To
+these formations he gave the name of Silurian (<i>q.v.</i>). From the
+key which his researches supplied, it was possible to recognize in
+other countries the same order of formations and the same sequence
+of fossils, so that, in the course of a few years, representatives of the
+Silurian system were found far and wide over the globe. While
+Murchison was thus engaged, Sedgwick devoted himself to the more
+difficult task of unravelling the complicated structure of North
+Wales. He eventually made out the order of the several formations
+there, with their vast intercalations of volcanic material. He named
+them the Cambrian system (<i>q.v.</i>), and found them to contain fossils,
+which, however, lay for some time unexamined by him. He at
+first believed, as Murchison also did, that his rocks were all older
+than any part of the Silurian series. It was eventually discovered
+that a portion of them was equivalent to the lower part of that
+series. The oldest of Sedgwick&rsquo;s groups, containing distinctive
+fossils, retain the name Cambrian, and are of high interest, as they
+enclose the remains of the earliest faunas which are yet well known.
+Sedgwick and Murchison rendered yet another signal service to
+stratigraphical geology by establishing, in 1839, on a basis of
+palaeontological evidence supplied by W. Lonsdale, the independence
+of the Devonian system (<i>q.v.</i>).</p>
+
+<p>For many years the rocks below the oldest fossiliferous deposits
+received comparatively little attention. They were vaguely described
+as the &ldquo;crystalline schists&rdquo; and were often referred to as parts of
+the primeval crust in which no chronology was to be looked for.
+W.E. Logan (1798-1875) led the way, in Canada, by establishing
+there several vast series of rocks, partly of crystalline schists and
+gneisses (Laurentian) and partly of slates and conglomerates
+(Huronian). Later observers, both in Canada and the United
+States, have greatly increased our knowledge of these rocks, and
+have shown their structure to be much more complex than was at
+first supposed (see <span class="sc"><a href="#artlinks">Archean System</a></span>).</p>
+
+<p>During the latter half of the 19th century the most important
+development of stratigraphical geology was the detailed working
+out and application of the principle of zonal classification to the
+fossiliferous formations&mdash;that is, the determination of the sequence
+and distribution of organic remains in these formations, and the
+arrangement of the strata into zones, each of which is distinguished
+by a peculiar assemblage of fossil species (see under Part VI.). The
+zones are usually named after one especially characteristic species.
+This system of classification was begun in Germany with reference
+to the members of the Jurassic system (<i>q.v.</i>) by A. Oppel (1856-1858)
+and F.A. von Quenstedt (1858), and it has since been extended
+through the other Mesozoic formations. It has even been found to
+be applicable to the Palaeozoic rocks, which are now subdivided
+into palaeontological zones. In the Silurian system, for example, the
+graptolites have been shown by C. Lapworth to furnish a useful
+basis for zonal subdivisions. The lowest fossiliferous horizon in the
+Cambrian rocks of Europe and North America is known as the
+<i>Olenellus</i> zone, from the prominence in it of that genus of trilobite.</p>
+
+<p>Another conspicuous feature in the progress of stratigraphy
+during the second half of the 19th century was displayed by the rise
+and rapid development of what is known as Glacial geology. The
+various deposits of &ldquo;drift&rdquo; spread over northern Europe, and the
+boulders scattered across the surface of the plains had long attracted
+notice, and had even found a place in popular legend and superstition.
+When men began to examine them with a view to ascertain
+their origin, they were naturally regarded as evidences of the
+Noachian deluge. The first observer who drew attention to the
+smoothed and striated surfaces of rock that underlie the Drifts was
+Hutton&rsquo;s friend, Sir James Hall, who studied them in the lowlands
+of Scotland and referred them to the action of great debacles of
+water, which, in the course of some ancient terrestrial convulsion,
+had been launched across the face of the country. Playfair, however,
+pointed out that the most potent geological agents for the transportation
+of large blocks of stone are the glaciers. But no one was
+then bold enough to connect the travelled boulders with glaciers
+on the plains of Germany and of Britain. Yet the transporting
+agency of ice was invoked in explanation of their diffusion. It
+came to be the prevalent belief among the geologists of the first
+half of the 19th century, that the fall of temperature, indicated by
+the gradual increase in the number of northern species of shells
+in the English Crag deposits, reached its climax during the time
+of the Drift, and that much of the north and centre of Europe was
+then submerged beneath a sea, across which floating icebergs and
+floes transported the materials of the Drift and dropped the scattered
+boulders. As the phenomena are well developed around the Alps,
+it was necessary to suppose that the submergence involved the
+lowlands of the Continent up to the foot of that mountain chain&mdash;a
+geographical change so stupendous as to demand much more
+evidence than was adduced in its support. At last Louis Agassiz
+(1807-1873), who had varied his palaeontological studies at Neuchātel
+by excursions into the Alps, was so much struck by the proofs of
+the former far greater extension of the Swiss glaciers, that he pursued
+the investigation and satisfied himself that the ice had formerly
+extended from the Alpine valleys right across the great plain of
+Switzerland, and had transported huge boulders from the central
+mountains to the flanks of the Jura. In the year 1840 he visited
+Britain and soon found evidence of similar conditions there. He
+showed that it was not by submergence in a sea cumbered with
+floating ice, but by the former presence of vast glaciers or sheets of
+ice that the Drift and erratic blocks had been distributed. The idea
+thus propounded by him did not at once command complete approval,
+though traces of ancient glaciers in Scotland and Wales were soon
+detected by native geologists, particularly by W. Buckland, Lyell,
+J.D. Forbes and Charles Maclaren. Robert Chambers (1802-1871)
+did good service in gathering additional evidence from Scotland and
+Norway in favour of Agassiz&rsquo;s views, which steadily gained adherents
+until, after some quarter of a century, they were adopted by the
+great majority of geologists in Britain, and subsequently in other
+countries. Since that time the literature of geology has been swollen
+by a vast number of contributions in which the history of the Glacial
+period, and its records both in the Old and New World, have been
+fully discussed.</p>
+
+<p><i>Rise and Progress of Palaeontological Geology.</i>&mdash;As this branch of
+the science deals with the evidence furnished by fossil organic
+remains as to former geographical conditions, it early attracted
+observers who, in the superficial beds of marine shells found at some
+distance from the coast, saw proofs of the former submergence of
+the land under the sea. But the occurrence of fossils embedded in
+the heart of the solid rocks of the mountains offered much greater
+difficulties of explanation, and further progress was consequently
+slow. Especially baneful was the belief that these objects were
+mere sports of nature, and had no connexion with any once living
+organisms. So long as the true organic origin of the fossil plants and
+animals contained in the rocks was in dispute, it was hardly possible
+that much advance could be made in their systematic study, or in
+the geological deductions to be drawn from them. One good result
+of the controversy, however, was to be seen in the large collections
+of these &ldquo;formed stones&rdquo; that were gathered together in the cabinets
+and museums of the 17th and 18th centuries. The accumulation
+and comparison of these objects naturally led to the production of
+treatises in which they were described and not unfrequently illustrated
+by good engravings. Switzerland was more particularly
+<span class="pagenum"><a name="page647" id="page647"></a>647</span>
+noted for the number and merit of its works of this kind, such as that
+of K.N. Lang (<i>Historia lapidum figuratorum Helvetiae</i>, 1708) and
+those of Johann Jacob Scheuchzer (1672-1733). In England, also,
+illustrated treatises were published both by men who looked on
+fossils as mere freaks of nature, and by those who regarded them as
+proofs of Noah&rsquo;s flood. Of the former type were the works of Martin
+Lister (1638-1712) and Robert Plot (<i>Natural History of Oxfordshire</i>,
+1677). The Celtic scholar Edward Llwyd (1660-1709) wrote a Latin
+treatise containing good plates of a thousand fossils in the Ashmolean
+Museum, Oxford, and J. Woodward, in 1728-1729, published his
+<i>Natural History of the Fossils of England</i>, already mentioned, wherein
+he described his own extensive collection, which he bequeathed to
+the University of Cambridge, where it is still carefully preserved.
+The most voluminous and important of all these works, however,
+appeared at a later date at Nuremberg. It was begun by G.W.
+Knorr (1705-1761), who himself engraved for it a series of plates,
+which for beauty and accuracy have seldom been surpassed. After
+his death the work was continued by J.E.I. Walch (1725-1778), and
+ultimately consisted of four massive folio volumes and nearly 300
+plates under the title of <i>Lapides diluvii universalis testes</i>. Although
+the authors supposed their fossils to be relics of Noah&rsquo;s flood, their
+work must be acknowledged to mark a distinct onward stage in the
+palaeontological department of geology.</p>
+
+<p>It was in France that palaeontological geology began to be cultivated
+in a scientific spirit. The potter Bernard Palissy, as far back
+as 1580, had dwelt on the importance of fossil shells as monuments
+of revolutions of the earth&rsquo;s surface; but the observer who first
+undertook the detailed study of the subject was Jean Etienne
+Guettard, who began in 1751 to publish his descriptions of fossils
+in the form of memoirs presented to the Academy of Sciences of
+Paris. To him they were not only of deep interest as monuments
+of former types of existence, but they had an especial value as
+records of the changes which the country had undergone from sea
+to land and from land to sea. More especially noteworthy was a
+monograph by him which appeared in 1765 bearing the title &ldquo;On
+the accidents that have befallen Fossil Shells compared with those
+which are found to happen to shells now living in the Sea.&rdquo; In this
+treatise he showed that the fossils have been encrusted with barnacles
+and serpulae, have been bored into by other organisms, and have
+often been rounded or broken before final entombment; and he
+inferred that these fossils must have lived and died on the sea-floor
+under similar conditions to those which obtain on the sea-floor
+to-day. His argument was the most triumphant that had ever
+been brought against the doctrine of <i>lusus naturae</i>, and that of the
+efficacy of Noah&rsquo;s flood&mdash;doctrines which still held their ground in
+Guettard&rsquo;s day. When Soulavie, Cuvier and Brongniart in France,
+and William Smith in England, showed that the rock formations
+of the earth&rsquo;s crust could be arranged in chronological order, and
+could be recognized far and wide by means of their enclosed organic
+remains, the vast significance of these remains in geological research
+was speedily realized, and palaeontological geology at once entered
+on a new and enlarged phase of development. But apart from
+their value as chronological monuments, and as witnesses of former
+conditions of geography, fossils presented in themselves a wide
+field of investigation as types of life that had formerly existed, but
+had now passed away. It was in France that this subject first took
+definite shape as an important branch of science. The mollusca of
+the Tertiary deposits of the Paris basin became, in the hands of
+Lamarck, the basis on which invertebrate palaeontology was founded.
+The same series of strata furnished to Cuvier the remains of extinct
+land animals, of which, by critical study of their fragmentary bones
+and skeletons, he worked out restorations that may be looked on
+as the starting-point of vertebrate palaeontology. These brilliant
+researches, rousing widespread interest in such studies, showed how
+great a flood of light could be thrown on the past history of the earth
+and its inhabitants. But the full significance of these extinct types
+of life could not be understood so long as the doctrine of the immutability
+of species, so strenuously upheld by Cuvier, maintained its
+sway among naturalists. Lamarck, as far back as the year 1800,
+had begun to propound his theory of evolution and the transformation
+of species; but his views, strongly opposed by Cuvier and the
+great body of naturalists of the day, fell into neglect. Not until
+after the publication in 1859 of the <i>Origin of Species</i> by Charles
+Darwin were the barriers of old prejudice in this matter finally
+broken down. The possibility of tracing the ancestry of living forms
+back into the remotest ages was then perceived; the time-honoured
+fiction that the stratified formations record a series of catastrophes
+and re-creations was finally dissipated; and the earth&rsquo;s crust was
+seen to contain a noble, though imperfect, record of the grand
+evolution of organic types of which our planet has been the theatre.</p>
+
+<p><i>Development of Petrographical Geology.</i>&mdash;Theophrastus, the favourite
+pupil of Aristotle, wrote a treatise <i>On Stones</i>, which has come
+down to our own day, and may be regarded as the earliest work on
+petrography. At a subsequent period Pliny, in his <i>Natural History</i>,
+collected all that was known in his day regarding the occurrence
+and uses of minerals and rocks. But neither of these works is
+of great scientific importance, though containing much interesting
+information. Minerals from their beauty and value attracted
+notice before much attention was paid to rocks, and their study
+gave rise to the science of mineralogy long before geology came
+into existence. When rocks began to be more particularly scrutinized,
+it was chiefly from the side of their usefulness for building
+and other economic purposes. The occurrence of marine shells in
+many of them had early attracted attention to them. But their
+varieties of composition and origin did not become the subject of
+serious study until after Linnaeus and J.G. Wallerius in the 18th
+century had made a beginning. The first important contribution
+to this department of the science was that of Werner, who in 1786
+published a classification and description of rocks in which he
+arranged them in two divisions, simple and compound, and further
+distinguished them by various external characters and by their
+relative age. The publication of this scheme may be said to mark
+the beginning of scientific petrography. Werner&rsquo;s system, however,
+had the serious defect that the chronological order in which he
+grouped the rocks, and the hypothesis by which he accounted for
+them as chemical precipitates from the original ocean, were both
+alike contrary to nature. It was hardly possible indeed that much
+progress could be made in this branch of geology until chemistry
+and mineralogy had made greater advances; and especially until
+it was possible to ascertain the intimate chemical and mineralogical
+composition, and the minute structure of rocks. The study, however,
+continued to be pursued in Germany, where the influence of Werner&rsquo;s
+enthusiasm still led men to enter the petrographical rather than the
+palaeontological domain. The resources of modern chemistry were
+pressed into the service, and analyses were made and multiplied to
+such a degree that it seemed as if the ultimate chemical constitution
+of every type of rock had now been thoroughly revealed. The
+condition of the science in the middle of the 19th century was well
+shown by J.L.A. Roth, who in 1861 collected about 1000 trustworthy
+analyses which up to that time had been made. But though
+the chemical elements of the rocks had been fairly well determined,
+the manner in which they were combined in the compound rocks
+could for the most part be only more or less plausibly conjectured.
+As far back as 1831 an account was published of a process devised by
+William Nicol of Edinburgh, whereby sections of fossil wood could be
+cut, mounted on glass, and reduced to such a degree of transparency
+as to be easily examined under a microscope. Henry Sorby, of
+Sheffield, having seen Nicol&rsquo;s preparations, perceived how admirably
+adapted the process was for the study of the minute structure and
+composition of rocks. In 1858 he published in the <i>Quarterly Journal
+of the Geological Society</i> a paper &ldquo;On the Microscopical Structure of
+Crystals.&rdquo; This essay led to a complete revolution of petrographical
+methods and gave a vast impetus to the study of rocks. Petrology
+entered upon a new and wider field of investigation. Not only were
+the mineralogical constituents of the rocks detected, but minute
+structures were revealed which shed new light on the origin and
+history of these mineral masses, and opened up new paths in
+theoretical geology. In the hands of H. Vogelsang, F. Zirkel,
+H. Rosenbusch, and a host of other workers in all civilized countries,
+the literature of this department of the science has grown to a
+remarkable extent. Armed with the powerful aid of modern optical
+instruments, geologists are now able with far more prospect of success
+to resume the experiments begun a century before by de Saussure
+and Hall. G.A. Daubrée, C. Friedel, E. Sarasin, F. Fouqué and
+A. Michel Lévy in France, C. Doelter y Cisterich and E. Hussak of
+Gratz, J. Morozewicz of Warsaw and others, have greatly advanced
+our knowledge by their synthetical analyses, and there is every
+reason to hope that further advances will be made in this field of
+research.</p>
+
+<p><i>Rise of Physiographical Geology.</i>&mdash;Until stratigraphical geology
+had advanced so far as to show of what a vast succession of rocks the
+crust of the earth is built up, by what a long and complicated series
+of revolutions these rocks have come to assume their present positions,
+and how enormous has been the lapse of time which all these changes
+represent, it was not possible to make a scientific study of the surface
+features of our globe. From ancient times it had been known that
+many parts of the land had once been under the sea; but down even
+to the beginning of the 19th century the vaguest conceptions continued
+to prevail as to the operations concerned in the submergence
+and elevation of land, and as to the processes whereby the present
+outlines of terrestrial topography were determined. We have seen,
+for instance, that according to the teaching of Werner the oldest
+rocks were first precipitated from solution in the universal ocean to
+form the mountains, that the vertical position of their strata was
+original, that as the waters subsided successive formations were
+deposited and laid bare, and that finally the superfluous portion of the
+ocean was whisked away into space by some unexplained co-operation
+of another planetary body. Desmarest, in his investigation of the
+volcanic history of Auvergne, was the first observer to perceive by
+what a long process of sculpture the present configuration of the land
+has been brought about. He showed conclusively that the valleys have
+been carved out by the streams that flow in them, and that while
+they have sunk deeper and deeper into the framework of the land,
+the spaces of ground between them have been left as intervening
+ridges and hills. De Saussure learnt a similar lesson from his studies
+of the Alps, and Hutton and Playfair made it a cardinal feature in
+their theory of the earth. Nevertheless the idea encountered so
+much opposition that it made but little way until after the middle
+of the 19th century. Geologists preferred to believe in convulsions
+of nature, whereby valleys were opened and mountains were
+<span class="pagenum"><a name="page648" id="page648"></a>648</span>
+upheaved. That the main features of the land, such as the great
+mountain-chains, had been produced by gigantic plication of the
+terrestrial crust was now generally admitted, and also that minor
+fractures and folds had probably initiated many of the valleys.
+But those who realized most vividly the momentous results achieved
+by ages of subaerial denudation perceived that, as Hutton showed,
+even without the aid of underground agency, the mere flow of water
+in streams across a mass of land must in course of time carve out
+just such a system of valleys as may anywhere be seen. It was
+J.B. Jukes who, in 1862, first revived the Huttonian doctrine,
+and showed how completely it explained the drainage-lines in the
+south of Ireland. Other writers followed in quick succession until,
+in a few years, the doctrine came to be widely recognized as one of
+the established principles of modern geology. Much help was derived
+from the admirable illustrations of land-sculpture and river-erosion
+supplied from the Western Territories and States of the American
+Union.</p>
+
+<p>Another branch of physiographical geology which could only come
+into existence after most of the other departments of the science
+had made large progress, deals with the evolution of the framework
+of each country and of the several continents and oceans of the globe.
+It is now possible, with more or less confidence, to trace backward
+the history of every terrestrial area, to see how sea and land have
+there succeeded each other, how rivers and lakes have come and
+gone, how the crust of the earth has been ridged up at widely
+separated intervals, each movement determining some line of
+mountains or plains, how the boundaries of the oceans have shifted
+again and again in the past, and thus how, after so prolonged a series
+of revolutions, the present topography of each country, and of the
+globe as a whole, has been produced. In the prosecution of this
+subject maps have been constructed to show what is conjectured
+to have been the distribution of sea and land during the various
+geological periods in different parts of the world, and thus to indicate
+the successive stages through which the architecture of the land has
+been gradually evolved. The most noteworthy contribution to this
+department of the science is the <i>Antlitz der Erde</i> of Professor Suess
+of Vienna. This important and suggestive work has been translated
+into French and English.</p>
+</div>
+
+<p class="pt2 center sc">Part II.&mdash;Cosmical Aspects</p>
+
+<p>Before geology had attained to the position of an inductive
+science, it was customary to begin investigations into the
+history of the earth by propounding or adopting some more
+or less fanciful hypothesis in explanation of the origin of our
+planet, or even of the universe. Such preliminary notions were
+looked upon as essential to a right understanding of the manner
+in which the materials of the globe had been put together. One
+of the distinguishing features of Hutton&rsquo;s Theory of the Earth
+consisted in his protest that it is no part of the province of
+geology to discuss the origin of things. He taught that in the
+materials from which geological evidence is to be compiled
+there can be found &ldquo;no traces of a beginning, no prospect of an
+end.&rdquo; In England, mainly to the influence of the school which
+he founded, and to the subsequent rise of the Geological Society
+of London, which resolved to collect facts instead of fighting
+over hypotheses, is due the disappearance of the crude and
+unscientific cosmologies by which the writings of the earlier
+geologists were distinguished.</p>
+
+<p>But there can now be little doubt that in the reaction against
+those visionary and often grotesque speculations, geologists
+were carried too far in an opposite direction. In allowing
+themselves to believe that geology had nothing to do with
+questions of cosmogony, they gradually grew up in the conviction
+that such questions could never be other than mere speculation,
+interesting or amusing as a theme for the employment of the
+fancy, but hardly coming within the domain of sober and
+inductive science. Nor would they soon have been awakened
+out of this belief by anything in their own science. It is still
+true that in the data with which they are accustomed to deal,
+as comprising the sum of geological evidence, there can be
+found no trace of a beginning, though the evidence furnished
+by the terrestrial crust shows a general evolution of organic
+forms from some starting-point which cannot be seen. The
+oldest rocks which have been discovered on any part of the
+globe have probably been derived from other rocks older than
+themselves. Geology by itself has not yet revealed, and is little
+likely ever to reveal, a trace of the first solid crust of our globe.
+If, then, geological history is to be compiled from direct evidence
+furnished by the rocks of the earth, it cannot begin at the
+beginning of things, but must be content to date its first chapter
+from the earliest period of which any record has been preserved
+among the rocks.</p>
+
+<p>Nevertheless, though geology in its usual restricted sense has
+been, and must ever be, unable to reveal the earliest history of
+our planet, it no longer ignores, as mere speculation, what is
+attempted in this subject by its sister sciences. Astronomy,
+physics and chemistry have in late years all contributed to cast
+light on the earlier stages of the earth&rsquo;s existence, previous to
+the beginning of what is commonly regarded as geological history.
+But whatever extends our knowledge of the former conditions
+of our globe may be legitimately claimed as part of the domain of
+geology. If this branch of inquiry, therefore, is to continue
+worthy of its name as the science of the earth, it must take
+cognizance of these recent contributions from other sciences.
+It must no longer be content to begin its annals with the records
+of the oldest rocks, but must endeavour to grope its way through
+the ages which preceded the formation of any rocks. Thanks
+to the results achieved with the telescope, the spectroscope and
+the chemical laboratory, the story of these earliest ages of our
+earth is every year becoming more definite and intelligible.</p>
+
+<p>Up to the present time no definite light has been thrown by
+physics on the origin and earliest condition of our globe. The
+famous nebular theory (<i>q.v.</i>) of Kant and Laplace sketched the
+supposed evolution of the solar system from a gaseous nebula,
+slowly rotating round a more condensed central portion of its
+mass, which eventually became the sun. As a consequence of
+increased rapidity of rotation resulting from cooling and contraction,
+the nebula acquired a more and more lenticular form,
+until at last it threw off from its equatorial protuberance a ring
+of matter. Subsequently the same process was repeated, and
+other similar rings successively separated from the parent mass.
+Each ring went through a corresponding series of changes until
+it ultimately became a planet, with or without one or more
+attendant satellites. The intimate relationship of our earth
+to the sun and the other planets was, in this way, shown. But
+there are some serious physical difficulties in the way of the
+acceptance of the nebular hypothesis. Another explanation
+is given by the meteoritic hypothesis, according to which, out
+of the swarms of meteorites with which the regions of space are
+crowded, the sun and planets have been formed by gradual
+accretion.</p>
+
+<p>According to these theoretical views we should expect to find
+a general uniformity of composition in the constituent matter
+of the solar system. For many years the only available evidence
+on this point was derived from the meteorites (<i>q.v.</i>) which so
+constantly fall from outer space upon the surface of the earth.
+These bodies were found to consist of elements, all of which had
+been recognized as entering into the constitution of the earth.
+But the discoveries of spectroscopic research have made known
+a far more widely serviceable method of investigation, which
+can be applied even to the luminous stars and nebulae that lie
+far beyond the bounds of the solar system. By this method
+information has been obtained regarding the constitution of the
+sun, and many of our terrestrial metals, such as iron, nickel and
+magnesium, have been ascertained to exist in the form of incandescent
+vapour in the solar atmosphere. The present
+condition of the sun probably represents one of the phases
+through which stars and planets pass in their progress towards
+becoming cool and dark bodies in space. If our globe was at
+first, like its parent sun, an incandescent mass of probably
+gaseous matter, occupying much more space than it now fills,
+we can conceive that it has ever since been cooling and contracting
+until it has reached its present form and dimensions, and that
+it still retains a high internal temperature. Its oblately spheroidal
+form is such as would be assumed by a rotating mass of matter
+in the transition from a vaporous and self-luminous or liquid
+condition to one of cool and dark solidity. But it has been
+claimed that even a solid spherical globe might develop, under
+the influence of protracted rotation, such a shape as the earth
+at present possesses.</p>
+
+<p>The observed increase of temperature downwards in our
+<span class="pagenum"><a name="page649" id="page649"></a>649</span>
+planet has hitherto been generally accepted as a relic and proof
+of an original high temperature and mobility of substance.
+Recently, however, the validity of this proof has been challenged
+on the ground that the ascertained amount of radium in the
+rocks of the outer crust is more than sufficient to account for
+the observed downward increase of temperature. Too little,
+however, is known of the history and properties of what is
+called radium to afford a satisfactory ground on which to
+discard what has been, and still remains, the prevalent belief
+on this subject.</p>
+
+<p>An important epoch in the geological history of the earth
+was marked by the separation of the moon from its mass (see
+<span class="sc"><a href="#artlinks">Tide</a></span>). Whether the severance arose from the rupture of a
+surrounding ring or the gradual condensation of matter in such
+a ring, or from the ejection of a single mass of matter from the
+rapidly rotating planet, it has been shown that our satellite
+was only a few thousand miles from the earth&rsquo;s surface, since
+when it has retreated to its present distance of 240,000 m. Hence
+the influence of the moon&rsquo;s attraction, and all the geological
+effects to which it gives rise, attained their maximum far back
+in the development of the globe, and have been slowly diminishing
+throughout geological history.</p>
+
+<p>The sun by virtue of its vast size has not yet passed out of
+the condition of glowing gas, and still continues to radiate heat
+beyond the farthest planet of the solar system. The earth,
+however, being so small a body in comparison, would cool down
+much more quickly. Underneath its hot atmosphere a crust
+would conceivably begin to form over its molten surface, though
+the interior might still possess a high temperature and, owing
+to the feeble conducting power of rocks, would remain intensely
+hot for a protracted series of ages.</p>
+
+<p>Full information regarding the form and size of the earth,
+and its relations to the other planetary members of the solar
+system, will be found in the articles <span class="sc"><a href="#artlinks">Planet</a></span> and <span class="sc"><a href="#artlinks">Solar System</a></span>.
+For the purposes of geological inquiry the reader will bear in
+mind that the equatorial diameter of our globe is estimated to
+be about 7925 m., and the polar diameter about 7899 m.; the
+difference between these two sums representing the amount of
+flattening at the poles (about 26½ m.). The planet has been
+compared in shape to an orange, but it resembles an orange
+which has been somewhat squeezed, for its equatorial circumference
+is not a regular circle but an ellipse, of which the major
+axis lies in long. 8° 15&prime; W.&mdash;on a meridian which cuts the north-west
+corner of America, passing through Portugal and Ireland,
+and the north-east corner of Asia in the opposite hemisphere.</p>
+
+<p>The rotation of the earth on its axis exerts an important
+influence on the movements of the atmosphere, and thereby
+affects the geological operations connected with these movements.
+The influence of rotation is most marked in the great aerial
+circulation between the poles and the equator. Currents of
+air, which set out in a meridional direction from high latitudes
+towards the equator, come from regions where the velocity due
+to rotation is small to where it is greater, and they consequently
+fall behind. Thus, in the northern hemisphere a north wind,
+as it moves away from its northern source of origin, is gradually
+deflected more and more towards the west and becomes a north-east
+current; while in the opposite hemisphere a wind making
+from high southern latitudes towards the equator becomes,
+from the same cause, a south-east current. Where, on the
+other hand, the air moves from the equatorial to the polar regions
+its higher velocity of rotation carries it eastward, so that on the
+south side of the equator it becomes a north-west current and
+on the north side a south-west current. It is to this cause that
+the easting and westing of the great atmospheric currents are
+to be attributed, as is familiarly exemplified in the trade winds.</p>
+
+<p>The atmospheric circulation thus deflected influences the
+circulation of the ocean. The winds which persistently blow
+from the north-east on the north side of the equator, and from
+the south-east on the south side, drive the superficial waters
+onwards, and give rise to converging oceanic currents which
+unite to form the great westerly equatorial current.</p>
+
+<p>A more direct effect of terrestrial rotation has been claimed
+in the case of rivers which flow in a meridional direction. It has
+been asserted that those, which in the northern hemisphere
+flow from north to south, like the Volga, by continually passing
+into regions where the velocity of rotation is increasingly greater,
+are thrown more against their western than their eastern banks,
+while those whose general course is in an opposite direction, like
+the Irtisch and Yenesei, press more upon their eastern sides.
+There cannot be any doubt that the tendency of the streams
+must be in the directions indicated. But when the comparatively
+slow current and constantly meandering course of most rivers
+are taken into consideration, it may be doubted whether the
+influence of rotation is of much practical account so far as
+river-erosion is concerned.</p>
+
+<p>One of the cosmical relations of our planet which has been
+more especially prominent in geological speculations relates to
+the position of the earth&rsquo;s axis of rotation. Abundant evidence
+has now been obtained to prove that at a comparatively late
+geological period a rich flora, resembling that of warm climates
+at the present day, existed in high latitudes even within less than
+9° of the north pole, where, with an extremely low temperature
+and darkness lasting for half of the year, no such vegetation could
+possibly now exist. It has accordingly been maintained by
+many geologists that the axis of rotation must have shifted,
+and that when the remarkable Arctic assemblage of fossil plants
+lived the region of their growth must have lain in latitudes much
+nearer to the equator of the time.</p>
+
+<p>The possibility of any serious displacement of the rotational
+axis since a very early period in the earth&rsquo;s history has been
+strenuously denied by astronomers, and their arguments have
+been generally, but somewhat reluctantly, accepted by geologists,
+who find themselves confronted with a problem which has
+hitherto seemed insoluble. That the axis is not rigidly stable,
+however, has been postulated by some physicists, and has now
+been demonstrated by actual observation and measurement.
+It is admitted that by the movement of large bodies of water
+the air over the surface of the globe, and more particularly by
+the accumulation of vast masses of snow and ice in different
+regions, the position of the axis might be to some extent shifted;
+more serious effects might follow from widespread upheavals
+or depressions of the surface of the lithosphere. On the assumption
+of the extreme rigidity of the earth&rsquo;s interior, however, the
+general result of mathematical calculation is to negative the
+supposition that in any of these ways within the period represented
+by what is known as the &ldquo;geological record,&rdquo; that is,
+since the time of the oldest known sedimentary formations, the
+rotational axis has ever been so seriously displaced as to account
+for such stupendous geological events as the spread of a luxuriant
+vegetation far up into polar latitudes. If, however, the inside
+of the globe possesses a great plasticity than has been allowed,
+the shifting of the axis might not be impossible, even to such an
+extent as would satisfy the geological requirements. This
+question is one on which the last word has not been said, and
+regarding which judgment must remain in suspense.</p>
+
+<p>In recent years fresh information bearing on the minor devagations
+of the pole has been obtained from a series of several
+thousand careful observations made in Europe and North
+America. It has thus been ascertained that the pole wanders
+with a curiously irregular but somewhat spiral movement,
+within an amplitude of between 40 and 50 ft., and completes
+its erratic circuit in about 428 days. It was not supposed that
+its movement had any geological interest, but Dr John Milne
+has recently pointed out that the times of sharpest curvature
+in the path of the pole coincide with the occurrence of large
+earthquakes, and has suggested that, although it can hardly be
+assumed that this coincidence shows any direct connexion
+between earthquake frequency and changes in the position of
+the earth&rsquo;s axis, both effects may not improbably arise from
+the same redistribution of surface material by ocean currents
+and meteorological causes.</p>
+
+<p>If for any reason the earth&rsquo;s centre of gravity were sensibly
+displaced, momentous geological changes would necessarily
+ensue. That the centre of gravity does not coincide with the
+<span class="pagenum"><a name="page650" id="page650"></a>650</span>
+centre of figure of the globe, but lies to the south of it, has long
+been known. This greater aggregation of dense material in the
+southern hemisphere probably dates from the early ages of the
+earth&rsquo;s consolidation, and it is difficult to believe that any
+readjustment of the distribution of this material in the earth&rsquo;s
+interior is now possible. But certain rearrangements of the
+hydrosphere on the surface of the globe may, from time to time,
+cause a shifting of the centre of gravity, which will affect the
+level of the ocean. The accumulation of enormous masses of
+ice around the pole will give rise to such a displacement, and
+will thus increase the body of oceanic water in the glaciated
+hemisphere. Various calculations have been made of the effect
+of the transference of the ice-cap from one pole to the other, a
+revolution which may possibly have occurred more than once
+in the past history of the globe. James Croll estimated that if
+the mass of ice in the southern hemisphere be assumed to be
+1000 ft. thick down to lat. 60°, its removal to the opposite
+hemisphere would raise the level of the sea 80 ft. at the north pole,
+while the Rev. Osmond Fisher made the rise as much as 409 ft.
+The melting of the ice would still further raise the sea-level by
+the addition of so large a volume of water to the ocean. To
+what extent superficial changes of this kind have operated in
+geological history remains an unsolved problem, but their
+probable occurrence in the past has to be recognized as one of
+the factors that must be considered in tracing the revolutions of
+the earth&rsquo;s surface.</p>
+
+<p><i>The Age of the Earth.</i>&mdash;Intimately connected with the relations
+of our globe to the sun and the other members of the solar system
+is the question of the planet&rsquo;s antiquity&mdash;a subject of great
+geological importance, regarding which much discussion has
+taken place since the middle of the 19th century. Though an
+account of this discussion necessarily involves allusion to departments
+of geology which are more appropriately referred to in
+later parts of this article, it may perhaps be most conveniently
+included here.</p>
+
+<p>Geologists were for many years in the habit of believing that
+no limit could be assigned to the antiquity of the planet, and that
+they were at liberty to make unlimited drafts on the ages of the
+past. In 1862 and subsequent years, however, Lord Kelvin
+(then Sir William Thomson) pointed out that these demands were
+opposed to known physical facts, and that the amount of time
+required for geological history was not only limited, but must
+have been comprised within a comparatively narrow compass.
+His argument rested on three kinds of evidence: (1) the internal
+heat and rate of cooling of the earth; (2) the tidal retardation
+of the earth&rsquo;s rotation; and (3) the origin and age of the sun&rsquo;s
+heat.</p>
+
+<p>1. Applying Fourier&rsquo;s theory of thermal conductivity, Lord
+Kelvin contended that in the known rate of increase of temperature
+downward and beneath the surface, and the rate of loss
+of heat from the earth, we have a limit to the antiquity of the
+planet. He showed, from the data available at the time, that
+the superficial consolidation of the globe could not have occurred
+less than 20 million years ago, or the underground heat would
+have been greater than it is; nor more than 400 million years
+ago, otherwise the underground temperature would have shown
+no sensible increase downwards. He admitted that very wide
+limits were necessary. In subsequently discussing the subject,
+he inclined rather towards the lower than the higher antiquity,
+but concluded that the limit, from a consideration of all the
+evidence, must be placed within some such period of past time
+as 100 millions of years.</p>
+
+<p>2. The argument from tidal retardation proceeds on the
+admitted fact that, owing to the friction of the tide-wave, the
+rotation of the earth is retarded, and is, therefore, much slower
+now than it must have been at one time. Lord Kelvin affirmed
+that had the globe become solid some 10,000 million years ago,
+or indeed any high antiquity beyond 100 million years, the
+centrifugal force due to the more rapid rotation must have given
+the planet a very much greater polar flattening than it actually
+possesses. He admitted, however, that, though 100 million
+years ago that force must have been about 3% greater than now,
+yet &ldquo;nothing we know regarding the figure of the earth, and
+the disposition of land and water, would justify us in saying
+that a body consolidated when there was more centrifugal
+force by 3% than now, might not now be in all respects like
+the earth, so far as we know it at present.&rdquo;</p>
+
+<p>3. The third argument, based upon the age of the sun&rsquo;s heat,
+is confessedly less to be relied on than the two previous ones.
+It proceeds upon calculations as to the amount of heat which
+would be available by the falling together of masses from space,
+which gave rise by their impact to our sun. The vagueness of
+the data on which this argument rests may be inferred from
+the fact that in one passage P.G. Tait placed the limit of time
+during which the sun has been illuminating the earth as, &ldquo;on
+the very highest computation, not more than about 15 or 20
+millions of years&rdquo;; while, in another sentence of the same
+volume, he admitted that, &ldquo;by calculations in which there is
+no possibility of large error, this hypothesis [of the origin of the
+sun&rsquo;s heat by the falling together of masses of matter] is
+thoroughly competent to explain 100 millions of years&rsquo; solar
+radiation at the present rate, perhaps more.&rdquo; In more recently
+reviewing his argument, Lord Kelvin expressed himself in
+favour of more strictly limiting geological time than he had at
+first been disposed to do. He insists that the time &ldquo;was more
+than 20 and less than 40 millions of years and probably much
+nearer 20 than 40.&rdquo; Geologists appear to have reluctantly
+brought themselves to believe that perhaps, after all, 100 millions
+of years might suffice for the evolution of geological history.
+But when the time was cut down to 15 or 20 millions they
+protested that such a restricted period was insufficient for that
+evolution, and though they did not offer any effective criticism
+of the arguments of the physicists they felt convinced that there
+must be some flaw in the premises on which these arguments
+were based.</p>
+
+<p>By degrees, however, there have arisen among the physicists
+themselves grave doubts as to the validity of the physical
+evidence on which the limitation of the earth&rsquo;s age has been
+founded, and at the same time greater appreciation has been
+shown of the signification and <span class="correction" title="amended from stength">strength</span> of the geological proofs
+of the high antiquity of our planet. In an address from the
+chair of the Mathematical Section of the British Association in
+1886, Professor (afterwards Sir) George Darwin reviewed the
+controversy, and pronounced the following deliberate judgment
+in regard to it: &ldquo;In considering these three arguments I have
+adduced some reasons against the validity of the first [tidal
+friction], and have endeavoured to show that there are elements
+of uncertainty surrounding the second [secular cooling of the
+earth]; nevertheless, they undoubtedly constitute a contribution
+of the first importance to physical geology. Whilst, then, we
+may protest against the precision with which Professor Tait
+seeks to deduce results from them, we are fully justified in
+following Sir William Thomson, who says that &lsquo;the existing
+state of things on the earth, life on the earth&mdash;all geological
+history showing continuity of life&mdash;must be limited within some
+such period of past time as 100 million years&rsquo;.&rdquo; Lord Kelvin
+has never dealt with the geological and palaeontological objections
+against the limitation of geological time to a few millions of years.
+But Professor Darwin, in the address just cited, uttered the
+memorable warning: &ldquo;At present our knowledge of a definite
+limit to geological time has so little precision that we should do
+wrong summarily to reject theories which appear to demand
+longer periods of time than those which now appear allowable.&rdquo;
+In his presidential address to the British Association at Cape
+Town in 1905 he returned to the subject, remarking that the
+argument derived from the increase of underground temperature
+&ldquo;seems to be entirely destroyed&rdquo; by the discovery of the
+properties of radium. He thinks that &ldquo;it does not seem extravagant
+to suppose that 500 to 1000 million years may have
+elapsed since the birth of the moon.&rdquo; He has &ldquo;always believed
+that the geologists were more nearly correct than the physicists,
+notwithstanding the fact that appearances were so strongly
+against them,&rdquo; and he concludes thus: &ldquo;It appears, then, that
+the physical argument is not susceptible of a greater degree of
+<span class="pagenum"><a name="page651" id="page651"></a>651</span>
+certainty than that of the geologists, and the scale of geological
+time remains in great measure unknown&rdquo; (see also Tide, chap.
+viii.).</p>
+
+<p>In an address to the mathematical section of the American
+Association for the Advancement of Science in 1889, the vice-president
+of the section, R.S. Woodward, thus expressed himself
+with regard to the physical arguments brought forward by Lord
+Kelvin and Professor Tait in limitation of geological time:
+&ldquo;Having been at some pains to look into this matter, I feel
+bound to state that, although the hypothesis appears to be the
+best which can be formulated at present, the odds are against
+its correctness. Its weak links are the unverified assumptions of
+an initial uniform temperature and a constant diffusivity. Very
+likely these are approximations, but of what order we cannot
+decide. Furthermore, if we accept the hypothesis, the odds
+appear to be against the present attainment of trustworthy
+numerical results, since the data for calculation, obtained
+mostly from observations on continental areas, are far too
+meagre to give satisfactory average values for the entire mass
+of the earth.&rdquo;</p>
+
+<p>Still more emphatic is the protest made from the physical
+side by Professor John Perry. He has attacked each of the
+three lines of argument of Lord Kelvin, and has impugned the
+validity of the conclusions drawn from them. The argument
+from tidal retardation he dismisses as fallacious, following in
+this contention the previous criticism of the Rev. Maxwell Close
+and Sir George Darwin. In dealing with the argument based on
+the secular cooling of the earth, he holds it to be perfectly
+allowable to assume a much higher conductivity for the interior
+of the globe, and that such a reasonable assumption would enable
+us greatly to increase our estimate of the earth&rsquo;s antiquity.
+As for the third argument, from the age of the sun&rsquo;s heat, he
+points out that the sun may have been repeatedly fed by a
+supply of meteorites from outside, while the earth may have been
+protected from radiation, and been able to retain much of its
+heat by being enveloped in a dense atmosphere. Remarking
+that &ldquo;almost anything is possible as to the present internal
+state of the earth,&rdquo; he concludes thus: &ldquo;To sum up, we can
+find no published record of any lower maximum age of life on
+the earth, as calculated by physicists, than 400 millions of years.
+From the three physical arguments Lord Kelvin&rsquo;s higher limits
+are 1000, 400 and 500 million years. I have shown that we have
+reasons for believing that the age, from all these, may be very
+considerably underestimated. It is to be observed that if we
+exclude everything but the arguments from mere physics, the
+<i>probable</i> age of life on the earth is much less than any of the above
+estimates; but if the palaeontologists have good reasons for
+demanding much greater times, I see nothing from the physicists&rsquo;
+point of view which denies them four times the greatest of these
+estimates.&rdquo;</p>
+
+<p>A fresh line of argument against Lord Kelvin&rsquo;s limitation of
+the antiquity of our globe has recently been started by the
+remarkable discoveries in radio-activity. From the ascertained
+properties of radium it appears to be possible that our estimates
+of solar heat, as derived from the theory of gravitation, may
+have to be augmented ten or twenty times; that stores of
+radium and similar bodies within the earth may have indefinitely
+deferred the establishment of the present temperature
+gradient from the surface inward; that consequently the earth
+may have remained for long ages at a temperature not greatly
+different from that which it now possesses, and hence that the
+times during which our globe has supported animal and vegetable
+life may be very much longer than that allowed in the estimates
+previously made by physicists from other data (see <span class="sc"><a href="#artlinks">Radioactivity</a></span>).</p>
+
+<p>The arguments from the geological side against the physical
+contention that would limit the age of our globe to some 10
+or 20 millions of years are mainly based on the observed rates of
+geological and biological changes at the present time upon land
+and sea, and on the nature, physical history and organic contents
+of the stratified crust of the earth. Unfortunately, actual
+numerical data are not obtainable in many departments of
+geological activity, and even where they can be procured they
+do not yet rest on a sufficiently wide collection of accurate and
+co-ordinated observations. But in some branches of dynamical
+geology, material exists for, at least, a preliminary computation
+of the rate of change. This is more especially the case in respect
+of the wide domain of denudation. The observational records
+of the action of the sea, of springs, rivers and glaciers are becoming
+gradually fuller and more trustworthy. A method of making
+use of these records for estimating the rate of denudation of
+the land has been devised. Taking the Mississippi as a general
+type of river action, it has been shown that the amount of
+material conveyed by this stream into the sea in one year is
+equivalent to the lowering of the general surface of the drainage
+basin of the river by <span class="spp">1</span>&frasl;<span class="suu">6000</span> of a foot. This would amount to one
+foot in 6000 years and 1000 ft. in 6 million years. So that at
+the present rate of waste in the Mississippi basin a whole continent
+might be worn away in a few millions of years.</p>
+
+<p>It is evident that as deposition and denudation are simultaneous
+processes, the ascertainment of the rate at which solid
+material is removed from the surface of the land supplies some
+necessary information for estimating the rate at which new
+sedimentary formations are being accumulated on the floor of
+the sea, and for a computation of the length of time that would
+be required at the present rate of change for the deposition of all
+the stratified rocks that enter into the composition of the crust
+of our globe. If the thickness of these rocks be assumed to be
+100,000 ft., and if we could suppose them to have been laid down
+over as wide an area as that of the drainage basins from the
+waste of which they were derived, then at the present rate of
+denudation their accumulation would require some 600 millions
+of years. But, as Dr A.R. Wallace has justly pointed out, the
+tract of sea-floor over which the material derived from the waste
+of the terrestrial surface is laid down is at present much less than
+that from which this material is worn away. We have no means,
+however, of determining what may have been the ratio between
+the two areas in past time. Certainly ancient marine sedimentary
+rocks cover at the present day a much more extensive area than
+that in which they are now being elaborated. If we take the
+ratio postulated by Dr Wallace&mdash;1 to 19&mdash;the 100,000 ft. of
+sedimentary strata would require 31 millions of years for their
+accumulation. It is quite possible, however, that this ratio may be
+much too high. There are reasons for believing that the proportion
+of coast-line to land area has been diminishing during geological
+time; in other words, that in early times the land was
+more insular and is now more continental. So that the 31
+millions of years may be much less than the period that would be
+required, even on the supposition of continuous uninterrupted
+denudation and sedimentation, during the whole of the time
+represented by the stratified formations.</p>
+
+<p>But no one who has made himself familiar with the actual
+composition of these formations and the detailed structure of the
+terrestrial crust can fail to recognize how vague, imperfect and
+misleading are the data on which such computations are founded.
+It requires no prolonged acquaintance with the earth&rsquo;s crust to
+impress upon the mind that one all-important element is omitted,
+and indeed can hardly be allowed for from want of sufficiently
+precise data, but the neglect of which must needs seriously
+impair the value of all numerical calculations made without it.
+The assumption that the stratified formations can be treated as
+if they consisted of a continuous unbroken sequence of sediments,
+indicating a vast and uninterrupted process of waste and deposition,
+is one that is belied on every hand by the actual structure
+of these formations. It can only give us a minimum of the time
+required; for, instead of an unbroken series, the sedimentary
+formations are full of &ldquo;unconformabilities&rdquo;&mdash;gaps in the
+sequence of the chronological records&mdash;as if whole chapters
+and groups of chapters had been torn out of a historical work.
+It can often be shown that these breaks of continuity must have
+been of vast duration, and actually exceeded in chronological
+importance thick groups of strata lying below and above them
+(see Part VI.). Moreover, even among the uninterrupted strata,
+where no such unconformabilities exist, but where the sediments
+<span class="pagenum"><a name="page652" id="page652"></a>652</span>
+follow each other in apparently uninterrupted sequence, and
+might be thought to have been deposited continuously at the
+same general rate, and without the intervention of any pause, it
+can be demonstrated that sometimes an inch or two of sediment
+<span class="correction" title="amended from much">might</span>, on certain horizons, represent the deposit of an enormously
+longer period than a hundred or a thousand times the same
+amount of sediment on other horizons. A prolonged study of
+these questions leads to a profound conviction that in many
+parts of the geological record the time represented by sedimentary
+deposits may be vastly less than the time which is not
+so represented.</p>
+
+<p>It has often been objected that the present rate of geological
+change ought not to be taken as a measure of the rate in past
+time, because the total sum of terrestrial energy has been steadily
+diminishing, and geological processes must consequently have
+been more vigorous in former ages than they are now. Geologists
+do not pretend to assert that there has been no variation
+or diminution in the activities of the various processes which
+they have to study. What they do insist on is that the
+present rate of change is the only one which we can watch and
+measure, and which will thus supply a statistical basis for any
+computations on the subject. But it has been dogmatically
+affirmed that because terrestrial energy has been diminishing
+therefore all kinds of geological work must have been more
+vigorously and more rapidly carried on in former times than
+now; that there were far more abundant and more stupendous
+volcanoes, more frequent and more destructive earthquakes,
+more gigantic upheavals and subsidences, more powerful oceanic
+waves and tides, more violent atmospheric disturbances with
+heavier rainfall and more active denudation.</p>
+
+<p>It is easy to make these assertions, and they look plausible;
+but, after all, they rest on nothing stronger than assumption.
+They can be tested by an appeal to the crust of the earth, in
+which the geological history of our planet has been so fully recorded.
+Had such portentous manifestations of geological
+activity ever been the normal condition of things since the
+beginning of that history, there ought to be a record of them in
+the rocks. But no evidence for them has been found there,
+though it has been diligently sought for in all quarters of the
+globe. We may confidently assert that while geological changes
+may quite possibly have taken place on a gigantic scale in the
+earliest ages of the earth&rsquo;s existence, of which no geological record
+remains, there is no proof that they have ever done so since the
+time when the very oldest of the stratified formations were
+deposited. There is no need to maintain that they have always
+been conducted precisely on the same scale as now, or to deny
+that they may have gradually become less vigorous as the general
+sum of terrestrial energy has diminished. But we may unhesitatingly
+affirm that no actual evidence of any such progressive
+diminution of activity has been adduced from the geological
+record in the crust of the earth: that, on the contrary, no appearances
+have been detected there which necessarily demand the
+assumption of those more powerful operations postulated by
+physicists, or which are not satisfactorily explicable by reference
+to the existing scale of nature&rsquo;s processes.</p>
+
+<p>That this conclusion is warranted even with regard to the innate
+energy of the globe itself will be seen if we institute a comparison
+between the more ancient and the more recent manifestations of
+that energy. Take, for example, the proofs of gigantic plication,
+fracture and displacement within the terrestrial crust. These,
+as they have affected the most ancient rocks of Europe, have
+been worked out in great detail in the north-west of Scotland.
+But they are not essentially different from or on a greater scale
+than those which have been proved to have affected the Alps,
+and to have involved strata of so recent a date as the older
+Tertiary formations. On the contrary, it may be doubted
+whether any denuded core of an ancient mountain-chain reveals
+traces of such stupendous disturbances of the crust as those
+which have given rise to the younger mountain-chains of the
+globe. It may, indeed, quite well have been the rule that instead
+of diminishing in intensity of effect, the consequences of terrestrial
+contraction have increased in magnitude, the augmenting
+thickness of the crust offering greater resistance to the stresses,
+and giving rise to vaster plications, faults, thrust-planes and
+metamorphism, as this growing resistance had to be overcome.</p>
+
+<p>The assertion that volcanic action must have been more
+violent and more persistent in ancient times than it is now has
+assuredly no geological evidence in its support. It is quite true
+that there are vastly more remains of former volcanoes scattered
+over the surface of the globe than there are active craters now,
+and that traces of copious eruptions of volcanic material can be
+followed back into some of the oldest parts of the geological
+record. But we have no proof that ever at any one time in
+geological history there have been more or larger or more vigorous
+volcanoes than those of recent periods. It may be said that the
+absence of such proof ought not to invalidate the assertion until
+a far wider area of the earth&rsquo;s surface has been geologically
+studied. But most assuredly, as far as geological investigation
+has yet gone, there is an overwhelming body of evidence to show
+that from the earliest epochs in geological history, as registered
+in the stratified rocks, volcanic action has manifested itself very
+much as it does now, but on a less rather than on a greater scale.
+Nowhere can this subject be more exhaustively studied than in
+the British Isles, where a remarkably complete series of volcanic
+eruptions has been chronicled ranging from the earliest Palaeozoic
+down to older Tertiary time. The result of a prolonged study
+of British volcanic geology has demonstrated that, even to
+minute points of detail, there has been a singular uniformity in
+the phenomena from beginning to end. The oldest lavas and
+ashes differ in no essential respect from the youngest. Nor have
+they been erupted more copiously or more frequently. Many
+successive volcanic periods have followed each other after prolonged
+intervals of repose, each displaying the same general
+sequence of phenomena and similar evidence of gradual diminution
+and extinction. The youngest, instead of being the feeblest,
+were the most extensive outbursts in the whole of this prolonged
+series.</p>
+
+<p>If now we turn for evidence of the alleged greater activity
+of all the epigene or superficial forces, and especially for proofs
+of more rapid denudation and deposition on the earth&rsquo;s surface,
+we search for it in vain among the stratified formations of the
+terrestrial crust. Had the oldest of these rocks been accumulated
+in a time of great atmospheric perturbation, of torrential rains,
+colossal tides and violent storms, we might surely expect to find
+among the sediments some proof of such disturbed meteorological
+and geographical conditions. We should look, on the one hand,
+for tumultuous accumulations of coarse unworn detritus, rapidly
+swept by rains, floods and waves from land to sea, and on the
+other hand, for an absence of any evidence of the tranquil and
+continuous deposit of such fine laminated silt as could only
+settle in quiet water. But an appeal to the geological record
+is made in vain for any such proofs. The oldest sediments, like
+the youngest, reveal the operation only of such agents and such
+rates of activity as are still to be witnessed in the accumulation
+of the same kind of deposits. If, for instance, we search the
+most ancient thick sedimentary formation in Britain&mdash;the
+Torridon Sandstone of north-west Scotland, which is older than
+the oldest fossiliferous deposits&mdash;we meet with nothing which
+might not be found in any Palaeozoic, Mesozoic or Cainozoic
+group of similar sediments. We see an accumulation, at least
+8000 or 10,000 ft. thick, of consolidated sand, gravel and mud,
+such as may be gathering now on the floor of any large mountain-girdled
+lake. The conglomerates of this ancient series are not
+pell-mell heaps of angular detritus, violently swept away from
+the land and huddled promiscuously on the sea-floor. They are,
+in general, built up of pebbles that have been worn smooth,
+rounded and polished by prolonged attrition in running water,
+and they follow each other on successive platforms with intervening
+layers of finer sediment. The sandstones are composed
+of well water-worn sand, some of which has been laid down so
+tranquilly that its component grains have been separated out in
+layers according to their specific gravity, in such manner that
+they now present dark laminae in which particles of magnetic
+iron, zircon and other heavy minerals have been sifted out
+<span class="pagenum"><a name="page653" id="page653"></a>653</span>
+together, just as iron-sand may be seen gathered into thin sheets
+on sandy beaches at the present day. Again, the same series
+of primeval sediments includes intercalations of fine silt, which
+has been deposited as regularly and intermittently there as it
+has been among the most recent formations. These bands of
+shale have been diligently searched for fossils, as yet without
+success; but they may eventually disclose organic remains older
+than any hitherto found in Europe.</p>
+
+<p>We now come to the consideration of the palaeontological
+evidence as to the value of geological time. Here the conclusions
+derived from a study of the structure of the sedimentary formations
+are vastly strengthened and extended. In the first place,
+the organization of the most ancient plants and animals furnishes
+no indication that they had to contend with any greater violence
+of storm, flood, wave or ocean-current than is familiar to their
+modern descendants. The oldest trees, shrubs, ferns and
+club-mosses display no special structures that suggest a difference
+in the general conditions of their environment. The most
+ancient crinoids, sponges, crustaceans, arachnids and molluscs
+were as delicately constructed as those of to-day, and their
+remains are often found in such perfect preservation as to show
+that neither during their lifetime nor after their death were they
+subject to any greater violence of the elements than their living
+representatives now experience. Of much more cogency,
+however, is the evidence supplied by the grand upward succession
+of organic forms, from the most ancient stratified rocks up to
+the present day. No biologist now doubts for a moment that
+this marvellous succession is the result of a gradual process of
+evolution from lower to higher types of organization. There
+may be differences of opinion as to the causes which have governed
+this process and the order of the steps through which it has
+advanced, but no one who is conversant with the facts will now
+venture to deny that it has taken place, and that, on any possible
+explanation of its progress, it must have demanded an enormous
+lapse of time. In the Cambrian or oldest fossiliferous formations
+there is already a large and varied fauna, in which the leading
+groups of invertebrate life are represented. On no tenable
+hypothesis can these be regarded as the first organisms that
+came into being on our planet. They must have had a long
+ancestry, and as Darwin first maintained, the time required for
+their evolution may have been &ldquo;as long as, or probably far
+longer than, the whole interval from the Silurian [Cambrian]
+age to the present day.&rdquo; The records of these earliest eras of
+organic development have unfortunately not survived the
+geological revolutions of the past; at least, they have not yet
+been recovered. But it cannot be doubted that they once
+existed and registered their testimony to the prodigious lapse of
+time prior to the deposition of the most ancient fossiliferous
+formations which have escaped destruction.</p>
+
+<p>The impressive character of the evidence furnished by the
+sequence of organic forms throughout the great series of fossiliferous
+strata can hardly be fully realized without a detailed and
+careful study of the subject. Professor E.B. Poulton, in an
+address to the zoological section of the British Association at
+the Liverpool Meeting in 1896, showed how overwhelming are
+the demands which this evidence makes for long periods of time,
+and how impossible it is of comprehension unless these demands
+be conceded. The history of life upon the earth, though it will
+probably always be surrounded with great and even insuperable
+difficulties, becomes broadly comprehensible in its general
+progress when sufficient time is granted for the evolution
+which it records; but it remains unintelligible on any other
+conditions.</p>
+
+<p>Taken then as a whole, the body of evidence, geological and
+palaeontological, in favour of the high antiquity of our globe
+is so great, so manifold, and based on such an ever-increasing
+breadth of observation and reflection, that it may be confidently
+appealed to in answer to the physical arguments which would
+seek to limit that antiquity to ten or twenty millions of years.
+In the present state of science it is out of our power to state
+positively what must be the lowest limit of the age of the earth.
+But we cannot assume it to be much less, and it may possibly
+have been much more, than the 100 millions of years which Lord
+Kelvin was at one time willing to concede.<a name="fa2c" id="fa2c" href="#ft2c"><span class="sp">2</span></a></p>
+
+<p class="pt2 center sc">Part III.&mdash;Geognosy. The Investigation of the Nature
+and Composition of the Materials of which the
+Earth Consists</p>
+
+<p>This division of the science is devoted to a description of the
+parts of the earth&mdash;of the atmosphere and ocean that surround
+the planet, and more especially of the solid materials that underlie
+these envelopes and extend downwards to an unknown distance
+into the interior. These various constituents of the globe are
+here considered as forms of matter capable of being analysed,
+and arranged according to their composition and the place they
+take in the general composition of the globe.</p>
+
+<p>Viewed in the simplest way the earth may be regarded as
+made up of three distinct parts, each of which ever since an
+early period of planetary history has been the theatre of important
+geological operations. (1) An envelope of air, termed
+the <i>atmosphere</i>, which surrounds the whole globe; (2) A lower
+and less extensive envelope of water, known as the <i>hydrosphere</i>
+(Gr. <span class="grk" title="hydōr">&#8021;&#948;&#969;&#961;</span>, water) which, constituting the oceans and seas,
+covers nearly three-fourths of the underlying solid surface of the
+planet; (3) A globe, called the <i>lithosphere</i> (Gr. <span class="grk" title="lithos">&#955;&#943;&#952;&#959;&#962;</span>, stone),
+the external part of which, consisting of solid stone, forms the
+<i>crust</i>, while underneath, and forming the vast mass of the
+interior, lies the <i>nucleus</i>, regarding the true constitution of
+which we are still ignorant.</p>
+
+<p>1. <i>The Atmosphere.</i>&mdash;The general characters of the atmosphere
+are described in separate articles (see especially <span class="sc"><a href="#artlinks">Atmosphere</a></span>;
+<span class="sc"><a href="#artlinks">Meteorology</a></span>). Only its relations to geology have here to be
+considered. As this gaseous envelope encircles the whole
+globe it is the most universally present and active of all the
+agents of geological change. Its efficacy in this respect arises
+partly from its composition, and the chemical reactions which
+it effects upon the surface of the land, partly from its great
+variations in temperature and moisture, and partly from its
+movements.</p>
+
+<div class="condensed">
+<p>Many speculations have been made regarding the chemical
+composition of the atmosphere during former geological periods.
+There can indeed be little doubt that it must originally have differed
+greatly from its present condition. If the whole mass of the planet
+originally existed in a gaseous state, there would be practically no
+atmosphere. The present outer envelope of air may be considered
+to be the surviving relic of this condition, after all the other constituents
+have been incorporated into the hydrosphere and lithosphere.
+The oxygen, which now forms fully a half of the outer
+crust of the earth, was doubtless originally, whether free or in
+combination, part of the atmosphere. So, too, the vast beds of coal
+found all over the world, in geological formations of many different
+ages, represent so much carbonic acid once present in the air. The
+chlorides and other salts in the sea may likewise partly represent
+materials carried down out of the atmosphere in the primitive
+condensation of the aqueous vapour, though they have been continually
+increased ever since by contributions from the drainage of
+the land. It has often been suggested that, during the Carboniferous
+period, the atmosphere must have been warmer and more charged
+with aqueous vapour and carbon dioxide than at the present day,
+to admit of so luxuriant a flora as that from which the coal-seams
+were formed. There seems, however, to be at present no method
+of arriving at any certainty on this subject. Lastly, the amount of
+carbonic acid absorbed in the weathering of rocks at the surface, and
+the consequent production of carbonates, represents an enormous
+abstraction of this gas.</p>
+
+<p>As at present constituted, the atmosphere is regarded as a
+<span class="pagenum"><a name="page654" id="page654"></a>654</span>
+mechanical mixture of nearly four volumes of nitrogen and one of
+oxygen, together with an average of 3.5 parts of carbon dioxide in
+every 10,000 parts of air, and minute quantities of various other
+gases and solid particles. Of the vapours contained in it by far the
+most important is that of water which, although always present,
+varies greatly in amount according to variations in temperature.
+By condensation the water vapour appears in visible form as dew,
+mist, cloud, rain, hail, snow and ice, and in these forms includes and
+carries down some of the other vapours, gases and solid particles
+present in the air. The circulation of water from the atmosphere to
+the land, from the land to the sea, and again from the sea to the
+land, forms the great geological process whereby the habitable
+condition of the planet is maintained and the surface of the land
+is sculptured (Part IV.).</p>
+</div>
+
+<p>2. <i>The Hydrosphere.</i>&mdash;The water envelope covers nearly
+three-fourths of the surface of the earth, and forms the various
+oceans and seas which, though for convenience of reference
+distinguished by separate names, are all linked together in one
+great body. The physical characters of this vast envelope are
+discussed in separate articles (see <span class="sc"><a href="#artlinks">Ocean</a></span> and <span class="sc"><a href="#artlinks">Oceanography</a></span>).
+Viewed from the geological standpoint, the features of the
+sea that specially deserve attention are first the composition of
+its waters, and secondly its movements.</p>
+
+<div class="condensed">
+<p>Sea-water is distinguished from that of ordinary lakes and rivers
+by its greater specific gravity and its saline taste. Its average
+density is about 1.026, but it varies even within the same ocean,
+being least where large quantities of fresh water are added from
+rain or melting snow and ice, and greatest where evaporation is most
+active. That sea-water is heavier than fresh arises from the greater
+proportion of salts which it contains in solution. These salts constitute
+about three and a half parts in every hundred of water.
+They consist mainly of chlorides of sodium and magnesium, the
+sulphates of magnesium, calcium and potassium, with minuter
+quantities of magnesium bromide and calcium carbonate. Still
+smaller proportions of other substances have been detected, gold for
+example having been found in the proportion of 1 part in 15,180,000.</p>
+
+<p>That many of the salts have existed in the sea from the time of
+its first condensation out of the primeval atmosphere appears to
+be probable. It is manifest, however, that, whatever may have
+been the original composition of the oceans, they have for a vast
+section of geological time been constantly receiving mineral matter
+in solution from the land. Every spring, brook and river removes
+various salts from the rocks over which it moves, and these substances,
+thus dissolved, eventually find their way into the sea.
+Consequently sea-water ought to contain more or less traceable
+proportions of every substance which the terrestrial waters can
+remove from the land, in short, of probably every element present
+in the outer shell of the globe, for there seems to be no constituent
+of this earth which may not, under certain circumstances, be held
+in solution in water. Moreover, unless there be some counteracting
+process to remove these mineral ingredients, the ocean water ought
+to be growing, insensibly perhaps, but still assuredly, <span class="correction" title="amended from salter">saltier</span>, for the
+supply of saline matter from the land is incessant.</p>
+
+<p>To the geologist the presence of mineral solutions in sea-water is
+a fact of much importance, for it explains the origin of a considerable
+part of the stratified rocks of the earth&rsquo;s crust. By evaporation
+the water has given rise to deposits of rock-salt, gypsum and other
+materials. The lime contained in solution, whether as sulphate or
+carbonate, has been extracted by many tribes of marine animals,
+which have thus built up out of their remains vast masses of solid
+limestone, of which many mountain-chains largely consist.</p>
+
+<p>Another important geological feature of the sea is to be seen
+in the fact that its basins form the great receptacles for the detritus
+worn away from the land. Besides the limestones, the visible parts
+of the terrestrial crust are, in large measure, composed of sedimentary
+rocks which were originally laid down on the sea-bottom. Moreover,
+by its various movements, the sea occupies a prominent place
+among the epigene or superficial agents which produce geological
+changes on the surface of the globe.</p>
+</div>
+
+<p>3. <i>The Lithosphere.</i>&mdash;Beneath the gaseous and liquid envelopes
+lies the solid part of the planet, which is conveniently regarded
+as consisting of two parts,&mdash;(<i>a</i>) the crust, and (<i>b</i>) the interior
+or nucleus.</p>
+
+<p>It was for a long time a prevalent belief that the interior of the
+globe is a molten mass round which an outer shell has gradually
+formed through cooling. Hence the term &ldquo;crust&rdquo;
+was applied to this external solid envelope, which
+<span class="sidenote">The crust.</span>
+was variously computed to be 10, 20, or more miles in thickness.
+The portion of this crust accessible to human observation was
+seen to afford abundant evidence of vast plications and corrugations
+of its substance, which were regarded as only explicable
+on the supposition of a thin solid collapsible shell floating on a
+denser liquid interior. When, however, physical arguments
+were adduced to show the great rigidity of the earth as a whole,
+the idea of a thin crust enclosing a molten nucleus was reluctantly
+abandoned by geologists, who found the problem of the earth&rsquo;s
+interior to be incapable of solution by any evidence which their
+science could produce. They continued, however, to use the
+term &ldquo;crust&rdquo; as a convenient word to denote the cool outer
+layer of the earth&rsquo;s mass, the structure and history of which
+form the main subjects of geological investigation. More
+recently, however, various lines of research have concurred in
+suggesting that, whatever may be the condition of the interior,
+its substance must differ greatly from that of the outer shell,
+and that there may be more reason than appeared for the
+retention of the name of crust. Observations on earthquake
+motion by Dr John Milne and others, show that the rate and
+character of the waves transmitted through the interior of the
+earth differ in a marked degree from those propagated along the
+crust. This difference indicates that rocky material, such as
+we know at the surface, may extend inwards for some 30 m.,
+below which the earth&rsquo;s interior rapidly becomes fairly homogeneous
+and possesses a high rigidity. From measurements
+of the force of gravity in India by Colonel S.G. Burrard, it has
+been inferred that the variations in density of the outer parts of
+the earth do not descend farther than 30 or 40 m., which might
+be assumed to be the limit of the thickness of the crust. Recent
+researches in regard to the radio-active substances present
+in rocks suggest that the crust is not more than 50 m. thick,
+and that the interior differs from it in possessing little or no
+radio-active material.</p>
+
+<p>Though we cannot hope ever to have direct acquaintance with
+more than the mere outside skin of our planet, we may be led
+to infer the irregular distribution of materials within
+the crust from the present distribution of land and
+<span class="sidenote">The interior.</span>
+water, and the observed differences in the amount of
+deflection of the plumb-line near the sea and near mountain-chains.
+The fact that the southern hemisphere is almost wholly
+covered with water appears explicable only on the assumption
+of an excess of density in the mass of that portion of the planet.
+The existence of such a vast sheet of water as that of the Pacific
+Ocean is to be accounted for, as Archdeacon J.H. Pratt pointed
+out, by the presence of &ldquo;some excess of matter in the solid
+parts of the earth between the Pacific Ocean and the earth&rsquo;s
+centre, which retains the water in its place, otherwise the ocean
+would flow away to the other parts of the earth.&rdquo; A deflection
+of the plumb-line towards the sea, which has in a number of
+cases been observed, indicates that &ldquo;the density of the crust
+beneath the mountains must be less than that below the plains,
+and still less than that below the ocean-bed.&rdquo; Apart therefore
+from the depression of the earth&rsquo;s surface in which the oceans
+lie, we must regard the internal density, whether of crust or
+nucleus, to be somewhat irregularly arranged, there being an
+excess of heavy materials in the water hemisphere, and beneath
+the ocean-beds, as compared with the continental masses.</p>
+
+<p>In our ignorance regarding the chemical constitution of the
+nucleus of our planet, an argument has sometimes been based
+upon the known fact that the specific gravity of the globe
+as a whole is about double that of the crust. This has been
+held by some writers to prove that the interior must consist of
+much heavier material and is therefore probably metallic. But
+the effect of pressure ought to make the density of the nucleus
+much higher, even if the interior consisted of matter no heavier
+than the crust. That the total density of the planet does not
+greatly exceed its observed amount seems only explicable on
+the supposition that some antagonistic force counteracts the
+effects of pressure. The only force we can suppose capable of so
+acting is heat. But comparatively little is yet known regarding
+the compression of gases, liquids and solids under such vast
+pressures as must exist within the nucleus.</p>
+
+<p>That the interior of the earth possesses a high temperature
+is inferred from the evidence of various sources. (1) Volcanoes,
+which are openings that constantly, or intermittently, give out
+hot vapours and molten lava from reservoirs beneath the crust.
+Besides active volcanoes, it is known that former eruptive vents
+<span class="pagenum"><a name="page655" id="page655"></a>655</span>
+have been abundantly and widely distributed over the globe
+from the earliest geological periods down to our own day.
+(2) Hot springs are found in many parts of the globe, with
+temperatures varying up to the boiling point of water. (3)
+From mines, tunnels and deep borings into the earth it has
+been ascertained that in all quarters of the globe below the
+superficial zone of invariable temperature, there is a progressive
+increase of heat towards the interior. The rate of this increase
+varies, being influenced, among other causes, by the varying
+conductivity of the rocks. But the average appears to be
+about 1° Fahr. for every 50 or 60 ft. of descent, as far down as
+observations have extended. Though the increase may not
+advance in the same proportion at great depths, the inference
+has been confidently drawn that the temperature of the nucleus
+must be exceedingly high.</p>
+
+<p>The probable condition of the earth&rsquo;s interior has been a fruitful
+source of speculation ever since geology came into existence;
+but no general agreement has been arrived at on the subject.
+Three chief hypotheses have been propounded: (1) that the
+nucleus is a molten mass enclosed within a solid shell; (2) that,
+save in local vesicular spaces which may be filled with molten
+or gaseous material, the globe is solid and rigid to the centre;
+(3) that the great body of the nucleus consists of incandescent
+vapours and gases, especially vaporous iron, which under the
+gigantic pressure within the earth are so compressed as to confer
+practical rigidity on the globe as a whole, and that outside this
+main part of the nucleus the gases pass into a shell of molten
+magma, which, in turn, shades off outwards into the comparatively
+thin, cool solidified crust. Recent seismological observations
+have led to the inference that the outer crust, some 30 to
+45 m. thick, must rapidly merge into a fairly homogeneous
+nucleus which, whatever be its constitution, transmits undulatory
+movements through its substance with uniform velocity and is
+believed to possess a high rigidity.</p>
+
+<p>The origin of the earth&rsquo;s high internal temperature has been
+variously accounted for. Most usually it has been assumed to
+be the residue of the original &ldquo;tracts of fluent heat&rdquo; out of
+which the planet shaped itself into a globe. According to another
+supposition the effects of the gradual gravitational compression
+of the earth&rsquo;s mass have been the main source of the high
+temperature. Recent researches in radio-activity, to which
+reference has already been made, have indicated another possible
+source of the internal heat in the presence of radium in the
+rocks of the crust. This substance has been detected in all
+igneous rocks, especially among the granites, in quantity
+sufficient, according to the Hon. R.J. Strutt, to account for the
+observed temperature-gradient in the crust, and to indicate
+that this crust cannot be more than 45 m. thick, otherwise the
+outflow of heat would be greater than the amount actually
+ascertained. Inside this external crust containing radio-active
+substances, it is supposed, as already stated, that the nucleus
+consists of some totally different matter containing little or no
+radium.</p>
+
+<div class="condensed">
+<p><i>Constitution of the Earth&rsquo;s Crust.</i>&mdash;As the crust of the earth contains
+the &ldquo;geological record,&rdquo; or stony chronicle from which geology
+interprets the history of our globe, it forms the main subject of study
+to the geologist. The materials of which this crust consists are
+known as minerals and rocks. From many chemical analyses,
+which have been made of these materials, the general chemical
+constitution of, at least, the accessible portion of the crust has been
+satisfactorily ascertained. This information becomes of much
+importance in speculations regarding the early history of the globe.
+Of the elements known to the chemist the great majority form but a
+small proportion of the composition of the crust, which is mainly
+built up of about twenty of them. Of these by far the most important
+are the non-metallic elements oxygen and silicon. The former
+forms about 47% and the latter rather more than 28% of the
+original crust, so that these two elements make up about three-fourths
+of the whole. Next after them come the metals aluminium
+(8.16%), iron (4.64), calcium (3.50), magnesium (2.62), sodium
+(2.63), and potassium (2.35). The other twelve elements included
+in the twenty vary in amount from a proportion of 0.41% in the
+case of titanium, to not more than 0.01% of chlorine, fluorine,
+chromium, nickel and lithium. The other fifty or more elements
+exist in such minute proportions in the crust that, probably, not
+one of them amounts to as much as 0.01%, though they include
+the useful metals, except iron. Taking the crust, and the external
+envelopes of the ocean and the air, we thus perceive that these
+outer parts of our planet consist of more than three-fourths of non-metals
+and less than one-fourth of metals.</p>
+
+<p>The combinations of the elements which are of most importance
+in the constitution of the terrestrial crust consist of oxides. From
+the mean of a large number of analyses of the rocks of the lower or
+primitive portion of the crust, it has been ascertained that silica
+(SiO<span class="su">2</span>) forms almost 60% and alumina (Al<span class="su">2</span>O<span class="su">3</span>) upwards of 15% of
+the whole. The other combinations in order of importance are
+lime (CaO) 4.90%, magnesia (MgO) 4.36, soda (Na<span class="su">2</span>O) 3.55, ferrous
+oxide (FeO) 3.52, potash (K<span class="su">2</span>O) 2.80, ferric oxide (Fe<span class="su">2</span>O<span class="su">3</span>) 2.63, water
+(H<span class="su">2</span>O) 1.52, titanium oxide (TiO<span class="su">2</span>) 0.60, phosphoric acid (P<span class="su">2</span>O<span class="su">5</span>)
+0.22; the other combinations of elements thus form less than 1%
+of the crust.</p>
+
+<p>These different combinations of the elements enter into further
+combinations with each other so as to produce the wide assortment
+of simple minerals (see <span class="sc"><a href="#artlinks">Mineralogy</a></span>). Thus, silica and alumina are
+combined to form the aluminous silicates, which enter so largely
+into the composition of the crust of the earth. The silicates of
+magnesia, potash and soda constitute other important families of
+minerals. A mass of material composed of one, but more usually
+of more than one mineral, is known as a <i>rock</i>. Under this term
+geologists are accustomed to class not only solid stone, such as
+granite and limestone, but also less coherent materials such as clay,
+peat and even loose sand. The accessible portion of the earth&rsquo;s
+crust consists of various kinds of rocks, which differ from each other
+in structure, composition and origin, and are therefore susceptible
+of diverse classifications according to the point of view from which
+they are considered. The details of this subject will be found in
+the article <span class="sc"><a href="#artlinks">Petrology</a></span>.</p>
+
+<p><i>Classification of Rocks.</i>&mdash;Various systems of classification of rocks
+have been proposed, but none of them is wholly satisfactory. The
+most useful arrangement for most purposes of the geologist is one
+based on the broad differences between them in regard to their mode
+of origin. From this point of view they may be ranged in three
+divisions:</p>
+
+<p>1. In the first place, a large number of rocks may be described
+as original or underived, for it is not possible to trace them back to
+any earlier source. They belong to the primitive constitution of the
+planet, and, as they have all come up from below through the crust,
+they serve to show the nature of the material which lies immediately
+below the outer parts of that crust. They include the numerous
+varieties of lava, which have been poured out in a molten state from
+volcanic vents, also a great series of other rocks which, though they
+may never have been erupted to the surface, have been forced
+upward in a melted condition into the other rocks of the crust and
+have solidified there. From their mode of origin this great class of
+rocks has been called &ldquo;igneous&rdquo; or &ldquo;eruptive.&rdquo; As they generally
+show no definite internal structure save such as may result from
+joints, they have been termed &ldquo;massive&rdquo; or &ldquo;unstratified,&rdquo; to
+distinguish them from those of the second division which are
+strongly marked out by the presence of a stratified structure. The
+igneous rocks present a considerable range of composition. For
+the most part they consist mainly of aluminous silicates, some of
+them being highly acid compounds with 75% or more of silica.
+But they also include highly basic varieties wherein the proportion
+of silica sinks to 40%, and where magnesia greatly predominates
+over alumina. The textures of igneous rocks likewise comprise a
+wide series of varieties. On the one hand, some are completely
+vitreous, like obsidian, which is a natural glass. From this extreme
+every gradation may be traced through gradual increase of the
+products of devitrification, until the mass may become completely
+crystalline. Again, some crystalline igneous rocks are so fine in
+grain as not to show their component crystals save under the microscope,
+while in others the texture is so coarse as to present the
+component minerals in separate crystals an inch or more in length.
+These differences indicate that, at first, the materials of the rock
+may have been as completely molten as artificial glass, and that
+the crystalline condition has been subsequently developed by cooling,
+and the separation of the chemical constituents into definite crystalline
+minerals. Many of the characters of igneous rocks have been
+reproduced experimentally by fusing together their minerals, or the
+constituents of their minerals, in the proper proportion. But it has
+not yet been found possible to imitate the structure of such rocks
+as granite. Doubtless these rocks consolidated with extreme
+slowness at great depths below the surface, under vast pressures
+and probably in the presence of water or water-vapour&mdash;conditions
+which cannot be adequately imitated in a laboratory.</p>
+
+<p>Though the igneous rocks occupy extensive areas in some countries,
+they nevertheless cover a much smaller part of the whole surface of
+the land than is taken up by the second division or stratified rocks.
+But they increase in quantity downwards and probably extend
+continuously round the globe below the other rocks. This important
+series brings before us the relations of the molten magma within the
+earth to the overlying crust and to the outer surface. On the one
+hand, it includes the oldest and most deep-seated extravasations
+of that magma, which have been brought to light by ruptures and
+upheavals of the crust and prolonged denudation. On the other,
+it presents to our study the varied outpourings of molten and
+fragmentary materials in the discharges of modern and ancient
+<span class="pagenum"><a name="page656" id="page656"></a>656</span>
+volcanoes. Between these two extremes of position and age, we
+find that the crust has been, as it were, riddled with injections of
+the magma from below. These features will be further noticed in
+Part V. of this article.</p>
+
+<p>2. The &ldquo;sedimentary&rdquo; or &ldquo;stratified rocks&rdquo; form by much the
+larger part of the dry land of the globe, and they are prolonged to
+an unknown distance from the shores under the bed of the sea.
+They include those masses of mineral matter which, unlike the
+igneous rocks, can be traced back to a definite origin on the surface
+of the earth. Three distinct types may be recognized among them:
+(<i>a</i>) By far the largest proportion of them consists of different kinds
+of sediment derived from the disintegration of pre-existing rocks.
+In this &ldquo;fragmental&rdquo; group are placed all the varieties of shingle,
+gravel, sand, clay and mud, whether these materials remain in a
+loose incoherent condition, or have been compacted into solid stone.
+(<i>b</i>) Another group consists of materials that have been deposited by
+chemical precipitation from solution in water. The white sinter
+laid down by calcareous springs is a familiar example on a small
+scale. Beds of rock-salt, gypsum and dolomite have, in some
+regions, been accumulated to a thickness of many thousand feet,
+by successive precipitations of the salt contained in the water of
+inland seas. (<i>c</i>) An abundant and highly important series of sedimentary
+formations has been formed from the remains of plants and
+animals. Such accumulations may arise either from the transport
+and deposit of these remains, as in the case of sheets of drift-wood,
+and banks of drifted sea-shells, or from the growth and decay of
+the organisms on the spot, as happens in peat bogs and in coral-reefs.</p>
+
+<p>As the sedimentary rocks have for the most part been laid down
+under water, and more especially on the sea-floor, they are often
+spoken of as &ldquo;aqueous,&rdquo; in contradistinction to the igneous rocks.
+Some of them, however, are accumulated by the drifting action of
+wind upon loose materials, and are known as &ldquo;aeolian&rdquo; formations.
+Familiar instances of such wind-formed deposits are the sand-dunes
+along many parts of the sea coast. Much more extensive in area are
+the sands of the great deserts in the arid regions of the globe.</p>
+
+<p>It is from the sedimentary rocks that the main portion of geological
+history is derived. They have been deposited one over another
+in successive strata from a remote period in the development of
+the globe down to the present time. From this arrangement they
+have been termed &ldquo;stratified,&rdquo; in contrast to the unstratified or
+igneous series. They have preserved memorials of the geographical
+revolutions which the surface of the earth has undergone; and
+above all, in the abundant fossils which they have enclosed, they
+furnish a momentous record of the various tribes of plants and
+animals which have successively flourished on land and sea. Their
+investigation is thus the most important task which devolves upon
+the geologist.</p>
+
+<p>3. In the third place comes a series of rocks which are not now
+in their original condition, but have undergone such alteration as
+to have acquired new characters that more or less conceal their
+first structures. Some of them can be readily recognized as altered
+igneous masses; others are as manifestly of sedimentary origin;
+while of many it is difficult to decide what may have been their
+pristine character. To this series the term &ldquo;metamorphic&rdquo; has
+been applied. Its members are specially distinguished by a prevailing
+fissile, or schistose, structure which they did not at first possess, and
+which differs from anything found in unaltered igneous or sedimentary
+rocks. This fissility is combined with a more or less pronounced
+crystalline structure. These changes are believed to be the result
+of movements within the crust of the earth, whereby the most solid
+rocks were crushed and sheared, while, at the same time, under the
+influence of a high temperature and the presence of water, they
+underwent internal chemical reactions, which led to a rearrangement
+and recomposition of their mineral constituents and the production
+of a crystalline structure (see <span class="sc"><a href="#artlinks">Metamorphism</a></span>).</p>
+
+<p>Among the less altered metamorphic rocks of sedimentary origin,
+the successive laminae of deposit of the original sediment can be
+easily observed; but they are also traversed by a new set of divisional
+planes, along which they split across the original bedding.
+Together with this superinduced cleavage there have been developed
+in them minute hairs, scales and rudimentary crystals. Further
+stages of alteration are marked by the increase of micaceous scales,
+garnets and other minerals, especially along the planes of cleavage,
+until the whole rock becomes crystalline, and displays its chief
+component minerals in successive discontinuous folia which merge
+into each other, and are often crumpled and puckered. Massive
+igneous rocks can be observed to have undergone intense crushing
+and cleavage, and to have ultimately assumed a crystalline foliated
+character. Rocks which present this aspect are known as schists
+(<i>q.v.</i>). They range from the finest silky slates, or phyllites, up to the
+coarsest gneisses, which in hand-specimens can hardly be distinguished
+from granites. There is indeed every reason to believe
+that such gneisses were probably originally true granites, and that
+their foliation and recrystallization have been the result of metamorphism.</p>
+
+<p>The schists are more especially to be found in the heart of
+mountain-chains, and in regions where the lowest and oldest parts
+of the earth&rsquo;s crust have, in the course of geological revolutions,
+been exposed to the light of day. They have been claimed by some
+writers to be part of the original or primitive surface of our globe
+that first consolidated on the molten nucleus. But the progress of
+investigation all over the world has shown that this supposition
+cannot be sustained. The oldest known rocks present none of the
+characters of molten material that has cooled and hardened in the
+air, like the various forms of recent lava. On the contrary, they
+possess many of the features characteristic of bodies of eruptive
+material that have been injected into the crust at some depth underground,
+and are now visible at the surface, owing to the removal
+by denudation of the rocks under which they consolidated. In their
+less foliated portions they can be recognized as true eruptive rocks.
+In many places gneisses that possess a thoroughly typical foliation
+have been found to pierce ancient sedimentary formations as intrusive
+bosses and veins.</p>
+</div>
+
+<p class="pt2 center sc">Part IV.&mdash;Dynamical Geology</p>
+
+<p>This section of the science includes the investigation of those
+processes of change which are at present in progress upon the
+earth, whereby modifications are made on the structure and
+composition of the crust, on the relations between the interior
+and the surface, as shown by volcanoes, earthquakes and other
+terrestrial disturbances, on the distribution of oceans and
+continents, on the outlines of the land, on the form and depth
+of the sea-bottom, on climate, and on the races of plants and
+animals by which the earth is tenanted. It brings before us,
+in short, the whole range of activities which it is the province of
+geology to study, and leads us to precise notions regarding their
+relations to each other and the results which they achieve. A
+knowledge of this branch of the subject is thus the essential
+groundwork of a true and fruitful acquaintance with the principles
+of geology, seeing that it necessitates a study of the present order
+of nature, and thus provides a key for the interpretation of the
+past.</p>
+
+<p>The whole range of operations included within the scope of
+inquiry in this branch of the science may be regarded as a vast
+cycle of change, into which we may break at any point, and
+round which we may travel, only to find ourselves brought
+back to our starting-point. It is a matter of comparatively
+small moment at what part of the cycle we begin our inquiries.
+We shall always find that the changes we see in action have
+resulted from some that preceded, and give place to others
+which follow them.</p>
+
+<p>At an early time in the earth&rsquo;s history, anterior to any of the
+periods of which a record remains in the visible rocks, the chief
+sources of geological action probably lay within the earth itself.
+If, as is generally supposed, the planet still retained a great
+store of its initial heat, it was doubtless the theatre of great
+chemical changes, giving rise, perhaps, to manifestations of
+volcanic energy somewhat like those which have so marvellously
+roughened the surface of the moon. As the outer layers of the
+globe cooled, and the disturbances due to internal heat and
+chemical action became less marked, the conditions would
+arise in which the materials for geological history were accumulated.
+The influence of the sun, which must always have
+operated, would then stand out more clearly, giving rise to that
+wide circle of superficial changes wherein variations of temperature
+and the circulation of air and water over the surface of the
+earth come into play.</p>
+
+<p>In the pursuit of his inquiries into the past history and into
+the present <i>régime</i> of the earth, the geologist must needs keep
+his mind ever open to the reception of evidence for kinds
+and especially for degrees of action which he had not before
+imagined. Human experience has been too short to allow him
+to assume that all the causes and modes of geological change
+have been definitively ascertained. On the earth itself there may
+remain for future discovery evidence of former operations by
+heat, magnetism, chemical change or otherwise, which may
+explain many of the phenomena with which geology has to deal.
+Of the influences, so many and profound, which the sun exerts
+upon our planet, we can as yet only perceive a little. Nor can
+we tell what other cosmical influences may have lent their aid in
+the evolution of geological changes.</p>
+
+<p>Much useful information regarding many geological processes
+has been obtained from experimental research in laboratories
+and elsewhere, and much more may be confidently looked for
+<span class="pagenum"><a name="page657" id="page657"></a>657</span>
+from future extensions of this method of inquiry. The early
+experiments of Sir James Hall, already noticed, formed the
+starting-point for numerous subsequent researches, which have
+elucidated many points in the origin and history of rocks. It
+is true that we cannot hope to imitate those operations of nature
+which demand enormous pressures and excessively high temperatures
+combined with a long lapse of time. But experience
+has shown that in regard to a large number of processes, it is
+possible to imitate nature&rsquo;s working with sufficient accuracy
+to enable us to understand them, and so to modify and control
+the results as to obtain a satisfactory solution of some geological
+problems.</p>
+
+<p>In the present state of our knowledge, all the geological
+energy upon and within the earth must ultimately be traced
+back to the primeval energy of the parent nebula or sun. There
+is, however, a certain propriety and convenience in distinguishing
+between that part of it which is due to the survival of some of
+the original energy of the planet and that part which arises
+from the present supply of energy received day by day from the
+sun. In the former case we have to deal with the interior of
+the earth, and its reaction upon the surface; in the latter, we
+deal with the surface of the earth and to some extent with its
+reaction on the interior. This distinction allows of a broad
+treatment of the subject under two divisions:</p>
+
+<p>I. Hypogene or Plutonic Action: The changes within the
+earth caused by internal heat, mechanical movement and
+chemical rearrangements.</p>
+
+<p>II. Epigene or Surface Action: The changes produced on the
+superficial parts of the earth, chiefly by the circulation of air
+and water set in motion by the sun&rsquo;s heat.</p>
+
+<p class="pt2 center"><i>DIVISION I.&mdash;HYPOGENE OR PLUTONIC ACTION</i></p>
+
+<p>In the discussion of this branch of the subject we must carry
+in our minds the conception of a globe still possessing a high
+internal temperature, radiating heat into space and consequently
+contracting in bulk. Portions of molten rocks from inside are
+from time to time poured out at the surface. Sudden shocks
+are generated by which destructive earthquakes are propagated
+through the diameter of the globe as well as to and along
+its surface. Wide geographical areas are pushed up or sink
+down. In the midst of these movements remarkable changes
+are produced upon the rocks of the crust; they are plicated,
+fractured, crushed, rendered crystalline and even fused.</p>
+
+<div class="condensed">
+<p class="pt2 center">(A) <i>Volcanoes and Volcanic Action.</i></p>
+
+<p>This subject is discussed in the article <span class="sc"><a href="#artlinks">Volcano</a></span>, and only a
+general view of its main features will be given here. Under the term
+volcanic action (vulcanism, vulcanicity) are embraced all the
+phenomena connected with the expulsion of heated materials from
+the interior of the earth to the surface. A volcano may be defined
+as a conical hill or mountain, built up wholly or mainly of materials
+which have been ejected from below, and which have accumulated
+around the central vent of eruption. As a rule its truncated summit
+presents a cup-shaped cavity, termed the crater, at the bottom of
+which is the opening of the main funnel or pipe whereby communication
+is maintained with the heated interior. From time to
+time, however, in large volcanoes rents are formed on the sides of
+the cone, whence steam and other hot vapours and also streams of
+molten lava are poured forth. On such rents smaller or parasitic
+cones are often formed, which imitate the operations of the parent
+cone and, after repeated eruptions, may rise to hills hundreds of
+feet in height. In course of centuries the result of the constant
+outpouring of volcanic materials may be to build up a large mountain
+like Etna, which towers above the sea to a height of 10,840 feet, and
+has some 200 minor cones along its flanks.</p>
+
+<p>But all volcanic eruptions do not proceed from central orifices.
+In Iceland it has been observed that, from fissures opened in the
+ground and extending for long distances, molten material has issued
+in such abundance as to be spread over the surrounding country
+for many miles, while along the lines of fissure small cones or hillocks
+of fragmentary material have accumulated round more active parts
+of the rent. There is reason to believe that in the geological past
+this fissure-type of eruption has repeatedly been developed, as well
+as the more common form of central cones like Vesuvius or Etna.</p>
+
+<p>In the operations of existing volcanoes only the superficial manifestations
+of volcanic action are observable. But when the rocks of
+the earth&rsquo;s crust are studied, they are found to enclose the relics
+of former volcanic eruptions. The roots of ancient volcanoes have
+thus been laid bare by geological revolutions; and some of the
+subterranean phases of volcanic action are thereby revealed which
+are wholly concealed in an active volcano. Hence to obtain as
+complete a conception as possible of the nature and history of
+volcanic action, regard must be had, not merely to modern volcanoes,
+but to the records of ancient eruptions which have been preserved
+within the crust.</p>
+
+<p>The substances discharged from volcanic vents consist of&mdash;(1)
+Gases and vapours: which, dissolved in the molten magma of the
+interior, take the chief share in volcanic activity. They include
+in greatest abundance water-gas, which condenses into the clouds
+of steam so conspicuous in volcanic eruptions. Hydrochloric acid
+and sulphuretted hydrogen are likewise plentiful, together with
+many other substances which, sublimed by the high internal temperature,
+take a solid form on cooling at the surface. (2) Molten
+rock or lava: which ranges from the extremely acid type of the
+obsidians and rhyolites with 70% or more of silica, to the more basic
+and heavy varieties such as basalts and leucite-lavas with much iron,
+and sometimes no more than 45% of silica. The specific gravity
+of lavas varies between 2.37 and 3.22, and the texture ranges from
+nearly pure glass, like obsidian, to a coarse granitoid compound,
+as in some rhyolites. (3) Fragmentary materials, which are sometimes
+discharged in enormous quantity and dispersed over a wide extent
+of country, the finer particles being transported by upper air-currents
+for hundreds of miles. These materials arise either from the explosion
+of lava by the sudden expansion of the dissolved vapours and gases,
+as the molten rock rises to the surface, or from the breaking up and
+expulsion of portions of the walls of the vent, or of the lava, which
+happens to have solidified within these walls. They vary from the
+finest impalpable dust and ashes, through increasing stages of
+coarseness up to huge &ldquo;bombs&rdquo; torn from the upper surface of the
+molten rock in the vent, and large blocks of already solidified lava,
+or of non-volcanic rock detached from the sides of the pipe up which
+the eruptions take place.</p>
+
+<p>Nothing is yet known as to the determining cause of any particular
+volcanic eruption. Some vents, like that of Stromboli, in the
+Mediterranean, are continually active, and have been so ever since
+man has observed them. Others again have been only intermittently
+in eruption, with intervals of centuries between their periods of
+activity. We are equally in the dark as to what has determined
+the sites on which volcanic action has manifested itself. There is
+reason, indeed, to believe that extensive fractures of the terrestrial
+crust have often provided passages up which the vapours, imprisoned
+in the internal magma, have been able to make their way, accompanied
+by other products. Where chains of volcanoes rise along
+definite lines, like those of Sumatra, Java, and many other tracts
+both in the Old and the New World, there appears to be little doubt
+that their linear distribution should be attributed to this cause.
+But where a volcano has appeared by itself, in a region previously
+exempt from volcanic action, the existence of a contributing fissure
+cannot be so confidently presumed. The study of certain ancient
+volcanoes, the roots of which have been exposed by long denudation,
+has shown an absence of any visible trace of their having availed
+themselves of fractures in the crust. The inference has been drawn
+that volcanic energy is capable of itself drilling an orifice through the
+crust, probably at some weaker part, and ejecting its products at
+the surface. The source of this energy is to be sought in the enormous
+expansive force of the vapours and gases dissolved in the magma.
+They are kept in solution by the enormous pressure within the earth;
+but as the lava approaches the surface and this pressure is relieved
+these dissolved vapours and gases rush out with explosive violence,
+blowing the upper part of the lava column into dust, and allowing
+portions of the liquid mass below to rise and escape, either from the
+crater or from some fissure which the vigour of explosion has opened
+on the side of the cone. So gigantic is the energy of these pent-up
+vapours, that, after a long period of volcanic quiescence, they
+sometimes burst forth with such violence as to blow off the whole of
+the upper part or even one side of a large cone. The history of
+Vesuvius, and the great eruptions of Krakatoa in 1883 and of
+Bandaizan in 1888 furnish memorable examples of great volcanic
+convulsions. It has been observed that such stupendous discharges
+of aeriform and fragmentary matter may be attended with the
+emission of little or no lava. On the other hand, some of the largest
+outflows of lava have been accompanied by comparatively little
+fragmentary material. Thus, the great lava-floods of Iceland in
+1783 spread for 40 m. away from their parent fissure, which was
+marked only by a line of little cones of slag.</p>
+
+<p>The temperature of lava as it issues from underground has been
+measured more or less satisfactorily, and affords an indication of
+that existing within the earth. At Vesuvius it has been ascertained
+to be more than 2000° Fahr. At first the molten rock glows with a
+white light, which rapidly reddens, and disappears under the rugged
+brown and black crust that forms on the surface. Underneath this
+badly conducting crust, the lava cools so slowly that columns of
+steam have been noticed rising from its surface more than 80 years
+after its eruption.</p>
+
+<p>Considerable alteration in the topography of volcanic regions
+may be produced by successive eruptions. The fragmentary
+materials are sometimes discharged in such abundance as to cover
+the ground for many miles around with a deposit of loose ashes,
+cinders and slag. Such a deposit accumulating to a depth of many
+<span class="pagenum"><a name="page658" id="page658"></a>658</span>
+feet may completely bury valleys and water-courses, and thus
+greatly affect the drainage. The coarsest materials accumulate
+nearest to the vent that emits them. The finer dust is not infrequently
+hurled forth with such an impetus as to be carried for
+thousands of feet into the tracks of upper air-currents, whereby it
+may be borne for hundreds of miles away from the vent so as ultimately
+to fall to the ground in countries far removed from any active
+volcano. Outflows of lava, from their greater solidity and durability,
+produce still more serious and lasting changes in the external features
+of the ground over which they flow. As they naturally seek the
+lowest levels, they find their way into the channels of streams.
+If they keep along the channels, they seal them up under a mass of
+compact stone which the running water, if not wholly diverted
+elsewhere, will take many long centuries to cut through. If, on the
+other hand, the lava crosses a stream, it forms a massive dam,
+above which the water is ponded back so as to form a lake.</p>
+
+<p>As the result of prolonged activity a volcanic cone is gradually
+built up by successive outflows of lava and showers of dust and
+stones. These materials are arranged in beds, or sheets, inclined
+outwards from the central vent. On surrounding level ground the
+alternating beds are flat. In course of time, deep gullies are cut on
+the outer slopes of the cone by rain, and by the heavy showers that
+arise from the condensation of the copious discharges of steam
+during eruptions. Along the sides of these ravines instructive
+sections may be studied of the volcanic strata. The larger rivers of
+some volcanic regions have likewise eroded vast gorges in the more
+horizontal lavas and ashes of the flatter country, and have thus laid
+bare stupendous cliffs, along which the successive volcanic sheets
+can be seen piled above each other for many hundred feet. On a
+small scale, some of these features are well displayed among the
+rivers that drain the volcanic tracts of central France; on a great
+scale, they are presented in the course of the Snake river, and other
+streams that traverse the great volcanic country of western North
+America. Similar volcanic scenery has been produced in western
+Europe by the action of denudation in dissecting the flat Tertiary
+lavas of Scotland, the Faeroe Isles and Iceland.</p>
+
+<p>Of special interest to the geologist are those volcanoes which have
+taken their rise on the sea-bottom; for the volcanic intercalations
+among the stratified formations of the earth&rsquo;s crust are almost
+entirely of submarine origin. Many active volcanoes situated on
+islands have begun their eruptions below sea-level. Both Vesuvius
+and Etna sprang up on the floor of the Mediterranean sea, and have
+gradually built up their cones into conspicuous parts of the dry land.
+Examples of a similar history are to be found among the volcanic
+islands of the Pacific Ocean. In some of these cases a movement
+of elevation has carried the submarine lavas, tuffs and agglomerates
+above sea-level, and has furnished opportunities of comparing these
+materials with those of recent subaerial origin, and also with the
+ancient records of submarine eruptions which have been preserved
+among the stratified formations. From the evidence thus supplied,
+it can be shown that the materials ejected from modern submarine
+volcanic vents closely resemble those accumulated by subaerial
+volcanoes; that the dust, ashes and stones become intermingled or
+interstratified with coral-mud, or other non-volcanic deposit of the
+sea-bottom, that vesicular lavas may be intercalated among them
+as on land, and that between the successive sheets of volcanic
+origin, layers of limestone may be laid down which are composed
+chiefly, or wholly, of the remains of calcareous marine organisms.</p>
+
+<p>Though active volcanoes are widely distributed over the globe,
+and are especially abundant around the vast basin of the Pacific
+Ocean, they afford an incomplete picture of the extent to which
+volcanic action has displayed itself on the surface of our planet.
+When the rocks of the land are attentively studied they disclose
+proofs of that action in many districts where there is now no outward
+sign of it. Not only so, but they reveal that volcanoes have been in
+eruption in some of these districts during many different periods of
+the past, back to the beginnings of geological history. The British
+Islands furnish a remarkable example of such a series of ancient
+eruptions. From the Cambrian period all through Palaeozoic times
+there rose at intervals in that country a succession of volcanic centres
+from some of which thousands of feet of lavas and tuffs were discharged.
+Again in older Tertiary times the same region witnessed
+a stupendous outpouring of basalt, the surviving relics of which
+are more than 3000 ft. thick, and cover many hundreds of square
+miles. Similar evidence is supplied in other countries both in the
+Old and the New world. Hence it is proved that, in the geological
+past, volcanic action has been vigorous at long intervals on the same
+sites during a vast series of ages, though no active vents are to be
+seen there now. The volcanoes now active form but a small proportion
+of the total number which has appeared on the surface of
+the earth.</p>
+
+<p>With regard to the cause of volcanic action much has been
+speculated, but little can be confidently affirmed. That water in
+the form of occluded gas plays the chief part in forcing the lava
+column up a volcanic chimney, and in the violent explosions that
+accompany the rise of the molten material, is generally admitted.
+But opinions differ as to the source of this water. According to
+some investigators, it should be regarded as in large measure of
+meteoric origin, derived from the descent of rain into the earth, and
+its absorption by the molten magma in the interior. Others, contending
+that the supply so furnished, even if it could reach and be
+dissolved in the magma, would yet be insufficient to furnish the
+prodigious quantity of aqueous vapour discharged during an eruption,
+maintain that the water belongs to the magma itself. They point
+to the admitted fact that many substances, particularly metals in
+a state of fusion, can absorb large quantities of vapours and gases
+without chemical combination, and on cooling discharge them with
+eruptive phenomena somewhat like those of volcanoes. This
+question must be regarded as one of the still unsolved problems of
+geology.</p>
+
+<p class="pt2 center">(B) <i>Movements of the Earth&rsquo;s Crust.</i></p>
+
+<p>Among the hypogene forces in geological dynamics an important
+place must be assigned to movements of the terrestrial crust. Though
+the expression &ldquo;the solid earth&rdquo; has become proverbial, it appears
+singularly inappropriate in the light of the results obtained in recent
+years by the use of delicate instruments of observation. With the
+facilities supplied by these instruments (see <span class="sc"><a href="#artlinks">Seismometer</a></span>), it has
+been ascertained that the ground beneath our feet is subject to
+continual slight tremors, and feeble pulsations of longer duration,
+some of which may be due to daily or seasonal variations of temperature,
+atmospheric pressure or other meteorological causes.
+The establishment of self-recording seismometers all over the world
+has led to the detection of many otherwise imperceptible shocks,
+over and above the appreciable earth-waves propagated from earthquake
+centres of disturbance. Moreover, it has been ascertained
+that some parts of the surface of the land are slowly rising, while
+others are falling with reference to the sea-level. From time to
+time the surface suffers calamitous devastation from earthquakes,
+when portions of the crust under great strain suddenly give way.
+Lastly, at intervals, probably separated from each other by vast
+periods of time, the terrestrial crust undergoes intense plication
+and fracture, and is consequently ridged up into mountain-chains.
+No event of this kind has been witnessed since man began to record
+his experiences. But from the structure of mountains, as laid open
+by prolonged denudation, it is possible to form a vivid conception
+of the nature and effects of these most stupendous of all geological
+revolutions.</p>
+
+<p>In considering this department of geological inquiry it will be
+convenient to treat it under the following heads: (1) Slow depression
+and upheaval; (2) Earthquakes; (3) Mountain-making; (4)
+Metamorphism of rocks.</p>
+
+<p>1. <i>Slow Depression and Upheaval.</i>&mdash;On the west side of Japan
+the land is believed to be sinking below the sea, for fields are replaced
+by beaches of sand or shingle, while the depth of the sea off shore
+has perceptibly increased. A subsidence of the south of Sweden has
+taken place in comparatively recent times, for streets and foundations
+of houses at successive levels are found below high-water mark.
+The west coast of Greenland over an extent of more than 600 m.
+is sinking, and old settlements are now submerged. Proofs of
+submergence of land are furnished by &ldquo;submerged forests,&rdquo; and
+beds of terrestrial peat now lying at various depths below the level
+of the sea, of which many examples have been collected along the
+shores of the British Isles, Holland and France. Interesting evidence
+that the west of Europe now stands at a lower level than it did at a
+late geological period is supplied in the charts of the North Sea and
+Atlantic, which show that the valleys of the land are prolonged
+under the sea. These valleys have been eroded out of the rocks by
+the streams which flow in them, and the depth of their submerged
+portions below the sea level affords an indication of the extent of the
+subsidence.</p>
+
+<p>The uprise of land has been detected in various parts of the world.
+One of the most celebrated instances is that of the shores of the Gulf
+of Bothnia, where, at Stockholm, the elevation, between the years
+1774 and 1875, appears to have been 48 centimetres (18½ in.) in
+a century. But on the west side of Sweden, fronting the Skager Rak,
+the coast, between the years 1820 and 1870, rose 30 centimetres,
+which is at the rate of 60 centimetres, or nearly 2 ft. in a century.
+In the region of the Great Lakes in the interior of Canada and the
+United States it has been ascertained that the land is undergoing a
+slow tilt towards the south-west, of which the mean rate appears to
+be rather less than 6 in. in a century. If this rate of change should
+continue the waters of Lake Michigan, owing to the progress of the
+tilt, will, in some 500 or 600 years, submerge the city of Chicago,
+and eventually the drainage of the lakes will be diverted into the
+basin of the Mississippi. Proof of recent emergence of land is supplied
+by what are called &ldquo;raised beaches&rdquo; or &ldquo;strand-lines,&rdquo; that is,
+lines of former shores marked by sheets of littoral deposits, or
+platforms cut by shore-waves in rock and flanked by old sea-cliffs
+and lines of sea-worn caves. Admirable examples of these features
+are to be seen along the west coast of Europe from the south of
+England to the north of Norway. These lines of old shores become
+fainter in proportion to their antiquity. In Britain they occur at
+various heights, the platforms at 25, 50 and 100 ft. being well
+marked.</p>
+
+<p>The cause of these slow upward and downward movements of the
+crust of the earth is still imperfectly understood. Upheaval might
+conceivably be produced by an ascent of the internal magma, and the
+consequent expansion of the overlying crust by heat; while depression
+might follow any subsidence of the magma, or its displacement
+<span class="pagenum"><a name="page659" id="page659"></a>659</span>
+to another district. If, as is generally believed, the globe is still
+contracting, the shrinkage of the surface may cause both these
+movements. Subsidence will be in excess, but between subsiding
+tracts lateral thrust may suffice to push upward intervening more
+solid and stable ground; but no solution of the problem yet proposed
+is wholly satisfactory.</p>
+
+<p>2. <i>Earthquakes.</i>&mdash;As this subject is discussed in a separate article
+it will be sufficient here to take note of its more important geological
+bearings. It was for many centuries taken for granted that earthquakes
+and volcanoes are due to a common cause. We have seen
+that in classical antiquity they were looked on as the results of the
+movements of wind imprisoned within the earth. Long after this
+notion was discarded, and a more scientific appreciation of volcanic
+action was reached, it was still thought that earthquakes should be
+regarded as manifestations of the same source of energy as that
+which displays itself in volcanic eruptions. It is true that earthquakes
+are frequent in districts of active volcanoes, and they may
+undoubtedly be often due there to the explosions of the magma,
+or to the rupture of rocks caused by its ascent towards the surface.
+But such shocks are comparatively local in their range and feeble
+in their effects. There is now a general agreement that between the
+great world-shaking earthquakes and volcanic phenomena, no
+immediate and intimate relationship can be traced, though they may
+be connected in ways which are not yet perceived. Some of the
+more recent great earthquakes on land have proved that the waves
+of shock are produced by the sudden rupture or collapse of rocks
+under great strain, either along lines of previous fracture or of new
+rents in the terrestrial crust; and that such ruptures may occur at
+a remote distance from any volcano. Thus the recent disastrous
+San Francisco earthquake has been recognized to have resulted from
+a slipping of ground along the line of an old fault, which has been
+traced for a long distance in California generally parallel to the
+coast. The position of this fault at the surface has long been clearly
+followed by its characteristic topography. After the earthquake
+these superficial features were found to have been removed by the
+same cause that had originated them. For some 300 m. on the track
+of this old fault-line a renewed slipping was seen to have taken place
+along one or both sides, and the ground at the surface was ruptured
+as well as displaced horizontally. Obviously, the jar occasioned by
+the sudden and simultaneous subsidence of a portion of the earth&rsquo;s
+crust several hundred miles long, must be far more serious than
+could be produced by an earthquake radiating from a single local
+volcanic focus.</p>
+
+<p>From their disastrous effects on buildings and human lives, an
+exaggerated importance has been imputed to earthquakes as agents
+of geological change. Experience shows that even after a severe
+shock which may have destroyed numerous towns and villages,
+together with thousands of their inhabitants, the face of the country
+has suffered scarcely any perceptible change, and that, in the course
+of a year or two, when the ruined houses and prostrate trees have
+been cleared away, little or no obvious trace of the catastrophe may
+remain. Among the more enduring records of a great earthquake
+may be enumerated (<i>a</i>) landslips, which lay bare hillsides, and sometimes
+pond back the drainage of valleys so as to give rise to lakes;
+(<i>b</i>) alterations of the topography, as in fissuring of the ground, or in
+the production of inequalities whereby the drainage is affected;
+new valleys and new lakes may thus be formed, while previously
+existing lakes may be emptied; (<i>c</i>) permanent changes of level,
+either in an upward or downward direction.</p>
+
+<p>3. <i>Mountain-making.</i>&mdash;This subject may be referred to here for
+the striking evidence which it supplies of the importance of movements
+of the earth&rsquo;s crust among geological processes. The structure
+of a great mountain-chain such as the Alps proves that the crust
+of the earth has been intensely plicated, crumpled and fractured.
+Vast piles of sedimentary strata have been folded to such an extent
+as to occupy now only half of their original horizontal extent. This
+compression in the case of the Alps has been computed to amount
+to as much as 120,000 metres or 74 English miles, so that two points
+on the opposite sides of that chain have been brought by so much
+nearer to each other than they were originally before the movements.
+Besides such intense plication, extensive rupturing of the crust has
+taken place in the same range of mountains. Not only have the
+most ancient rocks been squeezed up into the central axis of the
+chain, but huge slices of them have been torn away from the main
+body, and thrust forward for many miles, so as now actually to
+form the summits of mountains, which are almost entirely composed
+of much younger formations. If these colossal disturbances occurred
+rapidly, they would give rise to cataclysms of inconceivable
+magnitude over the surface of the globe. No record has been discovered
+of such accompanying devastation. But whether sudden
+and violent, or prolonged and gradual, such stupendous upturnings
+of the crust did undoubtedly take place, as is clearly revealed in
+innumerable natural sections, which have been laid open by the
+denudation of the crests and sides of the mountains.</p>
+
+<p>4. <i>Metamorphism of Rocks</i> (see <span class="sc"><a href="#artlinks">Metamorphism</a></span>).&mdash;During the
+movements to which the crust of the earth has been subject, not
+only have the rocks been folded and fractured, but they have likewise,
+in many regions, acquired new internal structures, and have
+thus undergone a process of &ldquo;regional metamorphism.&rdquo; This
+rearrangement of their substance has been governed by conditions
+which are probably not yet all recognized, but among them we should
+doubtless include a high temperature, intense pressure, mechanical
+movement resulting in crushing, shearing and foliation, and the
+presence of water in their pores. It is among igneous rocks that the
+progressive stages of metamorphism can be most easily traced.
+Their definite original structure and mineral composition afford a
+starting-point from which the investigation may be begun and
+pursued. Where an igneous rock has been invaded by metamorphic
+changes, it may be observed to have been first broken down into
+separate lenticles, the cores of which may still retain, with little or
+no alteration, the original characteristic minerals and crystalline
+structure of the rock. Between these lenticles, the intervening
+portions have been crushed down into a powder or paste, which
+seems to have been squeezed round and past them, and shows a
+laminated arrangement that resembles the flow-structure in lavas.
+As the degree of metamorphism increases, the lenticles diminish in
+size, and the intervening crushed and foliated matrix increases in
+amount, until at last it may form the entire mass of the rock. While
+the original minerals are thus broken down, new varieties make
+their appearance. Of these, among the earliest to present themselves
+are usually the micas, that impart their characteristic silvery sheen
+to the surfaces of the folia along which they spread. Younger
+felspars, as well as mica, are developed, and there arise also sillimanite,
+garnet, andalusite and many others. The texture becomes
+more coarsely crystalline, and the segregation of the constituent
+minerals more definite along the lines of foliation. From the finest
+silky phyllites a graduation may be traced through successively
+coarser mica-schists, until we reach the almost granitic texture of
+the coarsest gneisses.</p>
+
+<p>Regional metamorphism has arisen in the heart of mountain-chains,
+and in any other district where the deformation of the crust
+has been sufficiently intense. There is another type of alteration
+termed &ldquo;contact-metamorphism,&rdquo; which is developed around
+masses of igneous rock, especially where these have been intruded in
+large bosses among stratified formations. It is particularly displayed
+around masses of granite, where sandstones are found altered into
+quartzite, shales and grits into schistose compounds, and where sometimes
+fossils are still recognizable among the metamorphic minerals.</p>
+</div>
+
+<p class="pt2 center"><i>DIVISION II.&mdash;EPIGENE OR SUPERFICIAL ACTION</i></p>
+
+<p>It is on the surface of the globe, and by the operation of agents
+working there, that at present the chief amount of visible geological
+change is effected. In considering this branch of inquiry,
+we are not involved in a preliminary difficulty regarding the very
+nature of the agencies as is the case in the investigation of
+plutonic action. On the contrary, the surface agents are carrying
+on their work under our very eyes. We can watch it in all its
+stages, measure its progress, and mark in many ways how
+accurately it represents similar changes which, for long ages
+previously, must have been effected by the same means. But
+in the systematic treatment of this subject we encounter a
+difficulty of another kind. We discover that while the operations
+to be discussed are numerous and readily observable, they are so
+interwoven into one great network that any separation of them
+under different subdivisions is sure to be more or less artificial
+and to convey an erroneous impression. While, therefore, under
+the unavoidable necessity of making use of such a classification
+of subjects, we must always bear in mind that it is employed
+merely for convenience, and that in nature superficial geological
+action must be continually viewed as a whole, since the work of
+each agent has constant reference to that of the others, and is
+not properly intelligible unless that connexion be kept in view.</p>
+
+<p>The movements of the air; the evaporation from land and
+sea; the fall of rain, hail and snow; the flow of rivers and
+glaciers; the tides, currents and waves of the ocean; the growth
+and decay of organized existence, alike on land and in the depths
+of the sea;&mdash;in short, the whole circle of movement, which is
+continually in progress upon the surface of our planet, are the
+subjects now to be examined. It is desirable to adopt some
+general term to embrace the whole of this range of inquiry. For
+this end the word epigene (Gr. <span class="grk" title="epi">&#7952;&#960;&#943;</span>, upon) has been suggested as
+a convenient term, and antithetical to hypogene (Gr. <span class="grk" title="hypo">&#8017;&#960;&#972;</span>, under),
+or subterranean action.</p>
+
+<p>A simple arrangement of this part of Geological Dynamics is
+in three sections:</p>
+
+<div class="list">
+<p>A. <i>Air.</i>&mdash;The influence of the atmosphere in destroying and
+forming rocks.</p>
+
+<p>B. <i>Water.</i>&mdash;The geological functions of the circulation of
+water through the air and between sea and land, and the
+action of the sea.</p>
+
+<p><span class="pagenum"><a name="page660" id="page660"></a>660</span></p>
+
+<p>C. <i>Life.</i>&mdash;The part taken by plants and animals in preserving,
+destroying or reproducing geological formations.</p>
+</div>
+
+<p>The words destructive, reproductive and conservative,
+employed in describing the operations of the epigene agents, do
+not necessarily imply that anything useful to man is destroyed,
+reproduced or preserved. On the contrary, the destructive
+action of the atmosphere may turn barren rock into rich soil,
+while its reproductive effects sometimes turn rich land into
+barren desert. Again, the conservative influence of vegetation
+has sometimes for centuries retained as barren morass what
+might otherwise have become rich meadow or luxuriant woodland.
+The terms, therefore, are used in a strictly geological
+sense, to denote the removal and re-deposition of material, and
+its agency in preserving what lies beneath it.</p>
+
+<div class="condensed">
+<p class="pt2 center">(A) <i>The Air.</i></p>
+
+<p>As a geological agent, the air brings about changes partly by its
+component gases and partly by its movements. Its destructive
+action is both chemical and mechanical. The chemical changes are
+probably mainly, if not entirely, due to the moisture of the air,
+and particularly to the gases, vapours and organic matter which
+the moisture contains. Dry air seems to have little or no appreciable
+influence in promoting these reactions. As the changes in question
+are similar to those much more abundantly brought about by rain
+they are described in the following section under the division on rain.</p>
+
+<p>Among the more recognizable mechanical changes effected in
+the atmosphere, one of considerable importance is to be seen in the
+result of great and rapid changes of temperature. Heat expands
+rocks, while cold contracts them. In countries with a great annual
+range of temperature, considerable difficulty is sometimes experienced
+in selecting building materials liable to be little affected by the
+alternate expansion and contraction, which prevents the joints of
+masonry from remaining close and tight. In dry tropical climates,
+where the days are intensely hot and the nights extremely cold, the
+rapid nocturnal contraction produces a strain so great as to rival
+frost in its influence upon the surface of exposed rocks, disintegrating
+them into sand, or causing them to crack or peel off in skins or
+irregular pieces. Dr Livingstone found in Africa (12° S. lat., 34° E.
+long.) that surfaces of rock which during the day were heated up to
+137° Fahr., cooled so rapidly by radiation at night that, unable to
+sustain the strain of contraction, they split and threw off sharp
+angular fragments from a few ounces to 100 or 200 &#8468; in weight.
+In temperate regions this action, though much less pronounced,
+still makes itself felt. In these climates, however, and still more in
+high latitudes, somewhat similar results are brought about by frost.</p>
+
+<p>By its motion in wind the air drives loose sand over rocks, and in
+course of time abrades and smoothes them. &ldquo;Desert polish&rdquo; is
+the name given to the characteristic lustrous surface thus imparted.
+Holes are said to be drilled in window glass at Cape Cod by the same
+agency. Cavities are now and then hollowed out of rocks by the
+gyration in them of little fragments of stone or grains of sand kept
+in motion by the wind. Hurricanes form important geological
+agents upon land in uprooting trees, and thus sometimes impeding
+the drainage of a country and giving rise to the formation of peat
+mosses.</p>
+
+<p>The reproductive action of the air arises partly from the effect
+of the chemical and mechanical disintegration involved in the
+process of &ldquo;weathering,&rdquo; and partly from the transporting power
+of wind and of aerial currents. The layer of soil, which covers so
+much of the surface of the land, is the result of the decay of the
+underlying rocks, mingled with mineral matter blown over the ground
+by wind, or washed thither by rain, and with the mouldering remains
+of plants and animals. The extent to which fine dust may be
+transported over the surface of the land can hardly be realized in
+countries clothed with a covering of vegetation, though even there,
+in dry weather during spring, clouds of dust may often be seen
+blown away by wind from bare ploughed fields. Intercepted by the
+leaves of plants and washed down to their roots by rain, this dust
+goes to increase the soil below. In arid climates, where dust clouds
+are dense and frequent, enormous quantities of fine mineral particles
+are thus borne along and accumulated. The remarkable deposit
+of &ldquo;Loess,&rdquo; which is sometimes more than 1500 ft. thick and covers
+extensive areas in China and other countries, is regarded as due to
+the drifting of dust by wind. Again the dunes of sand so abundant
+along the inner side of sandy sea-beaches in many different parts
+of the world are attributable to the same action.</p>
+
+<p class="pt2 center">(B) <i>Water.</i></p>
+
+<p>In treating of the epigene action of water in geological processes
+it will be convenient to deal first with its operations in traversing
+the land, and then with those which it performs in the sea. The
+circulation of water from land to sea and again from sea to land
+constitutes the fundamental cause of most of the daily changes by
+which the surface of the land is affected.</p>
+
+<p>1. <i>Rain.</i>&mdash;Rain effects two kinds of changes upon the surface of
+the land. It acts <i>chemically</i> upon soils and stones, and sinking under
+ground continues a great series of similar reactions there. It acts
+<i>mechanically</i>, by washing away loose materials, and thus powerfully
+affecting the contours of the land. Its chemical action depends
+mainly upon the nature and proportion of the substances which, in
+descending to the earth, it abstracts from the atmosphere. Rain
+always absorbs a little air, which, in addition to its nitrogen and
+oxygen, contains carbonic acid, and in minute proportions, sodium
+chloride, sulphuric acid and other ingredients, especially inorganic
+dust, organic particles and living germs. Probably the most generally
+efficient of these constituents are oxygen, carbonic acid and organic
+matter. Armed with these reagents, rain effects a chemical decomposition
+of the rocks on which it falls, and through which it sinks
+underground. The principal changes thus produced are as follows:
+(<i>a</i>) Oxidation.&mdash;Owing to the prominence of oxygen in rain-water,
+and its readiness to unite with any substance which can contain
+more of it, a thin oxidized pellicle is formed on the surface of many
+rocks on which rain falls, and this oxidized layer if not at once
+washed off, sinks deeper until a crust is formed over the stone. A
+familiar illustration of this action is afforded by the rust, or oxide,
+which forms on iron when exposed to moisture, though this iron
+may be kept long bright if allowed to remain screened from moist
+air and rain. (<i>b</i>) Deoxidation.&mdash;Organic matter having an affinity
+for more oxygen decomposes peroxides by depriving them of some
+part of their share of that element and reducing them to protoxides.
+These changes are especially noticeable among the iron oxides so
+abundantly diffused among rocks. Hence rain-water, in sinking
+through soil and obtaining such organic matter, becomes thereby
+a reducing agent. (<i>c</i>) Solution.&mdash;This may take place either by the
+simple action of the water, as in the solution of rock-salt, or by the
+influence of the carbonic acid present in the rain. (<i>d</i>) Formation of
+Carbonates.&mdash;A familiar example of the action of carbonic acid
+in rain is to be seen in the corrosion of exposed marble slabs. The
+carbonic acid dissolves some of the lime, which, as a bicarbonate,
+is held in solution in the carbonated water, but is deposited again
+when the water loses its carbonic acid or evaporates. It is not
+merely carbonates, however, which are liable to this kind of destruction.
+Even silicates of lime, potash and soda, combinations existing
+abundantly as constituents of rocks, are attacked; their silica is
+liberated, and their alkalis or alkaline earths, becoming carbonates,
+are removed in solution. (<i>e</i>) Hydration.&mdash;Some minerals, containing
+little or no water, and therefore called anhydrous, when exposed to
+the action of the atmosphere, absorb water, or become hydrous,
+and are then usually more prone to further change. Hence the rocks
+of which they form part become disintegrated.</p>
+
+<p>Besides the reactions here enumerated, a considerable amount of
+decay may be observed as the result of the presence of sulphuric
+and nitric acid in the air, especially in that of large towns and
+manufacturing districts, where much coal is consumed. Metallic
+surfaces, as well as various kinds of stone, are there corroded, while
+the mortar of walls may often be observed to be slowly swelling out
+and dropping off, owing to the conversion of the lime into sulphate.
+Great injury is likewise done from a similar cause to marble monuments
+in exposed graveyards.</p>
+
+<p>The general result of the disintegrating action of the air and of
+rain, including also that of plants and animals, to be noticed in the
+sequel, is denoted by the term &ldquo;weathering.&rdquo; The amount of decay
+depends partly on conditions of climate, especially the range of
+temperature, the abundance of moisture, height above the sea and
+exposure to prevalent winds. Many rocks liable to be saturated
+with rain and rapidly dried under a warm sun are apt to disintegrate
+at the surface with comparative rapidity. The nature and progress
+of the weathering are mainly governed by the composition and
+texture of the rocks exposed to it. Rocks composed of particles
+liable to little chemical change from the influence of moisture are
+best fitted to resist weathering, provided they possess sufficient
+cohesion to withstand the mechanical processes of disintegration.
+Siliceous sandstones are excellent examples of this permanence.
+Consisting wholly or mainly of the durable mineral quartz, they are
+sometimes able so to withstand decay that buildings made of them
+still retain, after the lapse of centuries, the chisel-marks of the
+builders. Some rocks, which yield with comparative rapidity to
+the chemical attacks of moisture, may show little or no mark of
+disintegration on their surface. This is particularly the case with
+certain calcareous rocks. Limestone when pure is wholly soluble
+in acidulated water. Rain falling on such a rock removes some of it
+in solution, and will continue to do so until the whole is dissolved
+away. But where a limestone is full of impurities, a weathered crust
+of more or less insoluble particles remains after the solution of the
+calcareous part of the stone. Hence the relative purity of limestones
+may be roughly determined by examining their weathered surfaces,
+where, if they contain much sand, the grains will be seen projecting
+from the calcareous matrix, and where, should the rock be very
+ferruginous, the yellow hydrous peroxide, or ochre, will be found as
+a powdery crust. In limestones containing abundant encrinites,
+shells, or other organic remains, the weathered surface commonly
+presents the fossils standing out in relief. The crystalline arrangement
+of the lime in the organic structures enables them to resist
+disintegration better than the general mechanically aggregated
+matrix of the rock. An experienced fossil collector will always
+search well such weathered surfaces, for he often finds there, delicately
+<span class="pagenum"><a name="page661" id="page661"></a>661</span>
+picked out by the weather, minute and frail fossils which are wholly
+invisible on a freshly broken surface of the stone. Many rocks
+weather with a thick crust, or even decay inwards for many feet or
+yards. Basalt, for example, often shows a yellowish-brown ferruginous
+layer on its surface, formed by the conversion of its felspar
+into kaolin, and the removal of its calcium silicate as carbonate,
+by the hydration of its olivine and augite and their conversion into
+serpentine, or some other hydrous magnesian silicate, and by the
+conversion of its magnetite into limonite. Granite sometimes shows
+in a most remarkable way the distance to which weathering can
+reach. It may occasionally be dug into for a depth of 20 or 30 ft.,
+the quartz crystals and veins retaining their original positions, while
+the felspar is completely kaolinized. It is to the endlessly varied
+effects of weathering that the abundant fantastic shapes assumed
+by crags and other rocky masses are due. Most varieties of rock
+have their own characteristic modes of weathering, whereby they
+may be recognized even from a distance. To some of these features
+reference will be made in Part VIII.</p>
+
+<p>The mechanical action of rain, which is intimately bound up with
+its chemical action, consists in washing off the fine superficial
+particles of rocks which have been corroded and loosened by the
+process of weathering, and in thus laying open fresh portions to the
+same influences of decay. The detritus so removed is partly carried
+down into the soil which is thereby enriched, partly held in suspension
+in the little runnels into which the rain-drops gather as they begin
+to flow over the land, partly pushed downwards along the surface
+of sloping ground. A good deal of it finds its way into the nearest
+brooks and rivers, which are consequently made muddy by heavy
+rain.</p>
+
+<p>It is natural that a casual consideration of the subject should lead
+to an impression that, though the general result of the fall of rain
+upon a land-surface must lead to some amount of disintegration and
+lowering of that surface, the process must be so slow and slight as
+hardly to be considered of much importance among geological
+operations. But further attention will show such an impression to
+be singularly erroneous. It loses sight of the fact that a change
+which may be hardly appreciable within a human lifetime, or even
+within the comparatively brief span of geological time embraced in
+the compass of human history, may nevertheless become gigantic
+in its results in the course of immensely protracted periods. An
+instructive lesson in the erosive action of rain may be found in the
+pitted and channelled surface of ground lying under the drip of the
+eaves of a cottage. The fragments of stone and pebbles of gravel
+that form part of the soil can there be seen sticking out of the ground,
+because being hard they resist the impetus of the falling drops,
+protecting for a time the earth beneath them, while that which
+surrounded and covered them is washed away. From this familiar
+illustration the observer may advance through every stage in the
+disappearance of material which once covered the surface, until he
+comes to examples where once continuous and thick sheets of solid
+rock have been reduced to a few fragments or have been entirely
+removed. Since the whole land surface over which rain falls is
+exposed to this waste, the superficial covering of decayed rock or
+soil, as Hutton insisted, is constantly, though imperceptibly, travelling
+outward and downward to the sea. In this process of transport
+rain is an important carrying agent, while at the same time it serves
+to connect the work of the other disintegrating forces, and to make
+it conducive to the general degradation of the land. Though this
+decay is general and constant, it is obviously not uniform. In some
+places where, from the nature of the rock, from the flatness of the
+ground, or from other causes, rain works under great difficulties,
+the rate of waste may be extremely slow. In other places it may
+be rapid enough to be appreciable from year to year. A survey of
+this department of geological activity shows how unequal wasting
+by rain, combined with the operations of brooks and rivers, has
+produced the details of the present relief of the land, those tracts
+where the destruction has been greatest forming hollows and valleys,
+others, where it has been less, rising into ridges and hills (Part VIII.).</p>
+
+<p>Rain-action is not merely destructive, but is accompanied with
+reproductive effects, chief of which is the formation of soil. In
+favourable situations it has gathered together accumulations of loam
+and earth from neighbouring higher ground, such as the &ldquo;brick-earth,&rdquo;
+&ldquo;head,&rdquo; and &ldquo;rain-wash&rdquo; of the south of England&mdash;earthy
+deposits, sometimes full of angular stones, derived from the subaerial
+waste of the rocks of the neighbourhood.</p>
+
+<p>2. <i>Underground Water.</i>&mdash;Of the rain which falls upon the land
+one portion flows off into brooks and rivers by which the water is
+conducted back to the ocean; the larger part, however, sinks into
+the ground and disappears. It is this latter part which has now
+to be considered. Over and above the proportion of the rainfall
+which is absorbed by living vegetation and by the soil, there is a
+continual filtering down of the water from the surface into the rocks
+that lie below, where it partly lodges in pores and interstices, and
+partly finds its way into subterranean joints and fissures, in which
+it performs an underground circulation, and ultimately issues once
+more at the surface in the form of springs (<i>q.v.</i>). In the course of
+this circulation the water performs an important geological task.
+Not only carrying down with it the substances which the rain has
+abstracted from the air, but obtaining more acids and organic
+matter from the soil, it is enabled to effect chemical changes in the
+rocks underneath, and especially to dissolve limestone and other
+calcareous formations. So considerable is the extent of this solution
+in some places that the springs which come to the surface, and begin
+there to evaporate and lose some of their carbonic acid, contain more
+dissolved lime than they can hold. They consequently deposit it
+in the form of calcareous tuff or sinter (<i>q.v.</i>). Other subterranean
+waters issue with a large proportion of iron-salts in solution which
+form deposits of ochre. The various mineral springs so largely
+made use of for the mitigation or cure of diseases owe their properties
+to the various salts which they have dissolved out of rocks
+underground. As the result of prolonged subterranean solution in
+limestone districts, passages and caves (<i>q.v.</i>), sometimes of great
+width and length, are formed. When these lie near the surface their
+roofs sometimes fall in and engulf brooks and rivers, which then
+flow for some way underground until the tunnels conduct them back
+again to daylight on some lower ground.</p>
+
+<p>Besides its chemical activity water exerts among subterranean
+rocks a mechanical influence which leads to important changes in
+the topography of the surface. In removing the mineral matter,
+either in solution or as fine sediment, it sometimes loosens the support
+of overlying masses of rock which may ultimately give way on sloping
+ground, and rush down the declivities in the form of landslips.
+These destructive effects are specially frequent on the sides of valleys
+in mountainous countries and on lines of sea-cliff.</p>
+
+<p>3. <i>Brooks and Rivers.</i>&mdash;As geological agents the running waters
+on the face of the land play an important part in epigene
+changes. Like rain and springs they have both a chemical and a
+mechanical action. The latter receives most attention, as it undoubtedly
+is the more important; but the former ought not to be
+omitted in any survey of the general waste of the earth&rsquo;s surface.
+The water of rivers must possess the powers of a chemical solvent
+like rain and springs, though its actual work in this respect can be
+less easily measured, seeing that river water is directly derived from
+rain and springs, and necessarily contains in solution mineral substances
+supplied to it by them and not by its own operation. Nevertheless,
+it is sometimes easy to prove that streams dissolve chemically
+the rocks of their channels. Thus, in limestone districts the base
+of the cliffs of river ravines may be found eaten away into tunnels,
+arches, and overhanging projections, presenting in their smooth
+surfaces a great contrast to the angular jointed faces of the same
+rock, where now exposed to the influence only of the weather on the
+higher parts of the cliff.</p>
+
+<p>The mechanical action of rivers consists (<i>a</i>) in transporting mud,
+sand, gravel and blocks of stone from higher to lower levels; (<i>b</i>)
+in using these loose materials to widen and deepen their channels
+by erosion; (<i>c</i>) in depositing their load of detritus wherever possible
+and thus to make new geological formations.</p>
+
+<p>(<i>a</i>) <i>Transporting Power.</i>&mdash;River-water is distinguished from that
+of springs by being less transparent, because it contains more or less
+mineral matter in suspension, derived mainly from what is washed
+down by rain, or carried in by brooks, but partly also from the
+abrasion of the water-channels by the erosive action of the rivers
+themselves. The progress of this burden of detritus may be instructively
+followed from the mountain-tributaries of a river down to
+the mouth of the main stream. In the high grounds the water-courses
+may be observed to be choked with large fragments of rock
+disengaged from the cliffs and crags on either side. Traced downwards
+the blocks are seen to become gradually smaller and more rounded.
+They are ground against each other, and upon the rocky sides and
+bottom of the channel, getting more and more reduced as they
+descend, and at the same time abrading the rocks over or against
+which they are driven. Hence a great deal of débris is produced,
+and is swept along by the onward and downward movement of the
+water. The finer portions, such as mud and fine sand, are carried
+in suspension, and impart the characteristic turbidity to river-water;
+the coarser sand and gravel are driven along the river-bottom.
+The proportion of suspended mineral matter has been
+ascertained with more or less precision for a number of rivers. As
+an illustrative example of a river draining a vast area with different
+climates, forms of surface and geological structure the Mississippi
+may be cited. The average proportion of sediment in its water was
+ascertained by Humphreys and Abbot to be <span class="spp">1</span>&frasl;<span class="suu">1500</span> by weight or
+<span class="spp">1</span>&frasl;<span class="suu">2900</span> by volume. These engineers found that, in addition to this
+suspended material, coarse detritus is constantly being pushed
+forward along the bed of the river into the Gulf of Mexico, to an
+amount which they estimated at about 750,000,000 cubic ft. of
+sand, earth and gravel; they concluded that the Mississippi carries
+into the gulf every year an amount of mechanically transported
+sediment sufficient to make a prism one square mile in area and
+268 ft. in height.</p>
+
+<p>(<i>b</i>) <i>Excavating Power.</i>&mdash;It is by means of the sand, gravel and
+stones which they drive against the sides and bottoms of their
+channels that streams have hollowed out the beds in which they
+flow. Not only is the coarse detritus reduced in size by the friction
+of the stones against each other, but, at the same time, these materials
+abrade the rocks against which they are driven by the current.
+Where, owing to the shape of the bottom of the channel, the stones
+are caught in eddies, and are kept whirling round there, they become
+more and more worn down themselves, and at the same time scour
+out basin-shaped cavities, or &ldquo;pot-holes,&rdquo; in the solid rock below.
+<span class="pagenum"><a name="page662" id="page662"></a>662</span>
+The uneven bed of a swiftly flowing stream may in this way be
+honeycombed with such eroded basins which coalesce and thus
+appreciably lower the surface of the bed. The steeper the channel,
+other conditions being equal, the more rapid will be the erosion.
+Geological structure also affects the character and rate of the excavation.
+Where the rocks are so arranged as to favour the formation
+and persistence of a waterfall, a long chasm may be hollowed out
+like that of the Niagara below the falls, where a hard thick bed of
+nearly flat limestone lies on softer and more easily eroded shales.
+The latter are scooped out from underneath the limestone, which
+from time to time breaks off in large masses and the waterfall
+gradually retreats up stream, while the ravine is proportionately
+lengthened. To the excavating power of rivers the origin of the
+valley systems of the dry land must be mainly assigned (see Part VIII.).</p>
+
+<p>(<i>c</i>) <i>Reproductive Power.</i>&mdash;So long as a stream flows over a steep
+declivity its velocity suffices to keep the sediment in suspension,
+but when from any cause, such as a diminution of slope, the velocity
+is checked, the transporting power is lessened and the sediment
+begins to fall to the bottom and to remain there. Hence various
+river-formed or &ldquo;alluvial&rdquo; deposits are laid down. These sometimes
+cover considerable spaces at the foot of mountains. The
+floors of valleys are strewn with detritus, and their level may thereby
+be sensibly raised. In floods the ground inundated on either side
+of a stream intercepts some part of the detritus, which is then spread
+over the flood-plain and gradually heightens it. At the same time
+the stream continues to erode the channel, and ultimately is unable
+to reach the old flood-plain. It consequently forms a new plain at
+a lower level, and thus, by degrees, it comes to be flanked on either
+side by a series of successive terraces or platforms, each of which
+marks one of its former levels. Where a river enters a large body of
+water its current is checked. Some of its sediment is consequently
+dropped, and by slow accumulation forms a delta (<i>q.v.</i>). On land,
+every lake in mountain districts furnishes instances of this kind of
+alluvium. But the most important deltas are those formed in the
+sea at the mouths of the larger rivers of the globe. Off many coast-lines
+the detritus washed from the land gathers into bars, which
+enclose long strips of water more or less completely separated from
+the sea outside and known as lagoons. A chain of such lagoon-barriers
+stretches for hundreds of miles round the Gulf of Mexico
+and the eastern shores of the United States.</p>
+
+<p>4. <i>Lakes.</i>&mdash;These sheets of water, considered as a whole, do not
+belong to the normal system of drainage on the land whereby valleys
+are excavated. On the contrary they are exceptional to it; for
+the constant tendency of running water is to fill them up, or to drain
+them by wearing down the barriers that contain them at their
+outflow. Some of them are referable to movements of the terrestrial
+crust whereby depressions arise on the surface of the land, as has
+been noted after earthquakes. Others have arisen from solution
+such as that of rock-salt or of limestone, the removal of which by
+underground water causes a subsidence of the ground above. A
+third type of lake-basin occurs in regions that are now or have once
+been subject to the erosive action of glaciers (see under next subdivision,
+<i>Terrestrial Ice</i>). Many small lakes or tarns have been
+caused by the deposit of débris across a valley as by landslips or
+moraines. Considered from a geological point of view, lakes perform
+an important function in regulating the drainage of the ground below
+their outfall and diminishing the destructive effects of floods, in
+filtering the water received from their affluent streams, and in
+providing undisturbed areas of deposit in which thick and extensive
+lacustrine formations may be accumulated. In the inland basins
+of some dry climates the lakes are salt, owing to excess of evaporation,
+and their bottoms become the sites of chemical deposits, particularly
+of chlorides of sodium and magnesium, and calcium sulphate and
+carbonate.</p>
+
+<p>5. <i>Terrestrial Ice.</i>&mdash;Each of the forms assumed by frozen water
+has its own characteristic action in geological processes. Frost has
+a powerful influence in breaking up damp soils and surfaces of stone
+in the pores or cracks of which moisture has lodged. The water in
+freezing expands, and in so doing pushes asunder the component
+particles of soil or stone, or widens the space between the walls of
+joints or crevices. When the ice melts the loosened grains remain
+apart ready to be washed away by rain or blown off by wind, while
+by the widening of joints large blocks of rock are detached from
+the faces of cliffs. Where rivers or lakes are frozen over the ice
+exerts a marked pressure on their banks; and when it breaks up
+large sheets of it are driven ashore, pushing up quantities of gravel
+and stones above the level of the water. The piling up of the disrupted
+ice against obstructions in rivers ponds back the water, and
+often leads to destructive floods when the ice barriers break. Where
+the ice has formed round boulders in shallow water, or at the bottom
+(&ldquo;anchor-ice&rdquo;), it may lift these up when the frost gives way,
+and may transport them for some distance. Ice formed in the
+atmosphere, and descending to the ground in the form of hail, often
+causes great destruction to vegetation and not infrequently to
+animal life. Where the frozen moisture reaches the earth as snow,
+it serves to protect rock, soil and vegetation from the effects of
+frost; but on sloping ground it is apt to give rise to destructive
+avalanches or landslips, while indirectly, by its rapid melting, it
+may cause serious floods in rivers.</p>
+
+<p>But the most striking geological work performed by terrestrial
+ice is that achieved by glaciers (<i>q.v.</i>) and ice-sheets. These vast
+masses of moving ice, when they descend from mountains where the
+steeper rocks are clear of snow, receive on their surface the débris
+detached by frost from the declivities above, and bear these materials
+to lower levels or to the sea. Enormous quantities of rock-rubbish
+are thus transported in the Alps and other high mountain ranges.
+When the ice retreats the boulders carried by it are dropped where
+it melts, and left there as memorials of the former extension of the
+glaciers. Evidence of this nature proves the much wider extent of
+the Alpine ice at a comparatively recent geological date. It can
+also be shown that detritus from Scandinavia has been ice-borne to
+the south-east of England and far into the heart of Europe.</p>
+
+<p>The ice, by means of grains of sand and pieces of stone which it
+drags along, scores, scratches and polishes the surfaces of rock
+underneath it, and, in this way, produces the abundant fine sediment
+that gives the characteristic milky appearance to the rivers that
+issue from the lower ends of glaciers. By such long-continued
+attrition the rocks are worn down, portions of them of softer nature,
+or where the ice acts with especial vigour, are hollowed out into
+cavities which, on the disappearance of the ice, may be filled with
+water and become tarns or lakes. Rocks over which land-ice has
+passed are marked by a peculiar smooth, flowing outline, which
+forms a contrast to the more rugged surface produced by ordinary
+weathering. They are covered with groovings, which range from
+the finest striae left by sharp grains of sand to deep ruts ground out
+by blocks of stone. The trend of these markings shows the direction
+in which the ice flowed. By their evidence the position and movement
+of former glaciers in countries from which the ice has entirely
+vanished may be clearly determined (see <span class="sc"><a href="#artlinks">Glacial Period</a></span>).</p>
+
+<p>6. <i>The Sea.</i>&mdash;The physical features of the sea are discussed in
+separate articles (see <span class="sc"><a href="#artlinks">Ocean and Oceanography</a></span>). The sea must
+be regarded as the great regulator of temperature and climate over
+the globe, and as thus exerting a profound influence on the distribution
+of plant and animal life. Its distinctly geological work is partly
+erosive and partly reproductive. As an eroding agent it must to
+some extent effect chemical decompositions in the rocks and sediments
+over which it spreads; but these changes have not yet been
+satisfactorily studied. Undoubtedly, its chief destructive power
+is of a mechanical kind, and arises from the action of its waves in
+beating upon shore-cliffs. By the alternate compression and
+expansion of the air in crevices of the rocks on which heavy breakers
+fall, and by the hydraulic pressure which these masses of sea-water
+exert on the walls of the fissures into which they rush, large masses
+of rock are loosened and detached, and caves and tunnels are drilled
+along the base of sea-cliffs. Probably still more efficacious are the
+blows of the loose shingle, which, caught up and hurled forward by
+the waves, falls with great force upon the shore rocks, battering
+them as with a kind of artillery until they are worn away. The
+smooth surfaces of the rocks within reach of the waves contrasted
+with their angular forms above that limit bear witness to the amount
+of waste, while the rounded forms of the boulders and shingle show
+that they too are being continually reduced in size. Thus the sea,
+by its action on the coasts, produces much sediment, which is swept
+away by its waves and currents and strewn over its floor. Besides
+this material, it is constantly receiving the fine silt and sand carried
+down by rivers. As the floor of the ocean is thus the final receptacle
+for the waste of the land, it becomes the chief era on the surface of
+the globe for the accumulation of new stratified formations. And
+such has been one of its great functions since the beginning of
+geological time, as is proved by the rocks that form the visible part
+of the earth&rsquo;s crust, and consist in great part of marine deposits.
+Chemical precipitates take place more especially in enclosed parts
+of the sea, where concentration of the water by evaporation can take
+place, and where layers of sodium chloride, calcium sulphate and
+carbonate, and other salts are laid down. But the chief marine
+accumulations are of detrital origin. Near the land and for a variable
+distance extending sometimes to 200 or 300 m. from shore the
+deposits consist chiefly of sediments derived from the waste of the
+land, the finer silts being transported farthest from their source.
+At greater depths and distances the ocean floor receives a slow deposit
+of exceedingly fine clay, which is believed to be derived from the
+decomposition of pumice and volcanic dust from insular or submarine
+volcanoes. Wide tracts of the bottom are covered with
+various forms of ooze derived from the accumulation of the remains
+of minute organisms.</p>
+
+<p class="pt2 center">(C) <i>Life.</i></p>
+
+<p>Among the agents by which geological changes are carried on
+upon the surface of the globe living organisms must be enumerated.
+Both plants and animals co-operate with the inorganic agents in
+promoting the degradation of the land. In some cases, on the other
+hand, they protect rocks from decay, while, by the accumulation of
+their remains, they give rise to extensive formations both upon the
+land and in the sea. Their operations may hence be described as
+alike destructive, conservative and reproductive. Under this heading
+also the influence of Man as a geological agent deserves notice.</p>
+
+<p>(<i>a</i>) <i>Plants.</i>&mdash;Vegetation promotes the disintegration of rocks and
+soil in the following ways: (1) By keeping the surfaces of stone
+moist, and thus promoting both mechanical and chemical dissolution,
+as is especially shown by liverworts, mosses and other moisture-loving
+plants. (2) By producing through their decay carbonic and
+<span class="pagenum"><a name="page663" id="page663"></a>663</span>
+other acids, which, together with decaying organic matter taken up
+by passing moisture, become potent in effecting the chemical decomposition
+of rocks and in promoting the disintegration of soils. (3)
+By inserting their roots or branches between joints of rock, which
+are thereby loosened, so that large slices may be eventually wedged
+off. (4) By attracting rain, as thick woods, forests and peat-mosses
+do, and thus accelerating the general waste of a country by running
+water. (5) By promoting the decay of diseased and dead plants and
+animals, as when fungi overspread a damp rotting tree or the carcase
+of a dead animal.</p>
+
+<p>That plants also exert a conservative influence on the surface of
+the land is shown in various ways. (1) The formation of a stratum
+of turf protects the soil and rocks underneath from being rapidly
+disintegrated and washed away by atmospheric action. (2) Many
+plants, even without forming a layer of turf, serve by their roots or
+branches to protect the loose sand or soil on which they grow from
+being removed by wind. The common sand-carex and other arenaceous
+plants bind the loose sand-dunes of our coasts, and give them a
+permanence, which would at once be destroyed were the sand laid
+bare again to storms. The growth of shrubs and brushwood along
+the course of a stream not only keeps the alluvial banks from being
+so easily undermined and removed as would otherwise be the case,
+but serves to arrest the sediment in floods, filtering the water and
+thereby adding to the height of the flood plain. (3) Some marine
+plants, like the calcareous nullipores, afford protection to shore
+rocks by covering them with a hard incrustation. The tangles and
+smaller Fuci which grow abundantly on the littoral zone break the
+force of the waves or diminish the effects of ground swell. (4)
+Forests and brushwood protect the soil, especially on slopes, from
+being washed away by rain or ploughed up by avalanches.</p>
+
+<p>Plants contribute by the aggregation of their remains to the
+formation of stratified deposits. Some marine algae which secrete
+carbonate of lime not only encrust rocks but give rise to sheets of
+submarine limestone. An analogous part is played in fresh-water
+lakes by various lime-secreting plants, such as <i>Chara</i>. Long-continued
+growth of vegetation has, in some regions, produced thick
+accumulations of a dark loam, as in the black cotton soil (<i>regur</i>) of
+India, and the black earth (<i>tchernozom</i>) of Russia. Peat-mosses
+are formed in temperate and arctic climates by the growth of marsh-loving
+plants, sometimes to a thickness of 40 or 50 ft. In tropical
+regions the mangrove swamps on low moist shores form a dense
+jungle, sometimes 20 m. broad, which protects these shores from the
+sea until, by the arrest of sediment and the constant contribution of
+decayed vegetation, the spongy ground is at last turned into firm
+soil. Some plants (diatoms) can abstract silica and build it into
+their framework, so that their remains form a siliceous deposit or
+ooze which covers spaces of the deep sea-floor estimated at more
+than ten millions of square miles in extent.</p>
+
+<p>(<i>b</i>) <i>Animals.</i>&mdash;These exert a destructive influence in the following
+ways: (1) By seriously affecting the composition and arrangement
+of the vegetable soil. Worms bring up the lower portions of the
+soil to the surface, and while thus promoting its fertility increase
+its liability to be washed away by rain. Burrowing animals, by
+throwing up the soil and subsoil, expose these to be dried and blown
+away by the wind. At the same time their subterranean passages
+serve to drain off the superficial water and to injure the stability
+of the surface of the ground above them. In Britain the mole and
+rabbit are familiar examples. (2) By interfering with or even diverting
+the flow of streams. Thus beaver-dams check the current of
+water-courses, intercept floating materials, and sometimes turn
+streams into new channels. The embankments of the Mississippi
+are sometimes weakened to such an extent by the burrowings of the
+cray-fish as to give way and allow the river to inundate the surrounding
+country. Similar results have happened in Europe from
+subterranean operations of rats. (3) Some mollusca bore into stone
+or wood and by the number of contiguous perforations greatly
+weaken the material. (4) Many animals exercise a ruinously
+destructive influence upon vegetation. Of the numerous plagues
+of this kind the locust, phylloxera and Colorado beetle may be cited.</p>
+
+<p>The most important geological function performed by animals is
+the formation of new deposits out of their remains. It is chiefly by
+the lower grades of the animal kingdom that this work is accomplished,
+especially by molluscs, corals and foraminifera. Shell-banks
+are formed abundantly in such comparatively shallow and enclosed
+basins as that of the North Sea, and on a much more extensive scale
+on the floor of the West Indian seas. By the coral polyps thick
+masses of limestones have been built up in the warmer seas of the
+globe (see <span class="sc"><a href="#artlinks">Coral Reefs</a></span>). The floor of the Atlantic and other oceans
+is covered with a fine calcareous ooze derived mainly from the
+remains of foraminifera, while in other regions the bottom shows a
+siliceous ooze formed almost entirely of radiolaria. Vertebrate
+animals give rise to phosphatic deposits formed sometimes of their
+excrement, as in guano and coprolites, sometimes of an accumulation
+of their bones.</p>
+
+<p>(<i>c</i>) <i>Man.</i>&mdash;No survey of the geological workings of plant and
+animal life upon the surface of the globe can be complete which does
+not take account of the influence of man&mdash;an influence of enormous
+and increasing consequence in physical geography, for man has
+introduced, as it were, an element of antagonism to nature. His
+interference shows itself in his relations to climate, where he has
+affected the meteorological conditions of different countries: (1)
+By removing forests, and laying bare to the sun and winds areas
+which were previously kept cool and damp under trees, or which,
+lying on the lee side, were protected from tempests. It is supposed
+that the wholesale destruction of the woodlands formerly existing
+in countries bordering the Mediterranean has been in part the cause
+of the present desiccation of these districts. (2) By drainage, whereby
+the discharged rainfall is rapidly removed, and the evaporation is
+lessened, with a consequent diminution of rainfall and some increase
+in the general temperature of a country. (3) By the other processes
+of agriculture, such as the transformation of moor and bog into
+cultivated land, and the clothing of bare hillsides with green crops
+or plantations of coniferous and hardwood trees.</p>
+
+<p>Still more obvious are the results of human interference with the
+flow of water: (1) By increasing or diminishing the rainfall man
+directly affects the volume of rivers. (2) By his drainage operations
+he makes the rain to run off more rapidly than before, and thereby
+increases the magnitude of floods and of the destruction caused by
+them. (3) By wells, bores, mines, or other subterranean works he
+interferes with the underground waters, and consequently with the
+discharge of springs. (4) By embanking rivers he confines them to
+narrow channels, sometimes increasing their scour, and enabling
+them to carry their sediment further seaward, sometimes causing
+them to deposit it over the plains and raise their level. (5) By his
+engineering operations for water-supply he abstracts water from its
+natural basins and depletes the streams.</p>
+
+<p>In many ways man alters the aspect of a country: (1) By changing
+forest into bare mountain, or clothing bare mountains with forest.
+(2) By promoting the growth or causing the removal of peat-mosses.
+(3) By heedlessly uncovering sand-dunes, and thereby setting in
+motion a process of destruction which may convert hundreds of
+acres of fertile land into waste sand, or by prudently planting the
+dunes with sand-loving vegetation and thus arresting their landward
+progress. (4) By so guiding the course of rivers as to make them
+aid him in reclaiming waste land, and bringing it under cultivation.
+(5) By piers and bulwarks, whereby the ravages of the sea are
+stayed, or by the thoughtless removal from the beach of stones
+which the waves had themselves thrown up, and which would have
+served for a time to protect the land. (6) By forming new deposits
+either designedly or incidentally. The roads, bridges, canals,
+railways, tunnels, villages and towns with which man has covered
+the surface of the land will in many cases form a permanent record
+of his presence. Under his hand the whole surface of civilized
+countries is very slowly covered with a stratum, either formed
+wholly by him or due in great measure to his operations and containing
+many relics of his presence. The soil of ancient towns has
+been increased to a depth of many feet by their successive destructions
+and renovations.</p>
+
+<p>Perhaps the most subtle of human influences are to be seen in the
+distribution of plant and animal life upon the globe. Some of man&rsquo;s
+doings in this domain are indeed plain enough, such as the extirpation
+of wild animals, the diminution or destruction of some forms of
+vegetation, the introduction of plants and animals useful to himself,
+and especially the enormous predominance given by him to the
+cereals and to the spread of sheep and cattle. But no such extensive
+disturbance of the normal conditions of the distribution of life can
+take place without carrying with it many secondary effects, and
+setting in motion a wide cycle of change and of reaction in the
+animal and vegetable <span class="correction" title="amended from kindgoms">kingdoms</span>. For example, the incessant
+warfare waged by man against birds and beasts of prey in districts
+given up to the chase leads sometimes to unforeseen results. The
+weak game is allowed to live, which would otherwise be killed off
+and give more room for the healthy remainder. Other animals
+which feed perhaps on the same materials as the game are by the
+same cause permitted to live unchecked, and thereby to act as a
+further hindrance to the spread of the protected species. But the
+indirect results of man&rsquo;s interference with the régime of plants and
+animals still require much prolonged observation.</p>
+</div>
+
+<p class="pt2 center"><span class="sc">Part V.&mdash;Geotectonic or Structural Geology</span></p>
+
+<p>From a study of the nature and composition of minerals and
+rocks, and an investigation of the different agencies by which
+they are formed and modified, the geologist proceeds to inquire
+how these materials have been put together so as to build up the
+visible part of the earth&rsquo;s crust. He soon ascertains that they
+have not been thrown together wholly at random, but that they
+show a recognizable order of arrangement. Some of them,
+especially those of most recent growth, remain in their original
+condition and position, but, in proportion to their antiquity,
+they generally present increasing alteration, until it may no
+longer be possible to tell what was their pristine state. As by
+far the largest accessible portion of the terrestrial crust consists of
+stratified rocks, and as these furnish clear evidence of most of the
+modifications to which they have been subjected in the long
+course of geological history, it is convenient to take them into
+<span class="pagenum"><a name="page664" id="page664"></a>664</span>
+consideration first. They possess a number of structures which
+belong to the original conditions in which they were accumulated.
+They present in addition other structures which have been superinduced
+upon them, and which they share with the unstratified
+or igneous rocks.</p>
+
+<p class="pt2 center sc">1. Original Structures</p>
+
+<p>(<i>a</i>) <i>Stratified Rocks.</i>&mdash;This extensive and important series is
+above all distinguished by possessing a prevailing stratified
+arrangement. Their materials have been laid down in laminae,
+layers and strata, or beds, pointing generally to the intermittent
+deposition of the sediments of which they consist. As this
+stratification was, as a rule, originally nearly or quite horizontal,
+it serves as a base from which to measure any subsequent disturbance
+which the rocks have undergone. The occurrence of
+false-bedding, <i>i.e.</i> bands of inclined layers between the normal
+planes of stratification, does not form any real exception; but
+indicates the action of shifting currents whereby the sediment
+was transported and thrown down. Other important records of
+the original conditions of deposit are supplied by ripple-marks,
+sun-cracks, rain-prints and concretions.</p>
+
+<div class="condensed">
+<p>From the nature of the material further light is cast on the geographical
+conditions in which the strata were accumulated. Thus,
+conglomerates indicate the proximity of old shore-lines, sandstones
+mark deposits in comparatively shallow water, clays and shales
+point to the tranquil accumulation of fine silt at a greater depth
+and further from land, while fossiliferous limestones bear witness to
+clearer water in which organisms flourished at some distance from
+deposits of sand and mud. Again, the alternation of different kinds
+of sediment suggests a variability in the conditions of deposition,
+such as a shifting of the sediment-bearing currents and of the areas
+of muddy and clear water. A thick group of conformable strata,
+that is, a series of deposits which show no discordance in their
+stratification, may usually be regarded as having been laid down on
+a sea-floor that was gently sinking. Here and there evidence is
+obtainable of the limits or of the progress of the subsidence by what
+is called &ldquo;overlap.&rdquo; Of the absolute length of time represented by
+any strata or groups of strata no satisfactory estimates can yet be
+formed. Certain general conclusions may indeed be drawn, and
+comparisons may be made between different series of rocks. Sandstones
+full of false-bedding were probably accumulated more rapidly
+than finely-laminated shales or clays. It is not uncommon in certain
+Carboniferous formations to find coniferous and other trunks embedded
+in sandstone. Some of these trees seem to have been carried
+along and to have sunk, their heavier or root end touching the
+bottom and their upper end slanting upward in the direction of the
+current, exactly as in the case of the snags of the Mississippi. In
+other cases the trees have been submerged while still in their positions
+of growth. The continuous deposit of sand at last rose above the
+level of the trunks and buried them. It is clear then that the rate
+of deposit must have been sometimes sufficiently rapid to allow
+sand to accumulate to a depth of 30 ft. or more before the decay
+of the wood. Modern instances are known where, under certain
+circumstances, submerged trees may last for some centuries, but
+even the most durable must decay in what, after all, is a brief space
+of geological time. Since continuous layers of the same kind of
+deposit suggest a persistence of geological conditions, while numerous
+alternations of different kinds of sedimentary matter point to
+vicissitudes or alternations of conditions, it may be supposed that
+the time represented by a given thickness of similar strata was less
+than that shown by the same thickness of dissimilar strata, because
+the changes needed to bring new varieties of sediment into the area
+of deposit would usually require the lapse of some time for their
+completion. But this conclusion may often be erroneous. It will
+be best supported when, from the very nature of the rocks, wide
+variations in the character of the water-bottom can be established.
+Thus a group of shales followed by a fossiliferous limestone would
+almost always mark the lapse of a much longer period than an equal
+depth of sandy strata. A thick mass of limestone, made up of
+organic remains which lived and died upon the spot, and whose
+remains are crowded together generation above generation, must
+have demanded many years or centuries for its formation.</p>
+
+<p>But in all speculations of this kind we must bear in mind that the
+length of time represented by a given depth of strata is not to be
+estimated merely from their thickness or lithological character.
+The interval between the deposit of two successive laminae of shale
+may have been as long as, or even longer than, that required for
+the formation of one of the laminae. In like manner the interval
+needed for the transition from one stratum or kind of strata to
+another may often have been more than equal to the time required
+for the formation of the strata on either side. But the relative
+chronological importance of the bars or lines in the geological
+record can seldom be satisfactorily discussed merely on lithological
+grounds. This must mainly be decided on the evidence of organic
+remains, as shown in Part VI., where the grouping of the stratified
+rocks into formations and systems is described.</p>
+</div>
+
+<p>(<i>b</i>) <i>Igneous Rocks.</i>&mdash;As part of the earth&rsquo;s crust these rocks
+present characters by which they are strongly differentiated
+from the stratified series. While the broad petrographical
+distinctions of their several varieties remain persistent, they
+present sufficient local variations of type to point to the existence
+of what have been called petrographic provinces, in each of
+which the eruptive masses are connected by a general family
+relationship, differing more or less from that of a neighbouring
+province. In each region presenting a long chronological series
+of eruptive rocks a petrographical sequence can be traced, which
+is observed to be not absolutely the same everywhere, though its
+general features may be persistent. The earliest manifestations
+of eruptive material in any district appear to have been most
+frequently of an intermediate type between acid and basic,
+passing thence into a thoroughly acid series and concluding
+with an effusion of basic material.</p>
+
+<p>Considered as part of the architecture of the crust of the earth,
+igneous rocks are conveniently divisible into two great series:
+(1) those bodies of material which have been injected into the
+crust and have solidified there, and (2) those which have reached
+the surface and have been ejected there, either in a molten state
+as lava or in a fragmental form as dust, ashes and scoriae. The
+first of these divisions represents the plutonic, intrusive or
+subsequent phase of eruptivity; the second marks the volcanic,
+interstratified or contemporaneous phase.</p>
+
+<div class="condensed">
+<p>1. The plutonic or intrusive rocks, which have been forced into
+the crust and have consolidated there, present a wide range of texture
+from the most coarse-grained granites to the most perfect natural
+glass. Seeing that they have usually cooled with extreme slowness
+underground, they are as a general rule more largely crystalline
+than the volcanic series. The form assumed by each individual
+body of intrusive material has depended upon the shape of the space
+into which it has been injected, and where it has cooled and become
+solid. This shape has been determined by the local structure of
+the earth&rsquo;s crust on the one hand and by the energy of the eruptive
+force on the other. It offers a convenient basis for the classification
+of the intrusive rocks, which, as part of the framework of the crust,
+may thus be grouped according to the shape of the cavity which
+received them, as bosses, sills, dikes and necks.</p>
+
+<p>Bosses, or stocks, are the largest and most shapeless extravasations
+of erupted material. They include the great bodies of granite which,
+in most countries of the world, have risen for many miles through
+the stratified formations and have altered the rocks around them
+by contact-metamorphism. Sills, or intrusive sheets, are bed-like
+masses which have been thrust between the planes of sedimentary
+or even of igneous rocks. The term laccolite has been applied to
+sills which are connected with bosses. Intrusive sheets are distinguishable
+from true contemporaneously intercalated lavas by not
+keeping always to the same platform, but breaking across and
+altering the contiguous strata, and by the closeness of their texture
+where they come in contact with the contiguous rocks, which, being
+cold, chilled the molten material and caused it to consolidate on its
+outer margins more rapidly than in its interior. Dikes or veins
+are vertical walls or ramifying branches of intrusive material which
+has consolidated in fissures or irregular clefts of the crust. Necks
+are volcanic chimneys which have been filled up with erupted
+material, and have now been exposed at the surface after prolonged
+denudation has removed not only the superficial volcanic masses
+originally associated with them, but also more or less of the upper
+part of the vents. Plutonic rocks do not present evidence of their
+precise geological age. All that can be certainly affirmed from
+them is that they must be younger than the rocks into which they
+have been intruded. From their internal structure, however, and
+from the evidence of the rocks associated with them, some more or
+less definite conjectures may be made as to the limits of time within
+which they were probably injected.</p>
+
+<p>2. The interstratified or volcanic series is of special importance
+in geology, inasmuch as it contains the records of volcanic action
+during the past history of the globe. It was pointed out in Part I.
+that while towards the end of the 18th and in the beginning of the
+19th century much attention was paid by Hutton and his followers
+to the proofs of intrusion afforded by what they called the &ldquo;unerupted
+lavas&rdquo; within the earth&rsquo;s crust, these observers lost sight
+of the possibility that some of these rocks might have been erupted
+at the surface, and might thus be chronicles of volcanic action in
+former geological periods. It is not always possible to satisfactorily
+discriminate between the two types of contemporaneously intercalated
+and subsequently injected material. But rocks of the
+former type have not broken into or involved the overlying strata,
+and they are usually marked by the characteristic structures of
+superficial lavas and by their association with volcanic tuffs. By
+<span class="pagenum"><a name="page665" id="page665"></a>665</span>
+means of the evidence which they supply, it has been ascertained
+that volcanic action has been manifested in the globe since the
+earliest geological periods. In the British Isles, for example, the
+volcanic record is remarkably full for the long series of ages from
+Cambrian to Permian time, and again for the older Tertiary period.</p>
+</div>
+
+<p class="pt2 center sc">2. <span class="sc">Subsequently induced Structures</span></p>
+
+<p>After their accumulation, whether as stratified or eruptive
+masses, all kinds of rocks have been subject to various changes,
+and have acquired in consequence a variety of superinduced
+structures. It has been pointed out in the part of this article
+dealing with dynamical geology that one of the most important
+forms of energy in the evolution of geological processes is to be
+found in the movements that take place within the crust of the
+earth. Some of these movements are so slight as to be only
+recognizable by means of delicate instruments; but from this
+inferior limit they range up to gigantic convulsions by which
+mountain-chains are upheaved. The crust must be regarded as
+in a perpetual state of strain, and its component materials are
+therefore subject to all the effects which flow from that condition.
+It is the one great object of the geotectonic division of geology to
+study the structures which have been developed in consequence
+of earth-movements, and to discover from this investigation the
+nature of the processes whereby the rocks of the crust have been
+brought into the condition and the positions in which we now
+find them. The details of this subject will be found in separate
+articles descriptive of each of the technical terms applied to the
+several kinds of superinduced structures. All that need be
+offered here is a general outline connecting the several portions
+of the subject together.</p>
+
+<div class="condensed">
+<p>One of the most universal of these later structures is to be seen
+in the divisional planes, usually vertical or highly inclined, by which
+rocks are split into quadrangular or irregularly shaped blocks.
+To these planes the name of joints has been given. They are of
+prime importance from an industrial point of view, seeing that the
+art of quarrying consists mainly in detecting and making proper
+use of them. Their abundance in all kinds of rocks, from those of
+recent date up to those of the highest antiquity, affords a remarkable
+testimony to the strains which the terrestrial crust has suffered.
+They have arisen sometimes from tension, such as that caused by
+contraction from the drying and consolidation of an aqueous sediment
+or from the cooling of a molten mass; sometimes from torsion
+during movements of the crust.</p>
+
+<p>Although the stratified rocks were originally deposited in a more
+or less nearly horizontal position on the floor of the sea, where now
+visible on the dry land they are seldom found to have retained their
+flatness. On the contrary, they are seen to have been generally
+tilted up at various angles, sometimes even placed on end (crop,
+dip, strike). When a sufficiently large area of ground is examined,
+the inclination into which the strata have been thrown may be
+observed not to continue far in the same direction, but to turn over
+to the opposite or another quarter. It can then be seen that in
+reality the rocks have been thrown into undulations. From the
+lowest and flattest arches where the departure from horizontality
+may be only trifling, every step may be followed up to intense
+curvature, where the strata have been compressed and plicated as
+if they had been piles of soft carpets (anticline, syncline, monocline,
+geo-anticline, geo-syncline, isoclinal, plication, curvature, quaquaversal).
+It has further happened abundantly all over the surface of
+the globe that relief from internal strain in the crust has been obtained
+by fracture, and the consequent subsidence or elevation of one or
+both sides of the fissure. The differential movement between the
+two sides may be scarcely perceptible in the feeblest dislocation,
+but in the extreme cases it may amount to many thousand feet
+(fault, fissure, dislocation, hade, slickensides). The great faults in a
+country are among its most important structural features, and as
+they not infrequently continue to be lines of weakness in the crust
+along which sudden slipping may from time to time take place, they
+become the lines of origin of earthquakes. The San Francisco
+earthquake of 1906, already cited, affords a memorable illustration
+of this connexion.</p>
+
+<p>It is in a great mountain-chain that the extraordinary complication
+of plicated and faulted structures in the crust of the earth can
+be most impressively beheld. The combination of overturned folds
+with rupture has been already referred to as a characteristic feature
+in the Alps (Part IV.). The gigantic folds have in many places been
+pushed over each other so as to lie almost flat, while the upper limb
+has not infrequently been driven for many miles beyond the lower
+by a rupture along the axis. In this way successive slices of a thick
+series of formations have been carried northwards on the northern
+slope of the Alps, and have been piled so abnormally above each
+other that some of their oldest members recur several times on
+different thrust-planes, the whole being underlain by Tertiary
+strata (see <span class="sc"><a href="#artlinks">Alps</a></span>). Further proof of the colossal compression to
+which the rocks have been subjected is afforded by their intense
+crumpling and corrugation, and by the abundantly faulted and
+crushed condition to which they have been reduced. Similar
+evidence as to stresses in the terrestrial crust and the important
+changes which they produce among the rocks may also be obtained
+on a smaller scale in many non-mountainous countries.</p>
+
+<p>Another marked result of the compression of the terrestrial crust
+has been induced in some rocks by the production of the fissile
+structure which is typically shown in roofing-slate (cleavage).
+Closely connected with this internal rearrangement has been the
+development of microscopic microlites or crystals (rutile, mica, &amp;c.)
+in argillaceous slates which were undoubtedly originally fine marine
+mud and silt. From this incipient form of metamorphism successive
+stages may be traced through the various kinds of argillite and
+phyllite into mica-schist, and thence into more crystalline gneissoid
+varieties (foliation, slate, mica-schist, gneiss). The Alps afford
+excellent illustrations of these transformations.</p>
+
+<p>The fissures produced in the crust are sometimes clean, sharply
+defined divisional planes, like cracks across a pane of glass. Much
+more usually, however, the rocks on either side have been broken up
+by the friction of movement, and the fault is marked by a variable
+breadth of this broken material. Sometimes the walls have separated
+and molten rock has risen from below and solidified between them
+as a dike. Occasionally the fissures have opened to the surface,
+and have been filled in from above with detritus, as in the sandstone-dikes
+of Colorado and California. In mineral districts the fissures
+have been filled with various spars and ores, forming what are known
+as mineral veins.</p>
+
+<p>Where one series of rocks is covered by another without any
+break or discordance in the stratification they are said to be conformable.
+But where the older series has been tilted up or visibly
+denuded before being overlain by the younger, the latter is termed
+unconformable. This relation is one of the greatest value in
+structural geology, for it marks a gap in the geological record, which
+may represent a vast lapse of time not there recorded by strata.</p>
+</div>
+
+<p class="pt2 center sc">Part VI.&mdash;Paleontological Geology</p>
+
+<p>This division of the science deals with fossils, or the traces
+of plants and animals preserved in the rocks of the earth&rsquo;s crust,
+and endeavours to gather from them information as to the history
+of the globe and its inhabitants. The term &ldquo;fossil&rdquo; (Lat.
+<i>fossilis</i>, from <i>fodere</i>, to dig up), meaning literally anything
+&ldquo;dug up,&rdquo; was formerly applied indiscriminately to any mineral
+substance taken out of the earth&rsquo;s crust, whether organized or
+not. Since the time of Lamarck, however, the meaning of the
+word has been restricted, so as to include only the remains or
+traces of plants and animals preserved in any natural formation
+whether hard rock or superficial deposit. It includes not merely
+the petrified structures of organisms, but whatever was directly
+connected with or produced by these organisms. Thus the
+resin which was exuded from trees of long-perished forests
+is as much a fossil as any portion of the stem, leaves, flowers
+or fruit, and in some respects is even more valuable to the
+geologist than more determinable remains of its parent trees,
+because it has often preserved in admirable perfection the insects
+which flitted about in the woodlands. The burrows and trails
+of a worm preserved in sandstone and shale claim recognition as
+fossils, and indeed are commonly the only indications to be met
+with of the existence of annelid life among old geological formations.
+The droppings of fishes and reptiles, called coprolites,
+are excellent fossils, and tell their tale as to the presence and
+food of vertebrate life in ancient waters. The little agglutinated
+cases of the caddis-worm remain as fossils in formations from
+which, perchance, most other traces of life may have passed
+away. Nay, the very handiwork of man, when preserved in
+any natural manner, is entitled to rank among fossils; as
+where his flint-implements have been dropped into the pre-historic
+gravels of river-valleys or where his canoes have been
+buried in the silt of lake-bottoms.</p>
+
+<div class="condensed">
+<p>A study of the land-surfaces and sea-floors of the present time
+shows that there are so many chances against the conservation
+of the remains of either terrestrial or marine animals and plants
+that if, as is probable, the same conditions existed in former geological
+periods, we should regard the occurrence of organic remains among
+the stratified formations of the earth&rsquo;s crust as generally the result
+of various fortunate accidents.</p>
+
+<p>Let us consider, in the first place, the chances for the preservation
+of remains of the present fauna and flora of a country. The surface
+of the land may be densely clothed with forest and abundantly
+peopled with animal life. But the trees die and moulder into soil.
+<span class="pagenum"><a name="page666" id="page666"></a>666</span>
+The animals, too, disappear, generation after generation, and leave
+few or no perceptible traces of their existence. If we were not aware
+from authentic records that central and northern Europe were
+covered with vast forests at the beginning of our era, how could we
+know this fact? What has become of the herds of wild oxen, the
+bears, wolves and other denizens of primeval Europe? How could
+we prove from the examination of the surface soil of any country
+that those creatures had once abounded there? The conditions for
+the preservation of any relics of the plant and animal life of a terrestrial
+surface must obviously be always exceptional. They are
+supplied only where the organic remains can be protected from the
+air and superficial decay. Hence they may be observed in (1) the
+deposits on the floors of lakes; (2) in peat-mosses; (3) in deltas at
+river-mouths; and (4) under the stalagmite of caverns in limestone
+districts. But in these and other favourable places a mere infinitesimal
+fraction of the fauna or flora of a land-surface is likely to be
+entombed or preserved.</p>
+
+<p>In the second place, although in the sea the conditions for the
+preservation of organic remains are in many respects more favourable
+than on land, they are apt to be frustrated by many adverse circumstances.
+While the level of the land remains stationary, there can
+be but little effective entombment of marine organisms in littoral
+deposits; for only a limited accumulation of sediment will be formed
+until subsidence of the sea-floor takes place. In the trifling beds of
+sand or gravel thrown up on a stationary shore, only the harder and
+more durable forms of life, such as gastropods and lamellibranchs,
+which can withstand the triturating effects of the beach waves, are
+likely to remain uneffaced.</p>
+
+<p>Below tide-marks, along the margin of the land where sediment
+is gradually deposited, the conditions are more favourable for the
+preservation of marine organisms. In the sheets of sand and mud
+there laid down the harder parts of many forms of life may be
+entombed and protected from decay. But only a small proportion
+of the total marine fauna may be expected to appear in such deposits.
+At the best, merely littoral and shallow-water forms will occur, and,
+even under the most favourable conditions, they will represent but
+a fraction of the whole assemblage of life in these juxta-terrestrial
+parts of the ocean. As we recede from the land the rate of deposition
+of sediment on the sea-floor must become feebler, until, in the remote
+central abysses, it reaches a hardly appreciable minimum. Except,
+therefore, where some kind of ooze or other deposit is accumulating
+in these more pelagic regions, the conditions must be on the whole
+unfavourable for the preservation of any adequate representation
+of the deep-sea fauna. Hard durable objects, such as teeth and
+bones, may slowly accumulate, and be protected by a coating of
+peroxide of manganese, or of some of the silicates now forming here
+and there over the deep-sea bottom; or the rate of growth of the
+abysmal deposit may be so tardy that most of the remains of at
+least the larger animals will disappear, owing to decay, before they
+can be covered up and preserved. Any such deep-sea formation,
+if raised into land, would supply but a meagre picture of the whole
+life of the sea.</p>
+
+<p>It would thus appear that the portion of the sea-floor best suited
+for receiving and preserving the most varied assemblage of marine
+organic remains is the area in front of the land, to which rivers and
+currents bring continual supplies of sediment. The most favourable
+conditions for the accumulation of a thick mass of marine fossiliferous
+strata will arise when the area of deposit is undergoing a gradual
+subsidence. If the rate of depression and that of deposit were equal,
+or nearly so, the movement might proceed for a vast period without
+producing any great apparent change in marine geography, and even
+without seriously affecting the distribution of life over the sea-floor
+within the area of subsidence. Hundreds or thousands of feet of
+sedimentary strata might in this way be heaped up round the continents,
+containing a fragmentary series of organic remains belonging
+to those forms of comparatively shallow-water life which had hard
+parts capable of preservation. There can be little doubt that such
+has, in fact, been the history of the main mass of stratified formations
+in the earth&rsquo;s crust. By far the largest proportion of these piles
+of marine strata has unquestionably been laid down in water of no
+great depth within the area of deposit of terrestrial sediment.
+The enormous thickness to which they attain seems only explicable
+by prolonged and repeated movements of subsidence, interrupted,
+however, as we know, by other movements of a contrary kind.</p>
+
+<p>Since the conditions for the preservation of organic remains exist
+more favourably under the sea than on land, marine organisms must
+be far more abundantly conserved than those of the land. This is
+true to-day, and has, as far as known, been true in all past geological
+time. Hence for the purposes of the geologist the fossil remains of
+marine forms of life far surpass all others in value. Among them
+there will necessarily be a gradation of importance, regulated chiefly
+by their relative abundance. Now, of all the marine tribes which
+live within the juxta-terrestrial belt of sedimentation, unquestionably
+the Mollusca stand in the place of pre-eminence as regards their
+aptitude for becoming fossils. They almost all possess a hard, durable
+shell, capable of resisting considerable abrasion and readily passing
+into a mineralized condition. They are extremely abundant both as
+to individuals and genera. They occur on the shore within tide
+mark, and range thence down into the abysses. Moreover, they
+appear to have possessed these qualifications from early geological
+times. In the marine Mollusca, therefore, we have a common ground
+of comparison between the stratified formations of different periods.
+They have been styled the alphabet of palaeontological inquiry.</p>
+</div>
+
+<p>There are two main purposes to which fossils may be put in
+geological research: (1) to throw light upon former conditions
+of physical geography, such as the presence of land, rivers,
+lakes and seas, in places where they do not now exist, changes
+of climate, and the former distribution of plants and animals;
+and (2) to furnish a guide in geological chronology whereby
+rocks may be classified according to relative date, and the facts
+of geological history may be arranged and interpreted as a
+connected record of the earth&rsquo;s progress.</p>
+
+<div class="condensed">
+<p>1. As examples of the first of these two directions of inquiry
+reference may be made to (<i>a</i>) former land-surfaces revealed by the
+occurrence of layers of soil with tree-stumps and roots still in the
+position of growth (see <span class="sc"><a href="#artlinks">Purbeckian</a></span>); (<i>b</i>) ancient lakes proved by
+beds of marl or limestone full of lacustrine shells; (<i>c</i>) old sea-bottoms
+marked by the occurrence of marine organisms; (<i>d</i>) variations in
+the quality of the water, such as freshness or saltness, indicated by
+changes in the size and shape of the fossils; (<i>e</i>) proximity to former
+land, suggested by the occurrence of abundant drift-wood in the
+strata; (<i>f</i>) former conditions of climate, different from the present,
+as evidenced by such organisms as tropical types of plants and
+animals intercalated among the strata of temperate or northern
+countries.</p>
+
+<p>2. In applying fossils to the determination of geological chronology
+it is first necessary to ascertain the order of superposition of the
+rocks. Obviously, in a continuous series of undisturbed sedimentary
+deposits the lowest must necessarily be the oldest, and the plants or
+animals which they contain must have lived and died before any of
+the organisms that occur in the overlying strata. This order of
+superposition having been settled in a series of formations, it is
+found that the fossils at the bottom are not quite the same as those
+at the top of the series. Tracing the beds upward, we discover that
+species after species of the lowest platforms disappears, until perhaps
+not one of them is found. With the cessation of these older species
+others make their entrance. These, in turn, are found to die out,
+and to be replaced by newer forms. After patient examination of
+the rocks, it has been ascertained that every well-marked &ldquo;formation,&rdquo;
+or group of strata, is characterized by its own species or
+genera, or by a general assemblage, or <i>facies</i>, of organic forms.
+Such a generalization can only, of course, be determined by actual
+practical experience over an area of some size. When the typical
+fossils of a formation are known, they serve to identify that formation
+in its progress across a country. Thus, in tracts where the true
+order of superposition cannot be determined, owing to the want of
+sections or to the disturbed condition of the rocks, fossils serve as a
+means of identification and furnish a guide to the succession of the
+rocks. They even demonstrate that in some mountainous ground
+the beds have been turned completely upside down, where it
+can be shown that the fossils in what are now the uppermost
+strata ought properly to lie underneath those in the beds below
+them.</p>
+
+<p>It is by their characteristic fossils that the stratified rocks of the
+earth&rsquo;s crust can be most satisfactorily subdivided into convenient
+groups of strata and classed in chronological order. Each &ldquo;formation&rdquo;
+is distinguished by its own peculiar assemblage of organic
+remains, by means of which it can be followed and recognized, even
+amid the crumplings and dislocations of a disturbed region. The
+same general succession of organic types can be observed over a
+large part of the world, though, of course, with important modifications
+in different countries. This similarity of succession has been
+termed <i>homotaxis</i>, a term which expresses the fact that the order
+in which the leading types of organized existence have appeared
+upon the earth has been similar even in widely separated regions.
+It is evident that, in this way, a reliable method of comparison
+is furnished, whereby the stratified formations of different parts of
+the earth&rsquo;s crust can be brought into relation with each other.
+Had the geologist continued to remain, as in the days of Werner,
+hampered by the limitations imposed by a reliance on mere lithological
+characters, he would have made little or no progress in
+deciphering the record of the successive phases of the history of
+the globe chronicled in the crust. Just as, at the present time,
+sheets of gravel in one place are contemporaneous with sheets of
+mud at another, so in the past all kinds of sedimentation have been
+in progress simultaneously, and those of one period may not be
+distinguishable in themselves from those of another. Little or no
+reliance can be placed upon lithological resemblances or differences
+in comparing the sedimentary formations of different countries.</p>
+
+<p>In making use of fossil evidence for the purpose of subdividing
+the stratified rocks of the earth&rsquo;s crust, it is found to be applicable
+to the smaller details of stratigraphy as well as to the definition of
+large groups of strata. Thus a particular stratum may be marked
+by the occurrence in it of various fossils, one or more of which may
+be distinctive, either from occurring in no other bed above and
+below or from special abundance in that stratum. One or more of
+these species is therefore used as a guide to the occurrence of the bed
+<span class="pagenum"><a name="page667" id="page667"></a>667</span>
+in question, which is called by the name of the most abundant
+species. In this way what is called a &ldquo;geological horizon,&rdquo; or
+&ldquo;zone,&rdquo; is marked off, and its exact position in the series of formations
+is fixed.</p>
+
+<p>Perhaps the most distinctive feature in the progress of palaeontological
+geology during the last half century has been the recognition
+and wide application of this method of zonal stratigraphy, which,
+in itself, was only a further development of William Smith&rsquo;s famous
+idea, &ldquo;Strata identified by Organized Fossils.&rdquo; It was first carried
+out in detail by various palaeontologists in reference to the Jurassic
+formations, notably by F.A. von Quenstedt and C.A. Oppel in
+Germany and A.D. d&rsquo;Orbigny in France. The publication of
+Oppel&rsquo;s classic work <i>Die Juraformation Englands, Frankreichs und
+des südwestlichen Deutschlands</i> (1856-1858) marked an epoch in the
+development of stratigraphical geology. Combining what had been
+done by various observers with his own laborious researches in
+France, England, Württemberg and Bavaria, he drew up a classification
+of the Jurassic system, grouping its several formations into zones,
+each characterized by some distinctly predominant fossil after which
+it was named (see <span class="sc"><a href="#artlinks">Lias</a></span>). The same method of classification was
+afterwards extended to the Cretaceous series by A.D. d&rsquo;Orbigny,
+E. Hébert and others, until the whole Mesozoic rocks from the
+Trias to the top of the Chalk has now been partitioned into zones,
+each named after some characteristic species or genus of fossils.
+More recently the principle has been extended to the Palaeozoic
+formations, though as yet less fully than to the younger parts of the
+geological record. It has been successfully applied by Professor C.
+Lapworth to the investigation of the Silurian series (see <span class="sc"><a href="#artlinks">Silurian</a></span>;
+<span class="sc"><a href="#artlinks">Ordovician System</a></span>). He found that the species of graptolites
+have each a comparatively narrow vertical range, and they may
+consequently be used for stratigraphical purposes. Applying the
+method, in the first instance, to the highly plicated Silurian rocks of
+the south of Scotland, he found that by means of graptolites he was
+able to work out the structure of the ground. Each great group of
+strata was seen to possess its own graptolitic zones, and by their
+means could be identified not only in the original complex Scottish
+area, but in England and Wales and in Ireland. It was eventually
+ascertained that the succession of zones in Great Britain could be
+recognized on the Continent, in North America and even in Australia.
+The brachiopods and trilobites have likewise been made use of for
+zonal purposes among the oldest sedimentary formations. The
+most ancient of the Palaeozoic systems has as its fitting base the
+<i>Olenellus</i> zone.</p>
+
+<p>Within undefined and no doubt variable geographical limits
+palaeontological zones have been found to be remarkably persistent.
+They follow each other in the same general order, but not always
+with equal definiteness. The type fossil may appear in some districts
+on a higher or a lower platform than it does in others. Only to a
+limited degree is there any coincidence between lithological variations
+in the strata and the sequence of the zones. In the Jurassic formations,
+indeed, where frequent alternations of different sedimentary
+materials are to be met with, it is in some cases possible to trace a
+definite upward or downward limit for a zone by some abrupt
+change in the sedimentation, such as from limestone to shale. But
+such a precise demarcation is impossible where no distinct bands of
+different sediments are to be seen. The zones can then only be
+vaguely determined by finding their characteristic fossils, and noting
+where these begin to appear in the strata and where they cease.
+It would seem, therefore, that the sequence of palaeontological
+zones, or life-horizons, has not depended merely upon changes in
+the nature of the conditions under which the organisms lived. We
+should naturally expect that these changes would have had a marked
+influence; that, for instance, a difference should be perceptible
+between the character of the fossils in a limestone and that of those
+in a shale or a sandstone. The environment, when a limestone was
+in course of deposition, would generally be one of clear water,
+favourable for a more vigorous and more varied fauna than where
+a shale series was accumulating, when the water would be discoloured,
+and only such animals would continue to live in it, or on
+the bottom, as could maintain themselves in the midst of mud.
+But no such lithological reason, betokening geographical changes
+that would affect living creatures, can be adduced as a universally
+applicable explanation of the occurrence and limitation of palaeontological
+zones. One of these zones may be only a few inches, or
+feet or yards in vertical extent, and no obvious lithological or other
+cause can be seen why its specially characteristic fossils should
+not be found just as frequently in the similar strata above and
+below. There is often little or no evidence of any serious change
+in the conditions of sedimentation, still less of any widespread
+physical disturbance, such as the catastrophes by which the
+older geologists explained the extinction of successive types of
+life.</p>
+
+<p>It has been suggested that, where the life-zones are well defined,
+sedimentation has been extremely slow, and that though these zones
+follow each other with no break in the sedimentation, they were
+really separated by prolonged intervals of time during which organic
+evolution could come effectively into play. But it is not easy to
+explain how, for example in the Lower Lias, there could have been
+a succession of prodigious intervals, when practically no sediment
+was laid down, and yet that the strata should show no sign of contemporaneous
+disturbance or denudation, but succeed each other
+as if they had been accumulated by one continuous process of
+deposit. It must be admitted that the problem of life-zones in
+stratigraphical geology has not yet been solved.</p>
+
+<p>As Darwin first cogently showed, the history of life has been very
+imperfectly registered in the stratified parts of the earth&rsquo;s crust.
+Apart from the fact that, even under the most favourable conditions,
+only a small proportion of the total flora and fauna of any period
+would be preserved in the fossil state, enormous gaps occur where
+no record has survived at all. It is as if whole chapters and books
+were missing from a historical work. Some of these lacunae are
+sufficiently obvious. Thus, in some cases, powerful dislocations have
+thrown considerable portions of the rocks out of sight. Sometimes
+extensive metamorphism has so affected them that their original
+characters, including their organic contents, have been destroyed.
+Oftenest of all, denudation has come into play, and vast masses of
+fossiliferous rock have been entirely worn away, as is demonstrated
+by the abundant unconformabilities in the structure of the earth&rsquo;s
+crust.</p>
+
+<p>While the mere fact that one series of rocks lies unconformably
+on another proves the lapse of a considerable interval between their
+respective dates, the relative length of this interval may sometimes
+be proved by means of fossil evidence, and by this alone. Let us
+suppose, for example, that a certain group of formations has been
+disturbed, upraised, denuded and covered unconformably by a
+second group. In lithological characters the two may closely resemble
+each other, and there may be nothing to show that the gap represented
+by their unconformability is of an important character. In
+many cases, indeed, it would be quite impossible to pronounce any
+well-grounded judgment as to the amount of interval, even measured
+by the vague relative standards of geological chronology. But if
+each group contains a well-preserved suite of organic remains, it
+may not only be possible, but easy, to say exactly how much of the
+geological record has been left out between the two sets of formations.
+By comparing the fossils with those obtained from regions where the
+geological record is more complete, it may be ascertained, perhaps,
+that the lower rocks belong to a certain platform or stage in geological
+history which for our present purpose we may call D, and that the
+upper rocks can in like manner be paralleled with stage H. It would
+be then apparent that at this locality the chronicles of three great
+geological periods E, F, and G were wanting, which are elsewhere
+found to be intercalated between D and H. The lapse of time represented
+by this unconformability would thus be equivalent to that
+required for the accumulation of the three missing formations in
+those regions where sedimentation was more continuous.</p>
+
+<p>Fossil evidence may be made to prove the existence of gaps which
+are not otherwise apparent. As has been already remarked, changes
+in organic forms must, on the whole, have been extremely slow in
+the geological past. The whole species of a sea-floor could not pass
+entirely away, and be replaced by other forms, without the lapse
+of long periods of time. If then among the conformable stratified
+formations of former ages we encounter sudden and abrupt changes
+in the <i>facies</i> of the fossils, we may be certain that these must mark
+omissions in the record, which we may hope to fill in from a more
+perfect series elsewhere. The complete biological contrasts between
+the fossil contents of unconformable strata are sufficiently explicable.
+It is not so easy to give a satisfactory account of those which occur
+where the beds are strictly conformable, and where no evidence can
+be observed of any considerable change of physical conditions at the
+time of deposit. A group of strata having the same general lithological
+characters throughout may be marked by a great discrepance
+between the fossils above and below a certain line. A few species
+may pass from the one into the other, or perhaps every species may
+be different. In cases of this kind, when proved to be not merely
+local but persistent over wide areas, we must admit, notwithstanding
+the apparently undisturbed and continuous character of the original
+deposition of the strata, that the abrupt transition from the one <i>facies</i>
+of fossils to the other represents a long interval of time which has not
+been recorded by the deposit of strata. A.C. Ramsay, who called
+attention to these gaps, termed them &ldquo;breaks in the succession of
+organic remains.&rdquo; He showed that they occur abundantly among
+the Palaeozoic and Secondary rocks of England. It is obvious, of
+course, that such breaks, even though traceable over wide regions,
+were not general over the whole globe. There have never been any
+universal interruptions in the continuity of the chain of being,
+so far as geological evidence can show. But the physical changes
+which caused the breaks may have been general over a zoological
+district or minor region. They no doubt often caused the complete
+extinction of genera and species which had a small geographical
+range.</p>
+
+<p>From all these facts it is clear that the geological record, as it now
+exists, is at the best but an imperfect chronicle of geological history.
+In no country is it complete. The lacunae of one region must be
+supplied from another. Yet in proportion to the geographical
+distance between the localities where the gaps occur and those
+whence the missing intervals are supplied, the element of uncertainty
+in our reading of the record is increased. The most desirable
+method of research is to exhaust the evidence for each area or
+province, and to compare the general order of its succession as a
+whole with that which can be established for other provinces.</p>
+</div>
+
+<p><span class="pagenum"><a name="page668" id="page668"></a>668</span></p>
+
+<p class="pt2 center sc">Part VII.&mdash;Stratigraphical Geology</p>
+
+<p>This branch of the science arranges the rocks of the earth&rsquo;s
+crust in the order of their appearance, and interprets the sequence
+of events of which they form the records. Its province is to
+cull from the other departments of geology the facts which may
+be needed to show what has been the progress of our planet,
+and of each continent and country, from the earliest times of
+which the rocks have preserved any memorial. Thus from
+mineralogy and petrography it contains information regarding
+the origin and subsequent mutations of minerals and rocks.
+From dynamical geology it learns by what agencies the materials
+of the earth&rsquo;s crust have been formed, altered, broken, upheaved
+and melted. From geotectonic geology it understands the
+various processes whereby these materials were put together
+so as to build up the complicated crust of the earth. From
+palaeontological geology it receives in well-determined fossil
+remains a clue by which to discriminate the different stratified
+formations, and to trace the grand onward march of organized
+existence upon this planet. Stratigraphical geology thus
+gathers up the sum of all that is made known by the other
+departments of the science, and makes it subservient to the
+interpretation of the geological history of the earth.</p>
+
+<p>The leading principles of stratigraphy may be summed up
+as follows:</p>
+
+<p>1. In every stratigraphical research the fundamental requisite
+is to establish the order of superposition of the strata. Until
+this is accomplished it is impossible to arrange the dates, and
+make out the sequence of geological history.</p>
+
+<p>2. The stratified portion of the earth&rsquo;s crust, or what has been
+called the &ldquo;geological record,&rdquo; can be subdivided into natural
+groups, or series of strata, characterized by distinctive organic
+remains and recognizable by these remains, in spite of great
+changes in lithological character from place to place. A bed,
+or a number of beds, linked together by containing one or more
+distinctive species or genera of fossils is termed a <i>zone</i> or <i>horizon</i>,
+and usually bears the name of one of its more characteristic
+fossils, as the <i>Planorbis</i>-zone of the Lower Lias, which is so
+called from the prevalence in it of the ammonite <i>Psiloceras
+planorbis</i>. Two or more such zones related to each other by the
+possession of a number of the same characteristic species or
+genera have been designated <i>beds</i> or an <i>assise</i>. Two or more
+sets of beds or assises similarly related form a <i>group</i> or <i>stage</i>; a
+number of groups or stages make a <i>series</i>, <i>formation</i> or <i>section</i>,
+and a succession of formations may be united into a <i>system</i>.</p>
+
+<p>3. Some living species of plants and animals can be traced
+downwards through the more recent geological formations;
+but the number which can be so followed grows smaller as the
+examination is pursued into more ancient deposits. With their
+disappearance other species or genera present themselves which
+are no longer living. These in turn may be traced backward into
+earlier formations, till they too cease and their places are taken by
+yet older forms. It is thus shown that the stratified rocks contain
+the records of a gradual progression of organic forms. A species
+which has once died out does not seem ever to have reappeared.</p>
+
+<p>4. When the order of succession of organic remains among the
+stratified rocks has been determined, they become an invaluable
+guide in the investigation of the relative age of rocks and the
+structure of the land. Each zone and formation, being characterized
+by its own species or genera, may be recognized by their
+means, and the true succession of strata may thus be confidently
+established even in a country wherein the rocks have been
+shattered by dislocation, folded, inverted or metamorphosed.</p>
+
+<p>5. Though local differences exist in regard to the precise zone
+in which a given species of organism may make its first appearance,
+the general order of succession of the organic forms found in the
+rocks is never inverted. The record is nowhere complete in any
+region, but the portions represented, even though extremely
+imperfect, always follow each other in their proper chronological
+order, unless where disturbance of the crust has intervened to
+destroy the original sequence.</p>
+
+<p>6. The relative chronological value of the divisions of the
+geological record is not to be measured by mere depth of strata.
+While it may be reasonably assumed that, in general, a great
+thickness of stratified rock must mark the passage of a long
+period of time, it cannot safely be affirmed that a much less
+thickness elsewhere must represent a correspondingly diminished
+period. The need for this caution may sometimes be made
+evident by an unconformability between two sets of rocks, as
+has already been explained. The total depth of both groups
+together may be, say 1000 ft. Elsewhere we may find a single
+unbroken formation reaching a depth of 10,000 ft.; but it would
+be unwarrantable to assume that the latter represents ten times
+the length of time indicated by the former two. So far from
+this being the case, it might not be difficult to show that the
+minor thickness of rock really denotes by far the longer geological
+interval. If, for instance, it could be proved that the upper
+part of both the sections lies on one and the same geological
+platform, but that the lower unconformable series in the one
+locality belongs to a far lower and older system of rocks than the
+base of the thick conformable series in the other, then it would
+be clear that the gap marked by the unconformability really
+indicates a longer period than the massive succession of deposits.</p>
+
+<p>7. Fossil evidence furnishes the chief means of comparing the
+relative value of formations and groups of rock. A &ldquo;break in
+the succession of organic remains,&rdquo; as already explained, marks
+an interval of time often unrepresented by strata at the place
+where the break is found. The relative importance of these
+breaks, and therefore, probably, the comparative intervals
+of time which they mark, may be estimated by the difference
+of the <i>facies</i> or general character of the fossils on each side.
+If, for example, in one case we find every species to be dissimilar
+above and below a certain horizon, while in another locality only
+half of the species on each side are peculiar, we naturally infer,
+if the total number of species seems large enough to warrant
+the inference, that the interval marked by the former break
+was much longer than that marked by the second. But we may
+go further and compare by means of fossil evidence the relation
+between breaks in the succession of organic remains and the
+depth of strata between them.</p>
+
+<div class="condensed">
+<p>Three formations of fossiliferous strata, A, C, and H, may occur
+conformably above each other. By a comparison of the fossil
+contents of all parts of A, it may be ascertained that, while some
+species are peculiar to its lower, others to its higher portions, yet the
+majority extend throughout the formation. If now it is found that
+of the total number of species in the upper portion of A only one-third
+passes up into C, it may be inferred with some plausibility that the
+time represented by the break between A and C was really longer
+than that required for the accumulation of the whole of the formation
+A. It might even be possible to discover elsewhere a thick intermediate
+formation B filling up the gap between A and C. In like
+manner were it to be discovered that, while the whole of the formation
+C is characterized by a common suite of fossils, not one of the species
+and only one half of the genera pass up into H, the inference could
+hardly be resisted that the gap between the two formations marks
+the passage of a far longer interval than was needed for the deposition
+of the whole of C. And thus we reach the remarkable conclusion
+that, thick though the stratified formations of a country may be,
+in some cases they may not represent so long a total period of time
+as do the gaps in their succession,&mdash;in other words, that non-deposition
+was more frequent and prolonged than deposition, or that the
+intervals of time which have been recorded by strata have not been
+so long as those which have not been so recorded.</p>
+</div>
+
+<p>In all speculations of this nature, however, it is necessary
+to reason from as wide a basis of observation as possible, seeing
+that so much of the evidence is negative. Especially needful
+is it to bear in mind that the cessation of one or more species
+at a certain line among the rocks of a particular district may
+mean nothing more than that, onward from the time marked
+by that line, these species, owing to some change in the conditions
+of life, were compelled to migrate or became locally extinct or,
+from some alteration in the conditions of fossilization, were no
+longer imbedded and preserved as fossils. They may have
+continued to flourish abundantly in neighbouring districts for
+a long period afterward. Many examples of this obvious
+truth might be cited. Thus in a great succession of mingled
+marine, brackish-water and terrestrial strata, like that of the
+Carboniferous Limestone series of Scotland, corals, crinoids
+<span class="pagenum"><a name="page669" id="page669"></a>669</span>
+and brachiopods abound in the limestones and accompanying
+shales, but disappear as the sandstones, ironstones, clays, coals
+and bituminous shales supervene. An observer meeting for the
+first time with an instance of this disappearance, and remembering
+what he had read about breaks in succession, might be
+tempted to speculate about the extinction of these organisms,
+and their replacement by other and later forms of life, such as
+the ferns, lycopods, estuarine or fresh-water shells, ganoid
+fishes and other fossils so abundant in the overlying strata.
+But further research would show him that high above the plant-bearing
+sandstones and coals other limestones and shales might
+be observed, once more charged with the same marine fossils
+as before, and still farther overlying groups of sandstones, coals
+and carbonaceous beds followed by yet higher marine limestones.
+He would thus learn that the same organisms, after being
+locally exterminated, returned again and again to the same
+area. After such a lesson he would probably pause before too
+confidently asserting that the highest bed in which we can
+detect certain fossils marks their final appearance in the history
+of life. Some breaks in the succession may thus be extremely
+local, one set of organisms having been driven to a different part
+of the same region, while another set occupied their place until
+the first was enabled to return.</p>
+
+<p>8. The geological record is at the best but an imperfect
+chronicle of the geological history of the earth. It abounds
+in gaps, some of which have been caused by the destruction of
+strata owing to metamorphism, denudation or otherwise, others
+by original non-deposition, as above explained. Nevertheless
+from this record alone can the progress of the earth be traced.
+It contains the registers of the appearance and disappearance
+of tribes of plants and animals which have from time to time
+flourished on the earth. Only a small proportion of the total
+number of species which have lived in past time have been thus
+chronicled, yet by collecting the broken fragments of the record
+an outline at least of the history of life upon the earth can be
+deciphered.</p>
+
+<p>It cannot be too frequently stated, nor too prominently kept
+in view, that, although gaps occur in the succession of organic
+remains as recorded in the rocks, they do not warrant the conclusion
+that any such blank intervals ever interrupted the progress
+of plant and animal life upon the globe. There is every reason
+to believe that the march of life has been unbroken, onward and
+upward. Geological history, therefore, if its records in the
+stratified formations were perfect, ought to show a blending
+and gradation of epoch with epoch. But the progress has been
+constantly interrupted, now by upheaval, now by volcanic
+outbursts, now by depression. These interruptions serve as
+natural divisions in the chronicle, and enable the geologist to
+arrange his history into periods. As the order of succession
+among stratified rocks was first made out in Europe, and as many
+of the gaps in that succession were found to be widespread over
+the European area, the divisions which experience established
+for that portion of the globe came to be regarded as typical,
+and the names adopted for them were applied to the rocks of
+other and far distant regions. This application has brought out
+the fact that some of the most marked breaks in the European
+series do not exist elsewhere, and, on the other hand, that some
+portions of that series are much more complete than the corresponding
+sections in other regions. Hence, while the general
+similarity of succession may remain, different subdivisions and
+nomenclature are required as we pass from continent to continent.</p>
+
+<p>The nomenclature adopted for the subdivisions of the geological
+record bears witness to the rapid growth of geology. It is a
+patch-work in which no system nor language has been adhered
+to, but where the influences by which the progress of the science
+has been moulded may be distinctly traced. Some of the earliest
+names are lithological, and remind us of the fact that mineralogy
+and petrography preceded geology in the order of birth&mdash;Chalk,
+Oolite, Greensand, Millstone Grit. Others are topographical,
+and often recall the labours of the early geologists of England&mdash;London
+Clay, Oxford Clay, Purbeck, Portland, Kimmeridge beds.
+Others are taken from local English provincial names, and
+remind us of the debt we owe to William Smith, by whom so
+many of them were first used&mdash;Lias, Gault, Crag, Cornbrash.
+Others of later date recognize an order of superposition as
+already established among formations&mdash;Old Red Sandstone,
+New Red Sandstone. By common consent it is admitted that
+names taken from the region where a formation or group of rocks
+is typically developed are best adapted for general use.
+Cambrian, Silurian, Devonian, Permian, Jurassic are of this
+class, and have been adopted all over the globe.</p>
+
+<p>But whatever be the name chosen to designate a particular
+group of strata, it soon comes to be used as a chronological or
+homotaxial term, apart altogether from the stratigraphical
+character of the strata to which it is applied. Thus we speak
+of the Chalk or Cretaceous system, and embrace under that
+term formations which may contain no chalk; and we may
+describe as Silurian a series of strata utterly unlike in lithological
+characters to the formations in the typical Silurian country.
+In using these terms we unconsciously allow the idea of relative
+date to arise prominently before us. Hence such a word as
+&ldquo;chalk&rdquo; or &ldquo;cretaceous&rdquo; does not suggest so much to us the
+group of strata so called as the interval of geological history
+which these strata represent. We speak of the Cretaceous,
+Jurassic, and Cambrian periods, and of the Cretaceous fauna,
+the Jurassic flora, the Cambrian trilobites, as if these adjectives
+denoted simply epochs of geological time.</p>
+
+<p>The stratified formations of the earth&rsquo;s crust, or geological
+record, are classified into five main divisions, which in their
+order of antiquity are as follows: (1) Archean or Pre-Cambrian,
+called also sometimes Azoic (lifeless) or Eozoic (dawn of life);
+(2) Palaeozoic (ancient life) or Primary; (3) Mesozoic (middle
+life) or Secondary; (4) Cainozoic (recent life) or Tertiary;
+(5) Quaternary or Post-Tertiary. These divisions are further
+ranged into systems, formations, groups or stages, assises and
+zones. Accounts of the various subdivisions named are given
+in separate articles under their own headings. In order, however,
+that the sequence of the formations and their parallelism in
+Europe and North America may be presented together a stratigraphical
+table is given on next page.</p>
+
+<p class="pt2 center sc">Part VIII.&mdash;Physiographical Geology</p>
+
+<p>This department of geological inquiry investigates the origin
+and history of the present topographical features of the land.
+As these features must obviously be related to those of earlier
+time which are recorded in the rocks of the earth&rsquo;s crust, they
+cannot be satisfactorily studied until at least the main outlines
+of the history of these rocks have been traced. Hence physiographical
+research comes appropriately after the other branches
+of the science have been considered.</p>
+
+<p>From the stratigraphy of the terrestrial crust we learn that
+by far the largest part of the area of dry land is built up of marine
+formations; and therefore that the present land is not an
+aboriginal portion of the earth&rsquo;s surface, but has been overspread
+by the sea in which its rocks were mainly accumulated. We
+further discover that this submergence of the land did not
+happen once only, but again and again in past ages and in all
+parts of the world. Yet although the terrestrial areas varied
+much from age to age in their extent and in their distribution,
+being at one time more continental, at another more insular,
+there is reason to believe that these successive diminutions and
+expansions have on the whole been effected within, or not far
+outside, the limits of the existing continents. There is no
+evidence that any portion of the present land ever lay under the
+deeper parts of the ocean. The abysmal deposits of the ocean-floor
+have no true representatives among the sedimentary
+formations anywhere visible on the land. Nor, on the other
+hand, can it be shown that any part of the existing ocean
+abysses ever rose above sea-level into dry land. Hence geologists
+have drawn the inference that the ocean basins have probably
+been always where they now are; and that although the continental
+areas have often been narrowed by submergence and by
+denudation, there has probably seldom or never been a complete
+disappearance of land. The fact that the sedimentary formations
+of each successive geological period consist to so large an
+extent of mechanically formed terrigenous detritus, affords
+good evidence of the coexistence of tracts of land as well as of
+extensive denudation.</p>
+
+<p><span class="pagenum"><a name="page670" id="page670"></a>670</span></p>
+
+<p class="pt2 center"><i>The Geological Record or Order of Succession of the Stratified
+Formations of the Earth&rsquo;s Crust.</i></p>
+
+<table class="nobctr f90" summary="Contents">
+<tr><td class="tccm allb">&emsp;</td> <td class="tccm allb">&emsp;</td> <td class="tccm allb" colspan="2">Europe.</td> <td class="tccm allb">North America.</td></tr>
+
+<tr><td class="tccm allb" rowspan="2">Quaternary<br />or<br />Post-Tertiary.</td>
+
+<td class="tccm allb">Recent,<br />Post-glacial<br />or Human.</td>
+
+<td class="tcl rb bb" style="width: 40%;" colspan="2"><p>Historic, up to the present time.</p>
+<p>Prehistoric, comprising deposits of the Iron, Bronze, and later Stone Ages.</p>
+<p>Neolithic&mdash;alluvium, peat, lake-dwellings, loess, &amp;c.</p>
+<p>Palaeolithic&mdash;river-gravels, cave-deposits, &amp;c.</p></td>
+
+<td class="tcl rb bb" style="width: 40%;"><p>Similar to the European development, but with scantier traces of the presence of man.</p></td></tr>
+
+<tr><td class="tccm allb">Pleistocene or Glacial.</td>
+
+<td class="tcl rb bb" colspan="2"><p>Older Loess and valley-gravels; cave-deposits.</p>
+<p>Strand-lines or raised beaches; youngest moraines.</p>
+<p>Upper Boulder-clays; eskers; marine sands and clays.</p>
+<p>Interglacial deposits.</p>
+<p>Lower boulder-clay or Till, with striated rock-surfaces below.</p></td>
+
+<td class="tcl rb bb"><p>As in Europe, it is hardly possible to assign a definite chronological place to
+each of the various deposits of this period, terrestrial and marine. They generally resemble the
+European series. The characteristic marine, fluviatile and lacustrine terraces, which
+overlie the older drifts, have been classed as the Champlain Group.</p></td></tr>
+
+<tr><td class="tccm allb" rowspan="4">Cainozoic or Tertiary.</td>
+
+<td class="tccm allb">Pliocene.</td>
+
+<td class="tcl allb" colspan="2"><p>Newer:&mdash;English Forest-Bed Group; Red and Norwich Crag; Amstelian and Scaldesian groups
+ of Belgium and Holland; Sicilian and Astian of France and Italy.</p>
+<p>Older:&mdash;English Coralline Crag; Diestian of Belgium; Plaisancian of southern France and Italy.</p></td>
+
+<td class="tcl allb"><p>On the Atlantic border represented by the marine Floridian series; in the interior
+by a subaerial and lacustrine series; and on the Pacific border by the thick marine series of San Francisco.</p></td></tr>
+
+<tr><td class="tccm allb">Miocene.</td>
+
+<td class="tcl allb" colspan="2"><p>Wanting in Britain; well developed in France, S. E. Europe and Italy; divisible
+into the following groups in descending order: (1) Pontian; (2) Sarmatian; (3) Tortonian; (4) Helvetian;
+(5) Langhian (Burdigalian).</p></td>
+
+<td class="tcl allb"><p>Represented in the Eastern States by a marine series (Yorktown or Chesapeake, Chipola
+and Chattahoochee groups), and in the interior by the lacustrine Loup Fork (Nebraska), Deep
+River, and John Day groups.</p></td></tr>
+
+<tr><td class="tccm allb">Oligocene.</td>
+
+<td class="tcl allb" colspan="2"><p>In Britain the &ldquo;fluvio-marine series&rdquo; of the Isle of Wight;
+also the volcanic plateaux of Antrim and Inner Hebrides and those of the Faeroe Isles and Iceland. In
+continental Europe the following subdivisions have been established in descending order:
+(1) Aquitanian, (2) Stampian (Rupelian), (3) Tongrain (Sannoisian).</p></td>
+
+<td class="tcl allb"><p>On the Atlantic border no equivalents have been satisfactorily
+recognised, but on the Pacific side there are marine deposits in N. W. Oregon, which
+may represent this division. In the interior the equivalent is believed to be the fresh-water
+White River series, including (1) <i>Protoceras</i> beds, (2) <i>Oreodon</i> beds,
+and (3) <i>Titanothervum</i> beds.</p></td></tr>
+
+<tr><td class="tccm allb">Eocene.</td>
+
+<td class="tcl allb" colspan="2"><p>Barton sands and clays; Ludian series of France.</p>
+<p>Bracklesham Beds; Lutetian (Calcaire grossier and Caillasses) of Paris basin.</p>
+<p>London clay, Woolwich and Reading Beds; Thanet sands; Ypresian or Londinian of N. France and Belgium;
+ Sparnacian and Thanetian groups</p></td>
+
+<td class="tcl allb"><p>Woodstock and Aquia Creek groups of Potomac River; Vicksburg, Jackson,
+ Claiborne, Buhrstone, and Lignitic groups of Mississippi.</p>
+<p>In the interior a thick series of fresh-water formations, comprising, in descending order,
+ the Uinta, Bridger, Wind River, Wasatch, Torrejon, and Puerco groups.</p>
+<p>On the Pacific side the marine Tejon series of Oregon and California.</p></td></tr>
+
+<tr><td class="tccm allb" rowspan="5">Mesozoic or Secondary.</td>
+
+<td class="tccm cl bb">Cretaceous. Upper.</td>
+
+<td class="tcl bb" colspan="2"><p>Danian&mdash;wanting in Britain; uppermost limestone of Denmark.</p>
+<p>Senonian&mdash;Upper Chalk with Flints of England; Aturian and Emscherian stages on the European continent.</p>
+<p>Turonian&mdash;Middle Chalk with few flints, and comprising the Angoumian and Ligerian stages.</p>
+<p>Cenomanian&mdash;Lower Chalk and Chalk Marl.</p>
+<p>Albian&mdash;Upper Greensand and Gault.</p></td>
+
+<td class="tcl allb" rowspan="3"><p>On the Atlantic border both marine strata and others containing a
+ terrestrial flora represent the Cretaceous series of formations.</p>
+<p>In the interior there is also a commingling of marine with lacustrine deposits. At the top lies the
+ Laramie or Lignitic series with an abundant terrestrial flora, passing down into the lacustrine
+ and brackish-water Montana series. Of older date, the Colorado series contains an abundant
+ marine fauna, yet includes also some Niobrara marls and limestones  are likewise of marine
+ origin, but the lower members of the series (Benton and Dakota) show another great representation of
+ fresh-water sedimentation with lignites and coals.</p>
+<p>In California a vast succession of marine deposits (Shasta-Chico) represents the Cretaceous system;
+ and in western British N. America coal-seams also occur.</p></td></tr>
+
+<tr><td class="tccm cl">Cretaceous. Lower.</td>
+
+<td class="tcl" colspan="2"><p>Aptian&mdash;Lower Greensand; Marls and limestones of Provence, &amp;c.</p>
+<p>Urgonian (Barremian)&mdash;Atherfield clay; massive Hippurite limestones of southern France.</p>
+<p>Neocomian&mdash;Weald clay and Hastings sand; Hauterivian and Valanginian sub-stages of
+ Switzerland and France.</p></td></tr>
+
+<tr><td colspan="2">&nbsp;</td></tr>
+
+<tr><td class="tccm allb">Jurassic.</td>
+
+<td class="tcl allb" colspan="2"><p>Purbeckian&mdash;Purbeck beds; Münder Mergel; largely present in Westphalia.</p>
+<p>Portlandian&mdash;Portland group of England, represented in S. France by the thick Tithonian limestones.</p>
+<p>Kimmeridgian&mdash; Kimmeridge Clay of England; Virgulian and Pterocerian groups of
+ N. France; represented by thick limestones in the Mediterranean basin.</p>
+<p>Corallian&mdash;Coral Rag, Coralline Oolite; Sequanian stages of the Continent,
+ comprising the sub-stages of Astartian and Rauracian.</p>
+<p>Oxfordian&mdash;Oxford Clay; Axgovian and Neuvizyan stages.</p>
+<p>Callovian&mdash;Kellaways Rock, Divesian sub-stage of N. France.</p>
+<p>Bathonian&mdash;series of English strata from Cornbrash down to Fuller&rsquo;s Earth.</p>
+<p>Bajocian&mdash;Inferior Oolite of England.</p>
+<p>Lassic&mdash;divisible into (1) Upper Lias or Toarcian, (2) Middle Lias, Marlstone or Charmouthian, (3) Lower
+ Lias of Sinemurian and Hettangian.</p></td>
+
+<td class="tcl allb"><p>Representatives of the Middle and lower Jurassic formations have been found in
+ California and Oregon, and farther north among the Arctic islands.</p>
+<p>Strata containing Lower Jurassic marine fossils appear in Wyoming and Dakota; and above them come
+ the <i>Atlantosaurus</i> and <i>Baptanodon</i> beds, which have yielded so large a
+ variety of deinosaurs and other vertebrates, and especially the remains of a number of genera
+ of small mammals.</p></td></tr>
+
+<tr><td class="tccm allb">Triassic.</td>
+
+<td class="tcl allb" colspan="2"><p>In Germany and western Europe this division represents the deposits of
+ inland seas or lagoons, and is divisible into the following stages in descending
+ order: (1) Rhaetic, (2) Keuper, (3) Muschelkalk, (4) Bunter.  In the eastern Alps and the Mediterranean
+ basin the contemporaneous sedimentary formations are those of open clear sea, in which a thickness of many
+ thousand feet of strata was accumulated.</p></td>
+
+<td class="tcl allb"><p>In New York, Connecticut, New Brunswick, and Nova Scotia a series of red sandstone
+ (Newark series) contains land-plants and labyrinthodonts like the lagoon type of central
+ and western Europe. On the Pacific slope, however, marine equivalents occur, representing
+ the pelagic type of south-eastern Europe.</p></td></tr>
+
+<tr><td class="tccm allb" rowspan="9">Palaeozoic or Primary.</td>
+
+<td class="tccm allb">Permian.</td>
+
+<td class="tcl allb" colspan="2"><p>Thuringian&mdash;Zechstein, Magnesian Limestone; named from its development
+ in Thuringia; well represented also in Saxony, Bavaria and Bohemia.</p>
+<p>Saxonian&mdash;Rothliegendes Group; Red Sandstones, &amp;c.</p>
+<p>Autunian&mdash;where the strata present the lagoon facies, well displayed at Autun
+ in France; where the marine type is predominant, as in Russia, the group has been termed Artinskian.</p></td>
+
+<td class="tcl allb"><p>To this division of the geological record the Upper Barren
+ Measures of the coal-fields of Pennsylvania, Prince Edward Island, Nova Scotia and
+ New Brunswick have been assigned.</p>
+<p>Farther south in Kansas, Texas, and Nebraska the representatives of the division have an
+ abundant marine fauna.</p></td></tr>
+
+<tr><td class="tccm allb">Carboniferous.</td>
+
+<td class="tcl allb" colspan="2"><p>Stephanian or Uralian&mdash;represented in Russia by marine formations, and in
+ central and western Europe by numerous small basins containing a peculiar
+ flora and in some places a great variety of insects.</p>
+<p>Westphalian or Moscovian&mdash;Coal-measures, Millstone Grit.</p>
+<p>Culm or Dinantian&mdash;Carboniferous Limestone and Calciferous Sandstone series.</p></td>
+
+<td class="tcl allb"><p>Upper productive Coal-measures.</p>
+<p>Lower Barren measures.</p>
+<p>Lower productive Coal-measures.</p>
+<p>Pottsville conglomerate.</p>
+<p>Mauch Chunk shales; limestones of Chester, St Louis, &amp;c.</p>
+<p>Pocono series; Kinderhook limestone.</p></td></tr>
+
+<tr><td class="tccm allb" rowspan="4">Devonian and Old Red Sandstone.</td>
+
+<td class="tccm allb">Devonian type.</td>
+
+<td class="tccm allb">Old Red Sandstone type.</td>
+
+<td class="rb">&nbsp;</td></tr>
+
+<tr><td class="tclm rb cl"><p>Upper</p>
+ <p> &emsp;&emsp; Famennian.</p>
+ <p> &emsp;&emsp; Frasnian.</p></td>
+
+<td class="tclm rb cl"><p>Yellow and red sandstone with <i>Holoptychius</i>,
+ <i>Bothriolepis</i>, &amp;c.</p></td>
+
+<td class="tcl rb cl"><p>Catskill red sandstone; Old Red Sandstone type: the strata below show the
+ Devonian type.</p>
+<p>Chemung Group.</p>
+<p>Genesee Group.</p></td></tr>
+
+<tr><td class="tclm rb">Middle<br />
+  &emsp;&emsp; Givetian.<br />
+  &emsp;&emsp; Eifelian.</td>
+
+<td class="tcl rb"><p>Caithness Flagstones with <i>Osteolepus</i>, <i>Dipterus</i>,
+ <i>Homosteus</i>, &amp;c.</p></td>
+
+<td class="tclm rb"><p>Hamilton Group.</p>
+<p>Marcellus Group.</p></td></tr>
+
+<tr><td class="tclm rb cl"><p>Lower</p>
+ <p> &emsp;&emsp; Coblentizian.</p>
+ <p> &emsp;&emsp; Gedinnian.</p></td>
+
+<td class="tclm rb cl"><p>Red and purple sandstones and conglomerates with <i>Cephalaspis</i>,
+ <i>Pteraspis</i>,</p></td>
+
+<td class="tcl rb cl"><p>Corniferous Limestone.</p>
+<p>Onondaga Limestone.</p>
+  <p> &emsp;&emsp; Upper Helderberg Group.</p>
+<p>Oriskany Sandstone.</p></td></tr>
+
+<tr><td class="tccm allb" rowspan="2">Silurian.</td>
+
+<td class="tclm rb tb" colspan="2">Upper<br />
+  &emsp;&emsp; Ludlow  Group.<br />
+  &emsp;&emsp; Wenlock Group.<br />
+  &emsp;&emsp; Llandovery Group.</td>
+
+<td class="tclm rb tb"><p>Lower Helderberg Group.</p>
+<p>Water-Lime.</p>
+<p>Niagara Shale and Limestone.</p>
+<p>Clinton Group.</p>
+<p>Medina Group.</p></td></tr>
+
+<tr><td class="tclm rb cl" colspan="2">Lower (Ordovician)<br />
+  &emsp;&emsp; Ludlow Group.<br />
+  &emsp;&emsp; Wenlock Group.<br />
+  &emsp;&emsp; Llandovery Group.</td>
+
+<td class="tclm rb cl"><p>Cincinnati Group.</p>
+<p>Utica Group.</p>
+<p>Trenton Group.</p>
+<p>Chazy Group.</p>
+<p>Calciferous Group.</p></td></tr>
+
+<tr><td class="tccm allb">Cambrian.</td>
+
+<td class="tcl allb" colspan="2"><p>Upper or <i>Olenus</i> series&mdash;Tremadoc slates and <i>Lingula</i> Flags.</p>
+<p>Middle or <i>Pardoxides</i> series&mdash;Menevian Group.</p>
+<p>Lower or <i>Olenellus</i> series&mdash;Llanberis and Harlech Group, and <i>Olenellus</i>-zone.</p></td>
+
+<td class="tcl allb"><p>Upper or Potsdam series with <i>Olenus</i> and <i>Dicelocephalus</i> fauna.</p>
+<p>Middle or Acadian series with <i>Paradoxides</i> fauna.</p>
+<p>Lower or Georgian series with <i>Olenellus</i> fauna.</p></td></tr>
+
+<tr><td class="tccm allb">Archean, Pre-Cambrian Eozoic.</td>
+
+<td class="tccm allb">&nbsp;</td>
+
+<td class="tcl allb" colspan="2"><p>In Scotland, underneath the Cambrian Olenellus group, lies unconformably
+ a mass of red sandstone and conglomerate (Torridonian) 8000 or 10,000 ft. thick, which rests with a strong
+ gneisses and schists (Lewisian).  A thick series of slates and phyllites lies below the oldest Palaeozoic rocks
+ in central Europe, with coarse gneisses below.</p></td>
+
+<td class="tcl allb"><p>In Canada and the Lake Superior region of the United States a vast succession of
+ rocks of Pre-Cambrian age has been grouped into the following subdivisions in descending order: (1) Keweenwan,
+ lying unconformably on (2) Animikie, separated by a strong unconformability from
+ (3) Upper Huronian, (4) Lower Huronian with an unconformable base, (5) Goutchiching,
+ (6) Laurentian. In the eastern part of Canada, Newfoundland, &amp;c., and also in Montana,
+ sedimentary formations of great thickness below the lowest Cambrian zone have
+ been found to contain some obscure organisms.</p></td></tr>
+
+</table>
+
+<p><span class="pagenum"><a name="page671" id="page671"></a>671</span></p>
+
+<p class="pt2">From these general considerations we proceed to inquire how
+the existing topographical features of the land arose. Obviously
+the co-operation of the two great geological agencies of hypogene
+and epigene energy, which have been at work from the beginning
+of our globe&rsquo;s decipherable history, must have been the cause
+to which these features are to be assigned; and the task of the
+geologist is to ascertain, if possible, the part that has been taken
+by each. There is a natural tendency to see in a stupendous
+piece of scenery, such as a deep ravine, a range of hills, a line of
+precipice or a chain of mountains, evidence only of subterranean
+convulsion; and before the subject was taken up as a matter
+of strict scientific induction, an appeal to former cataclysms
+was considered a sufficient solution of the problems presented
+by such features of landscape. The rise of the modern
+Huttonian school, however, led to a more careful examination
+of these problems. The important share taken by erosion in the
+determination of the present features of landscape was then
+recognized, while a fuller appreciation of the relative parts
+played by the hypogene and epigene causes has gradually been
+reached.</p>
+
+<p>1. The study of the progress of denudation at the present
+time has led to the conclusion that even if the rate of waste
+were not more rapid than it is to-day, it would yet suffice in a
+comparatively brief geological period to reduce the dry land to
+below the sea-level. But not only would the area of the land be
+diminished by denudation, it could hardly fail to be more or
+less involved in those widespread movements of subsidence,
+during which the thick sedimentary formations of the crust
+appear to have been accumulated. It is thus manifest that there
+must have been from time to time during the history of our
+globe upward movements of the crust, whereby the balance
+between land and sea was redressed. Proofs of such movements
+have been abundantly preserved among the stratified formations.
+We there learn that the uplifts have usually followed each other
+at long intervals between which subsidence prevailed, and thus
+that there has been a prolonged oscillation of the crust over the
+great continental areas of the earth&rsquo;s surface.</p>
+
+<p>An examination of that surface leads to the recognition of two
+great types of upheaval. In the one, the sea-floor, with all its
+thick accumulations of sediment, has been carried upwards,
+sometimes for several thousand feet, so equably that the strata
+retain their original flatness with hardly any sensible disturbance
+for hundreds of square miles. In the other type the solid crust
+has been plicated, corrugated and dislocated, especially along
+particular lines, and has attained its most stupendous disruption
+in lofty chains of mountains. Between these two phases of uplift
+many intermediate stages have been developed, according to
+the direction and intensity of the subterranean force and the
+varying nature and disposition of the rocks Of the crust.</p>
+
+<p>(<i>a</i>) Where the uplift has extended over wide spaces, without
+appreciable deformation of the crust, the flat strata have given
+rise to low plains, or if the amount of uprise has been great
+enough, to high plains, plateaux or tablelands. The plains of
+Russia, for example, lie for the most part on such tracts of
+equably uplifted strata. The great plains of the western interior
+of the United States form a great plateau or tableland, 5000 or
+6000 ft. above the sea, and many thousands of square miles in
+extent, on which the Rocky Mountains have been ridged up.</p>
+
+<p>(<i>b</i>) It is in a great mountain-chain that the complicated
+structures developed during disturbances of the earth&rsquo;s crust
+can best be studied (see Parts IV. and V. of this article), and
+where the influence of these structures on the topography of the
+surface is most effectively displayed. Such a chain may be the
+result of one colossal disturbance; but those of high geological
+antiquity usually furnish proofs of successive uplifts with more
+or less intervening denudation. Formed along lines of continental
+displacement in the crust, they have again and again given
+relief from the strain of compression by fresh crumpling, fracture
+and uprise. The chief guide in tracing these successive stages
+of growth is supplied by unconformability. If, for example, a
+mountain-range consists of upraised Silurian rocks, upon the
+upturned and denuded edges of which the Carboniferous Limestone
+lies transgressively, it is clear that its original upheaval
+must have taken place in the period of geological time represented
+by the interval between the Silurian and the Carboniferous
+Limestone formations. If, as the range is followed along its
+course, the Carboniferous Limestone is found to be also highly
+inclined and covered unconformably by the Upper Coal-measures,
+a second uplift of that portion of the ground can be proved to
+have taken place between the time of the Limestone and that of
+the Upper Coal-measures. By this simple and obvious kind of
+evidence the relative ages of different mountain-chains may
+be compared. In most great chains, however, the rocks have
+been so intensely crumpled, and even inverted, that much
+labour may be required before their true relations can be determined.</p>
+
+<p>The Alps furnish an instructive example of the long series of
+revolutions through which a great mountain-system may have
+passed before reaching its present development. The first
+beginnings of the chain may have been upraised before the
+oldest Palaeozoic formations were laid down. There are at
+least traces of land and shore-lines in the Carboniferous period.
+Subsequent submergences and uplifts appear to have occurred
+during the Mesozoic periods. There is evidence that thereafter
+the whole region sank deep under the sea, in which the older
+Tertiary sediments were accumulated, and which seems to
+have spread right across the heart of the Old World. But after
+the deposition of the Eocene formations came the gigantic
+disruptions whereby all the rocks of the Alpine region were
+folded over each other, crushed, corrugated, fractured and
+displaced, some of their older portions, including the fundamental
+gneisses and schists, being squeezed up, torn off, and pushed
+horizontally for many miles over the younger rocks. But this
+upheaval, though the most momentous, was not the last which
+the chain has undergone, for at a later epoch in Tertiary time
+renewed disturbance gave rise to a further series of ruptures
+and plications. The chain thus successively upheaved has
+been continuously exposed to denudation and has consequently
+lost much of its original height. That it has been left in a state
+of instability is indicated by the frequent earthquakes of the
+Alpine region, which doubtless arise from the sudden snapping
+of rocks under intense strain.</p>
+
+<p>A distinct type of mountain due to direct hypogene action is
+to be seen in a volcano. It has been already pointed out (Part IV.
+sect. 1) that at the vents which maintain a communication
+between the molten magma of the earth&rsquo;s interior and the
+surface, eruptions take place whereby quantities of lava and
+fragmentary materials are heaped round each orifice of
+discharge. A typical volcanic mountain takes the form of a
+perfect cone, but as it grows in size and its main vent is choked,
+while the sides of the cone are unable to withstand the force of
+the explosions or the pressure of the ascending column of lava,
+eruptions take place laterally, and numerous parasitic cones
+arise on the flanks of the parent mountain. Where lava flows
+out from long fissures, it may pile up vast sheets of rock, and
+bury the surrounding country under several thousand feet of
+solid stone, covering many hundreds of square miles. In this
+way volcanic tablelands have been formed which, attacked by
+the denuding forces, are gradually trenched by valleys and
+ravines, until the original level surface of the lava-field may be
+almost or wholly lost. As striking examples of this physiographical
+type reference may be made to the plateau of Abyssinia,
+the Ghats of India, the plateaux of Antrim, the Inner Hebrides
+and Iceland, and the great lava-plains of the western territories
+of the United States.</p>
+
+<p>2. But while the subterranean movements have upraised
+portions of the surface of the lithosphere above the level of the
+ocean, and have thus been instrumental in producing the existing
+tracts of land, the detailed topographical features of a landscape
+<span class="pagenum"><a name="page672" id="page672"></a>672</span>
+are not solely, nor in general even chiefly, attributable to these
+movements. From the time that any portion of the sea-floor
+appears above sea-level, it undergoes erosion by the various
+epigene agents. Each climate and geological region has its own
+development of these agents, which include air, aridity, rapid and
+frequent alternations of wetness and dryness or of heat and
+cold, rain, springs, frosts, rivers, glaciers, the sea, plant and
+animal life. In a dry climate subject to great extremes of
+temperature the character and rate of decay will differ from
+those of a moist or an arctic climate. But it must be remembered
+that, however much they may vary in activity and in the results
+which they effect, the epigene forces work without intermission,
+while the hypogene forces bring about the upheaval of land only
+after long intervals. Hence, trifling as the results during a
+human life may appear, if we realize the multiplying influence
+of time we are led to perceive that the apparently feeble superficial
+agents can, in the course of ages, achieve stupendous
+transformations in the aspect of the land. If this efficacy may
+be deduced from what can be seen to be in progress now, it
+may not less convincingly be shown, from the nature of the
+sedimentary rocks of the earth&rsquo;s crust, to have been in progress
+from the early beginnings of geological history. Side by side
+with the various upheavals and subsidences, there has been a
+continuous removal of materials from the land, and an equally
+persistent deposit of these materials under water, with the
+consequent growth of new rocks. Denudation has been aptly
+compared to a process of sculpturing wherein, while each of the
+implements employed by nature, like a special kind of graving
+tool, produces its own characteristic impress on the land, they
+all combine harmoniously towards the achievement of their
+one common task. Hence the present contours of the land
+depend partly on the original configuration of the ground, and
+the influence it may have had in guiding the operations of the
+erosive agents, partly on the vigour with which these agents
+perform their work, and partly on the varying structure and
+powers of resistance possessed by the rocks on which the erosion
+is carried on.</p>
+
+<p>Where a new tract of land has been raised out of the sea
+by such an energetic movement as broke up the crust and
+produced the complicated structure and tumultuous external
+forms of a great mountain chain, the influence of the hypogene
+forces on the topography attains its highest development.
+But even the youngest existing chain has suffered so greatly
+from denudation that the aspect which it presented at the time
+of its uplift can only be dimly perceived. No more striking
+illustration of this feature can be found than that supplied by
+the Alps, nor one where the geotectonic structures have been
+so fully studied in detail. On the outer flanks of these mountains
+the longitudinal ridges and valleys of the Jura correspond with
+lines of anticline and syncline. Yet though the dominant
+topographical elements of the region have obviously been
+produced by the plication of the stratified formations, each
+ridge has suffered so large an amount of erosion that the younger
+rocks have been removed from its crest where the older members
+of the series are now exposed to view, while on every slope
+proofs may be seen of extensive denudation. If from these
+long wave-like undulations of the ground, where the relations
+between the disposition of the rocks below and the forms of
+the surface are so clearly traceable, the observer proceeds
+inwards to the main chain, he finds that the plications and
+displacements of the various formations assume an increasingly
+complicated character; and that although proofs of great
+denudation continue to abound, it becomes increasingly difficult
+to form any satisfactory conjecture as to the shape of the ground
+when the upheaval ended or any reliable estimate of the amount
+of material which has since then been removed. Along the
+central heights the mountains lift themselves towards the sky
+like the storm-swept crests of vast earth-billows. The whole
+aspect of the ground suggests intense commotion, and the
+impression thus given is often much intensified by the twisted
+and crumpled strata, visible from a long distance, on the crags
+and crests. On this broken-up surface the various agents of
+denudation have been ceaselessly engaged since it emerged
+from the sea. They have excavated valleys, sometimes along
+depressions provided for them by the subterranean disturbances,
+sometimes down the slopes of the disrupted blocks of ground.
+So powerful has been this erosion that valleys cut out along
+lines of anticline, which were natural ridges, have sometimes
+become more important than those in lines of syncline, which
+were structurally depressions. The same subaerial forces have
+eroded lake-basins, dug out corries or cirques, notched the
+ridges, splintered the crests and furrowed the slopes, leaving
+no part of the original surface of the uplifted chain
+unmodified.</p>
+
+<p>It has often been noted with surprise that features of
+underground structure which, it might have been confidently
+anticipated, should have exercised a marked influence on the
+topography of the surface have not been able to resist the
+levelling action of the denuding agents, and do not now affect
+the surface at all. This result is conspicuously seen in coal-fields
+where the strata are abundantly traversed by faults. These
+dislocations, having sometimes a displacement of several hundred
+feet, might have been expected to break up the surface into
+a network of cliffs and plains; yet in general they do not modify
+the level character of the ground above. One of the most
+remarkable faults in Europe is the great thrust which bounds
+the southern edge of the Belgian coal-field and brings the
+Devonian rocks above the Coal-measures. It can be traced
+across Belgium into the Boulonnais, and may not improbably
+run beneath the Secondary and Tertiary rocks of the south of
+England. It is crossed by the valleys of the Meuse and other
+northerly-flowing streams. Yet so indistinctly is it marked
+in the Meuse valley that no one would suspect its existence from
+any peculiarity in the general form of the ground, and even an
+experienced geologist, until he had learned the structure of the
+district, would scarcely detect any fault at all.</p>
+
+<p>Where faults have influenced the superficial topography,
+it is usually by giving rise to a hollow along which the subaerial
+agents and especially running water can act effectively. Such
+a hollow may be eventually widened and deepened into a valley.
+On bare crags and crests, lines of fault are apt to be marked by
+notches or clefts, and they thus help to produce the pinnacles
+and serrated outlines of these exposed uplands.</p>
+
+<p>It was cogently enforced by Hutton and Playfair, and independently
+by Lamarck, that no co-operation of underground
+agency is needed to produce such topography as may be seen
+in a great part of the world, but that if a tract of sea-floor were
+upraised into a wide plain, the fall of rain and the circulation
+of water over its surface would in the end carve out such a system
+of hills and valleys as may be seen on the dry land now. No
+such plain would be a dead-level. It would have inequalities
+on its surface which would serve as channels to guide the drainage
+from the first showers of rain. And these channels would be
+slowly widened and deepened until they would become ravines
+and valleys, while the ground between them would be left projecting
+as ridges and hills. Nor would the erosion of such a system
+of water-courses require a long series of geological periods for
+its accomplishment. From measurements and estimates of the
+amount of erosion now taking place in the basin of the Mississippi
+river it has been computed that valleys 800 ft. deep might be
+carved out in less than a million years. In the vast tablelands
+of Colorado and other western regions of the United States an
+impressive picture is presented of the results of mere subaerial
+erosion on undisturbed and nearly level strata. Systems of
+stream-courses and valleys, river gorges unexampled elsewhere
+in the world for depth and length, vast winding lines of escarpment,
+like ranges of sea-cliffs, terraced slopes rising from plateau
+to plateau, huge buttresses and solitary stacks standing like
+islands out of the plains, great mountain-masses towering into
+picturesque peaks and pinnacles cleft by innumerable gullies,
+yet everywhere marked by the parallel bars of the horizontal
+strata out of which they have been carved&mdash;these are the orderly
+symmetrical characteristics of a country where the scenery is
+due entirely to the action of subaerial agents on the one hand and
+<span class="pagenum"><a name="page673" id="page673"></a>673</span>
+the varying resistance of perfectly regular stratified rocks on the
+other.</p>
+
+<p>The details of the sculpture of the land have mainly depended
+on the nature of the materials on which nature&rsquo;s erosive tools
+have been employed. The joints by which all rocks are traversed
+have been especially serviceable as dominant lines down which
+the rain has filtered, up which the springs have risen and into
+which the frost wedges have been driven. On the high bare
+scarps of a lofty mountain the inner structure of the mass is laid
+open, and there the system of joints even more than faults is
+seen to have determined the lines of crest, the vertical walls of
+cliff and precipice, the forms of buttress and recess, the position
+of cleft and chasm, the outline of spire and pinnacle. On the
+lower slopes, even under the tapestry of verdure which nature
+delights to hang where she can over her naked rocks, we may
+detect the same pervading influence of the joints upon the forms
+assumed by ravines and crags. Each kind of stone, too, gives
+rise to its own characteristic form of scenery. Massive crystalline
+rocks, such as granite, break up along their joints and often
+decay into sand or earth along their exposed surfaces, giving
+rise to rugged crags with long talus slopes at their base. The
+stratified rocks besides splitting at their joints are especially
+distinguished by parallel ledges, cornices and recesses, produced
+by the irregular decay of their component strata, so that they
+often assume curiously architectural types of scenery. But
+besides this family feature they display many minor varieties of
+aspect according to their lithological composition. A range of
+sandstone hills, for example, presents a marked contrast to one
+of limestone, and a line of chalk downs to the escarpments
+formed by alternating bands of harder and softer clays and
+shales.</p>
+
+<p>It may suffice here merely to allude to a few of the more
+important parts of the topography of the land in their relation
+to physiographical geology. A true mountain-chain, viewed
+from the geological side, is a mass of high ground which owes its
+prominence to a ridging-up of the earth&rsquo;s crust, and the intense
+plication and rupture of the rocks of which it is composed. But
+ranges of hills almost mountainous in their bulk may be formed
+by the gradual erosion of valleys out of a mass of original high
+ground, such as a high plateau or tableland. Eminences which
+have been isolated by denudation from the main mass of the
+formations of which they originally formed part are known as
+&ldquo;outliers&rdquo; or &ldquo;hills of circumdenudation.&rdquo;</p>
+
+<p>Tablelands, as already pointed out, may be produced either
+by the upheaval of tracts of horizontal strata from the sea-floor
+into land; or by the uprise of plains of denudation, where rocks
+of various composition, structure and age have been levelled
+down to near or below the level of the sea by the co-operation
+of the various erosive agents. Most of the great tablelands
+of the globe are platforms of little-disturbed strata which have
+been upraised bodily to a considerable elevation. No sooner,
+however, are they placed in that position than they are attacked
+by running water, and begin to be hollowed out into systems of
+valleys. As the valleys sink, the platforms between them grow
+into narrower and more definite ridges, until eventually the
+level tableland is converted into a complicated network of hills
+and valleys, wherein, nevertheless, the key to the whole arrangement
+is furnished by a knowledge of the disposition and effects
+of the flow of water. The examples of this process brought to
+light in Colorado, Wyoming, Nevada and the other western
+regions by Newberry, King, Hayden, Powell and other explorers,
+are among the most striking monuments of geological operations
+in the world.</p>
+
+<p>Examples of ancient and much decayed tablelands formed by
+the denudation of much disturbed rocks are furnished by the
+Highlands of Scotland and of Norway. Each of these tracts of
+high ground consists of some of the oldest and most dislocated
+formations of Europe, which at a remote period were worn down
+into a plain, and in that condition may have lain long submerged
+under the sea and may possibly have been overspread there
+with younger formations. Having at a much later time been
+raised several thousand feet above sea-level the ancient platforms
+of Britain and Scandinavia have been since exposed to denudation,
+whereby each of them has been so deeply channeled into
+glens and fjords that it presents to-day a surface of rugged
+hills, either isolated or connected along the flanks, while only
+fragments of the general surface of the tableland can here and
+there be recognized amidst the general destruction.</p>
+
+<p>Valleys have in general been hollowed out by the greater
+erosive action of running water along the channels of drainage.
+Their direction has been probably determined in the great
+majority of cases by irregularities of the surface along which
+the drainage flowed on the first emergence of the land. Sometimes
+these irregularities have been produced by folds of the
+terrestrial crust, sometimes by faults, sometimes by the irregularities
+on the surface of an uplifted platform of deposition or of
+denudation. Two dominant trends may be observed among
+them. Some are longitudinal and run along the line of flexures
+in the upraised tract of land, others are transverse where the
+drainage has flowed down the slopes of the ridges into the longitudinal
+valleys or into the sea. The forms of valleys have been
+governed partly by the structure and composition of the rocks,
+and partly by the relative potency of the different denuding
+agents. Where the influence of rain and frost has been slight,
+and the streams, supplied from distant sources, have had
+sufficient declivity, deep, narrow, precipitous ravines or gorges
+have been excavated. The canyons of the arid region of the
+Colorado are a magnificent example of this result. Where, on
+the other hand, ordinary atmospheric action has been more
+rapid, the sides of the river channels have been attacked, and
+open sloping glens and valleys have been hollowed out. A
+gorge or defile is usually due to the action of a waterfall, which,
+beginning with some abrupt declivity or precipice in the course
+of the river when it first commenced to flow, or caused by some
+hard rock crossing the channel, has eaten its way backward.</p>
+
+<p>Lakes have been already referred to, and their modes of origin
+have been mentioned. As they are continually being filled up
+with the detritus washed into them from the surrounding
+regions they cannot be of any great geological antiquity, unless
+where by some unknown process their basins are from time to
+time widened and deepened.</p>
+
+<p>In the general subaerial denudation of a country, innumerable
+minor features are worked out as the structure of the rocks
+controls the operations of the eroding agents. Thus, among
+comparatively undisturbed strata, a hard bed resting upon
+others of a softer kind is apt to form along its outcrop a line of
+cliff or escarpment. Though a long range of such cliffs resembles
+a coast that has been worn by the sea, it may be entirely due to
+mere atmospheric waste. Again, the more resisting portions of
+a rock may be seen projecting as crags or knolls. An igneous
+mass will stand out as a bold hill from amidst the more decomposable
+strata through which it has risen. These features,
+often so marked on the lower grounds, attain their most conspicuous
+development among the higher and barer parts of the
+mountains, where subaerial disintegration is most rapid. The
+torrents tear out deep gullies from the sides of the declivities.
+Corries or cirques are scooped out on the one hand and naked
+precipices are left on the other. The harder bands of rock
+project as massive ribs down the slopes, shoot up into prominent
+<i>aiguilles</i>, or help to give to the summits the notched saw-like
+outlines they so often present.</p>
+
+<p>The materials worn from the surface of the higher are spread
+out over the lower grounds. The streams as they descend begin
+to drop their freight of sediment when, by the lessening of their
+declivity, their carrying power is diminished. The great plains
+of the earth&rsquo;s surface are due to this deposit of gravel, sand and
+loam. They are thus monuments at once of the destructive and
+reproductive processes which have been in progress unceasingly
+since the first land rose above the sea and the first shower of rain
+fell. Every pebble and particle of their soil, once part of the
+distant mountains, has travelled slowly and fitfully to lower
+levels. Again and again have these materials been shifted,
+ever moving downward and sea-ward. For centuries, perhaps,
+they have taken their share in the fertility of the plains and
+<span class="pagenum"><a name="page674" id="page674"></a>674</span>
+have ministered to the nurture of flower and tree, of the bird of
+the air, the beast of the field and of man himself. But their
+destiny is still the great ocean. In that bourne alone can they
+find undisturbed repose, and there, slowly accumulating in
+massive beds, they will remain until, in the course of ages,
+renewed upheaval shall raise them into future land, there once
+more to pass through the same cycle of change.</p>
+<div class="author">(A. Ge.)</div>
+
+<div class="condensed">
+<p><span class="sc">Literature.</span>&mdash;<i>Historical</i>: The standard work is Karl A. von
+Zittel&rsquo;s <i>Geschichte der Geologie und Paläontologie</i> (1899), of which
+there is an abbreviated, but still valuable, English translation;
+D&rsquo;Archiac, <i>Histoire des progrčs de la géologie</i>, deals especially with
+the period 1834-1850; Keferstein, <i>Geschichte und Literatur der
+Geognosie</i>, gives a summary up to 1840; while Sir A. Geikie&rsquo;s
+<i>Founders of Geology</i> (1897; 2nd ed., 1906) deals more particularly
+with the period 1750-1820. General treatises: Sir Charles Lyell&rsquo;s
+<i>Principles of Geology</i> is a classic. Of modern English works, Sir A.
+Geikie&rsquo;s <i>Text Book of Geology</i> (4th ed., 1903) occupies the first place;
+the work of T.C. Chamberlin and R.D. Salisbury, <i>Geology</i>; <i>Earth
+History</i> (3 vols., 1905-1906), is especially valuable for American
+geology. A. de Lapparent&rsquo;s <i>Traité de géologie</i> (5th ed., 1906), is the
+standard French work. H. Credner&rsquo;s <i>Elemente der Geologie</i> has gone
+through several editions in Germany. Dynamical and physiographical
+geology are elaborately treated by E. Suess, <i>Das Antlitz
+der Erde</i>, translated into English, with the title <i>The Face of the Earth</i>.
+The practical study of the science is treated of by F. von Richthofen,
+<i>Führer für Forschungsreisende</i> (1886); G.A. Cole, <i>Aids in Practical
+Geology</i> (5th ed., 1906); A. Geikie, <i>Outlines of Field Geology</i> (5th ed.,
+1900). The practical applications of Geology are discussed by
+J.V. Elsden, <i>Applied Geology</i> (1898-1899). The relations of Geology
+to scenery are dealt with by Sir A. Geikie, <i>Scenery of Scotland</i> (3rd ed.,
+1901); J.E. Marr, <i>The Scientific Study of Scenery</i> (1900); Lord
+Avebury, <i>The Scenery of Switzerland</i> (1896); <i>The Scenery of England</i>
+(1902); and J. Geikie, <i>Earth Sculpture</i> (1898). A detailed bibliography
+is given in Sir A. Geikie&rsquo;s <i>Text Book of Geology</i>. See also
+the separate articles on geological subjects for special references to
+authorities.</p>
+</div>
+
+<hr class="foot" /> <div class="note">
+
+<p><a name="ft1c" id="ft1c" href="#fa1c"><span class="fn">1</span></a> In De Luc&rsquo;s <i>Lettres physiques et morales sur les montagnes</i> (1778),
+the word &ldquo;cosmology&rdquo; is used for our science, the author stating
+that &ldquo;geology&rdquo; is more appropriate, but it &ldquo;was not a word in use.&rdquo;
+In a completed edition, published in 1779, the same statement is
+made, but &ldquo;geology&rdquo; occurs in the text; in the same year De
+Saussure used the word without any explanation, as if it were
+well known.</p>
+
+<p><a name="ft2c" id="ft2c" href="#fa2c"><span class="fn">2</span></a> The subject of the age of the earth has also been discussed by
+Professor J. Joly and Professor W.J. Sollas. The former geologist,
+approaching the question from a novel point of view, has estimated
+the total quantity of sodium in the water of the ocean and the
+quantity of that element received annually by the ocean from the
+denudation of the land. Dividing the one sum by the other, he
+arrives at the result that the probable age of the earth is between
+90 and 100 millions of years (<i>Trans. Roy. Dublin Soc.</i> ser. ii. vol. vii.,
+1899, p. 23: <i>Geol. Mag.</i>, 1900, p. 220). Professor Sollas believes
+that this limit exceeds what is required for the evolution of geological
+history, that the lower limit assigned by Lord Kelvin falls short of
+what the facts demand, and that geological time will probably be
+found to have been comprised within some indeterminate period
+between these limits. (Address to Section C, <i>Brit. Assoc. Report</i>,
+1900; <i>Age of the Earth</i>, London, 1905.)</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">GEOMETRICAL CONTINUITY.<a name="ar19" id="ar19"></a></span> In a report of the Institute
+prefixed to Jean Victor Poncelet&rsquo;s <i>Traité des propriétés projectives
+des figures</i> (Paris, 1822), it is said that he employed &ldquo;ce
+qu&rsquo;il appelle le principe de continuité.&rdquo; The law or principle
+thus named by him had, he tells us, been tacitly assumed as
+axiomatic by &ldquo;les plus savans géomčtres.&rdquo; It had in fact been
+enunciated as &ldquo;lex continuationis,&rdquo; and &ldquo;la loi de la continuité,&rdquo;
+by Gottfried Wilhelm Leibnitz (Oxf. N.E.D.), and previously
+under another name by Johann Kepler in cap. iv. 4 of his <i>Ad
+Vitellionem paralipomena quibus astronomiae pars optica traditur</i>
+(Francofurti, 1604). Of sections of the cone, he says, there are
+five species from the &ldquo;recta linea&rdquo; or line-pair to the circle.
+From the line-pair we pass through an infinity of hyperbolas to
+the parabola, and thence through an infinity of ellipses to the
+circle. Related to the sections are certain remarkable points
+which have no name. Kepler calls them foci. The circle has
+one focus at the centre, an ellipse or hyperbola two foci equidistant
+from the centre. The parabola has one focus within it,
+and another, the &ldquo;caecus focus,&rdquo; which may be imagined to be
+<i>at infinity</i> on the axis <i>within or without the curve</i>. The line from it
+to any point of the section is parallel to the axis. To carry out
+the analogy we must speak paradoxically, and say that the line-pair
+likewise has foci, which in this case coalesce as in the circle
+and fall upon the lines themselves; for our geometrical terms
+should be subject to analogy. Kepler dearly loves analogies, his
+most trusty teachers, acquainted with all the secrets of nature,
+&ldquo;<i>omnium naturae arcanorum conscios</i>.&rdquo; And they are to be
+especially regarded in geometry as, by the use of &ldquo;however
+absurd expressions,&rdquo; classing extreme limiting forms with an
+infinity of intermediate cases, and placing the whole essence of a
+thing clearly before the eyes.</p>
+
+<p>Here, then, we find formulated by Kepler the doctrine of the
+concurrence of parallels at a single point at infinity and the
+principle of continuity (under the name analogy) in relation to the
+infinitely great. Such conceptions so strikingly propounded in
+a famous work could not escape the notice of contemporary
+mathematicians. Henry Briggs, in a letter to Kepler from
+Merton College, Oxford, dated &ldquo;10 Cal. Martiis 1625,&rdquo; suggests
+improvements in the <i>Ad Vitellionem paralipomena</i>, and gives
+the following construction: Draw a line CBADC, and let an
+ellipse, a parabola, and a hyperbola have B and A for focus and
+vertex. Let CC be the other foci of the ellipse and the hyperbola.
+Make AD equal to AB, and with centres CC and radius in each
+case equal to CD describe circles. Then any point of the ellipse
+is equidistant from the focus B and one circle, and any point of
+the hyperbola from the focus B and the other circle. Any point
+P of the parabola, in which the second focus is missing or infinitely
+distant, is equidistant from the focus B and the line
+through D which we call the directrix, this taking the place of
+either circle when its centre C is at infinity, and every line CP
+being then parallel to the axis. Thus Briggs, and we know not
+how many &ldquo;savans géomčtres&rdquo; who have left no record, had
+already taken up the new doctrine in geometry in its author&rsquo;s
+lifetime. Six years after Kepler&rsquo;s death in 1630 Girard Desargues,
+&ldquo;the Monge of his age,&rdquo; brought out the first of his remarkable
+works founded on the same principles, a short tract entitled
+<i>Méthode universelle de mettre en perspective les objets donnés
+réellement ou en devis</i> (Paris, 1636); but &ldquo;Le privilége étoit de
+1630.&rdquo; (Poudra, <i>&OElig;uvres de Des.</i>, i. 55). Kepler as a modern
+geometer is best known by his <i>New Stereometry of Wine Casks</i>
+(Lincii, 1615), in which he replaces the circuitous Archimedean
+method of exhaustion by a direct &ldquo;royal road&rdquo; of infinitesimals,
+treating a vanishing arc as a straight line and regarding a curve
+as made up of a succession of short chords. Some 2000 years
+previously one Antipho, probably the well-known opponent of
+Socrates, has regarded a circle in like manner as the limiting
+form of a many-sided inscribed rectilinear figure. Antipho&rsquo;s
+notion was rejected by the men of his day as unsound, and when
+reproduced by Kepler it was again stoutly opposed as incapable
+of any sort of geometrical demonstration&mdash;not altogether without
+reason, for it rested on an assumed law of continuity rather
+than on palpable proof.</p>
+
+<p>To complete the theory of continuity, the one thing needful
+was the idea of imaginary points implied in the algebraical
+geometry of René Descartes, in which equations between variables
+representing co-ordinates were found often to have imaginary
+roots. Newton, in his two sections on &ldquo;Inventio orbium&rdquo;
+(<i>Principia</i> i. 4, 5), shows in his brief way that he is familiar with
+the principles of modern geometry. In two propositions he uses
+an auxiliary line which is supposed to cut the conic in X and Y,
+but, as he remarks at the end of the second (prop. 24), it may not
+cut it at all. For the sake of brevity he passes on at once with the
+observation that the required constructions are evident from the
+case in which the line cuts the trajectory. In the scholium
+appended to prop. 27, after saying that an asymptote is a tangent
+at infinity, he gives an unexplained general construction for the
+axes of a conic, which seems to imply that it has asymptotes.
+In all such cases, having equations to his loci in the background,
+he may have thought of elements of the figure as passing into the
+imaginary state in such manner as not to vitiate conclusions
+arrived at on the hypothesis of their reality.</p>
+
+<p>Roger Joseph Boscovich, a careful student of Newton&rsquo;s works,
+has a full and thorough discussion of geometrical continuity in
+the third and last volume of his <i>Elementa universae matheseos</i>
+(ed. prim. Venet, 1757), which contains <i>Sectionum conicarum
+elementa nova quadam methodo concinnata et dissertationem de
+transformatione locorum geometricorum, ubi de continuitatis
+lege, et de quibusdam infiniti mysteriis</i>. His first principle is
+that all varieties of a defined locus have the same properties, so
+that what is demonstrable of one should be demonstrable in like
+manner of all, although some artifice may be required to bring
+out the underlying analogy between them. The opposite
+extremities of an infinite straight line, he says, are to be regarded
+as joined, as if the line were a circle having its centre at the
+infinity on either side of it. This leads up to the idea of a <i>veluti
+plus quam infinita extensio</i>, a line-circle containing, as we say,
+the line infinity. Change from the real to the imaginary state is
+contingent upon the passage of some element of a figure through
+zero or infinity and never takes place <i>per saltum</i>. Lines being
+some positive and some negative, there must be negative rectangles
+and negative squares, such as those of the exterior
+diameters of a hyperbola. Boscovich&rsquo;s first principle was that
+of Kepler, by whose <i>quantumvis absurdis locutionibus</i> the boldest
+<span class="pagenum"><a name="page675" id="page675"></a>675</span>
+applications of it are covered, as when we say with Poncelet
+that all concentric circles in a plane touch one another in two
+imaginary fixed points at infinity. In G.K. Ch. von Staudt&rsquo;s
+<i>Geometrie der Lage and Beiträge zur G. der L.</i> (Nürnberg, 1847,
+1856-1860) the geometry of position, including the extension of
+the field of pure geometry to the infinite and the imaginary, is
+presented as an independent science, &ldquo;welche des Messens nicht
+bedarf.&rdquo; (See <span class="sc"><a href="#artlinks">Geometry</a></span>: <i>Projective</i>.)</p>
+
+<p>Ocular illusions due to distance, such as Roger Bacon notices
+in the <i>Opus majus</i> (i. 126, ii. 108, 497; Oxford, 1897), lead up to
+or illustrate the mathematical uses of the infinite and its reciprocal
+the infinitesimal. Specious objections can, of course, be
+made to the anomalies of the law of continuity, but they are
+inherent in the higher geometry, which has taught us so much
+of the &ldquo;secrets of nature.&rdquo; Kepler&rsquo;s excursus on the &ldquo;analogy&rdquo;
+between the conic sections hereinbefore referred to is given at
+length in an article on &ldquo;The Geometry of Kepler and Newton&rdquo;
+in vol. xviii. of the <i>Transactions of the Cambridge Philosophical
+Society</i> (1900). It had been generally overlooked, until attention
+was called to it by the present writer in a note read in 1880 (<i>Proc.
+C.P.S.</i> iv. 14-17), and shortly afterwards in <i>The Ancient and
+Modern Geometry of Conics, with Historical Notes and Prolegomena</i>
+(Cambridge 1881).</p>
+<div class="author">(C. T.*)</div>
+
+
+<hr class="art" />
+<p><span class="bold">GEOMETRY,<a name="ar20" id="ar20"></a></span> the general term for the branch of mathematics
+which has for its province the study of the properties of
+space. From experience, or possibly intuitively, we characterize
+existent space by certain fundamental qualities, termed axioms,
+which are insusceptible of proof; and these axioms, in conjunction
+with the mathematical entities of the point, straight line,
+curve, surface and solid, appropriately defined, are the premises
+from which the geometer draws conclusions. The geometrical
+axioms are merely conventions; on the one hand, the system
+may be based upon inductions from experience, in which case
+the deduced geometry may be regarded as a branch of physical
+science; or, on the other hand, the system may be formed by
+purely logical methods, in which case the geometry is a phase
+of pure mathematics. Obviously the geometry with which we
+are most familiar is that of existent space&mdash;the three-dimensional
+space of experience; this geometry may be termed Euclidean,
+after its most famous expositor. But other geometries exist,
+for it is possible to frame systems of axioms which definitely
+characterize some other kind of space, and from these axioms
+to deduce a series of non-contradictory propositions; such
+geometries are called non-Euclidean.</p>
+
+<p>It is convenient to discuss the subject-matter of geometry
+under the following headings:</p>
+
+<p>I. <i>Euclidean Geometry</i>: a discussion of the axioms of existent
+space and of the geometrical entities, followed by a synoptical
+account of Euclid&rsquo;s Elements.</p>
+
+<p>II. <i>Projective Geometry</i>: primarily Euclidean, but differing
+from I. in employing the notion of geometrical continuity (<i>q.v.</i>)&mdash;points
+and lines at infinity.</p>
+
+<p>III. <i>Descriptive Geometry</i>: the methods for representing upon
+planes figures placed in space of three dimensions.</p>
+
+<p>IV. <i>Analytical Geometry</i>: the representation of geometrical
+figures and their relations by algebraic equations.</p>
+
+<p>V. <i>Line Geometry</i>: an analytical treatment of the line regarded
+as the space element.</p>
+
+<p>VI. <i>Non-Euclidean Geometry</i>: a discussion of geometries
+other than that of the space of experience.</p>
+
+<p>VII. <i>Axioms of Geometry</i>: a critical analysis of the foundations
+of geometry.</p>
+
+<div class="condensed">
+<p>Special subjects are treated under their own headings: <i>e.g.</i>
+<span class="sc"><a href="#artlinks">Projection</a></span>, <span class="sc"><a href="#artlinks">Perspective</a></span>; <span class="sc"><a href="#artlinks">Curve</a></span>, <span class="sc"><a href="#artlinks">Surface</a></span>; <span class="sc"><a href="#artlinks">Circle</a></span>, <span class="sc"><a href="#artlinks">Conic
+Section</a></span>; <span class="sc"><a href="#artlinks">Triangle</a></span>, <span class="sc"><a href="#artlinks">Polygon</a></span>, <span class="sc"><a href="#artlinks">Polyhedron</a></span>; there are also
+articles on special curves and figures, <i>e.g.</i> <span class="sc"><a href="#artlinks">Ellipse</a></span>, <span class="sc"><a href="#artlinks">Parabola</a></span>,
+<span class="sc"><a href="#artlinks">Hyperbola</a></span>; <span class="sc"><a href="#artlinks">Tetrahedron</a></span>, <span class="sc"><a href="#artlinks">Cube</a></span>, <span class="sc"><a href="#artlinks">Octahedron</a></span>, <span class="sc"><a href="#artlinks">Dodecahedron</a></span>,
+<span class="sc"><a href="#artlinks">Icosahedron</a></span>; <span class="sc"><a href="#artlinks">Cardioid</a></span>, <span class="sc"><a href="#artlinks">Catenary</a></span>, <span class="sc"><a href="#artlinks">Cissoid</a></span>, <span class="sc"><a href="#artlinks">Conchoid</a></span>, <span class="sc"><a href="#artlinks">Cycloid</a></span>,
+<span class="sc"><a href="#artlinks">Epicycloid</a></span>, <span class="sc"><a href="#artlinks">Limaēon</a></span>, <span class="sc"><a href="#artlinks">Oval</a></span>, <span class="sc"><a href="#artlinks">Quadratrix</a></span>, <span class="sc"><a href="#artlinks">Spiral</a></span>, &amp;c.</p>
+</div>
+
+<p><i>History.</i>&mdash;The origin of geometry (Gr. <span class="grk" title="gź">&#947;&#8134;</span>, earth, <span class="grk" title="metron">&#956;&#941;&#964;&#961;&#959;&#957;</span>, a
+measure) is, according to Herodotus, to be found in the etymology
+of the word. Its birthplace was Egypt, and it arose from the
+need of surveying the lands inundated by the Nile floods. In
+its infancy it therefore consisted of a few rules, very rough and
+approximate, for computing the areas of triangles and quadrilaterals;
+and, with the Egyptians, it proceeded no further, the
+geometrical entities&mdash;the point, line, surface and solid&mdash;being
+only discussed in so far as they were involved in practical affairs.
+The point was realized as a mark or position, a straight line as a
+stretched string or the tracing of a pole, a surface as an area;
+but these units were not abstracted; and for the Egyptians
+geometry was only an art&mdash;an auxiliary to surveying.<a name="fa1d" id="fa1d" href="#ft1d"><span class="sp">1</span></a> The
+first step towards its elevation to the rank of a science was made
+by Thales (<i>q.v.</i>) of Miletus, who transplanted the elementary
+Egyptian mensuration to Greece. Thales clearly abstracted
+the notions of points and lines, founding the geometry of the
+latter unit, and discovering <i>per saltum</i> many propositions concerning
+areas, the circle, &amp;c. The empirical rules of the Egyptians
+were corrected and developed by the Ionic School which he
+founded, especially by Anaximander and Anaxagoras, and in
+the 6th century <span class="scs">B.C.</span> passed into the care of the Pythagoreans.
+From this time geometry exercised a powerful influence on
+Greek thought. Pythagoras (<i>q.v.</i>), seeking the key of the
+universe in arithmetic and geometry, investigated logically the
+principles underlying the known propositions; and this resulted
+in the formulation of definitions, axioms and postulates which,
+in addition to founding a <i>science</i> of geometry, permitted a
+crystallization, fractional, it is true, of the amorphous collection
+of material at hand. Pythagorean geometry was essentially a
+geometry of areas and solids; its goal was the regular solids&mdash;the
+tetrahedron, cube, octahedron, dodecahedron and icosahedron&mdash;which
+symbolized the five elements of Greek cosmology.
+The geometry of the circle, previously studied in Egypt and
+much more seriously by Thales, was somewhat neglected, although
+this curve was regarded as the most perfect of all plane figures
+and the sphere the most perfect of all solids. The circle, however,
+was taken up by the Sophists, who made most of their discoveries
+in attempts to solve the classical problems of squaring the circle,
+doubling the cube and trisecting an angle. These problems,
+besides stimulating pure geometry, <i>i.e.</i> the geometry of constructions
+made by the ruler and compasses, exercised considerable
+influence in other directions. The first problem led to the
+discovery of the method of <i>exhaustion</i> for determining areas.
+Antiphon inscribed a square in a circle, and on each side an
+isosceles triangle having its vertex on the circle; on the sides
+of the octagon so obtained, isosceles triangles were again constructed,
+the process leading to inscribed polygons of 8, 16 and
+32 sides; and the areas of these polygons, which are easily
+determined, are successive approximations to the area of the
+circle. Bryson of Heraclea took an important step when he
+circumscribed, in addition to inscribing, polygons to a circle,
+but he committed an error in treating the circle as the mean of
+the two polygons. The method of Antiphon, in assuming that
+by continued division a polygon can be constructed coincident
+with the circle, demanded that magnitudes are not infinitely
+divisible. Much controversy ranged about this point; Aristotle
+supported the doctrine of infinite divisibility; Zeno attempted
+to show its absurdity. The mechanical tracing of loci, a principle
+initiated by Archytas of Tarentum to solve the last two problems,
+was a frequent subject for study, and several mechanical curves
+were thus discovered at subsequent dates (cissoid, conchoid,
+quadratrix). Mention may be made of Hippocrates, who,
+besides developing the known methods, made a study of similar
+figures, and, as a consequence, of proportion. This step is
+important as bringing into line discontinuous number and
+continuous magnitude.</p>
+
+<p>A fresh stimulus was given by the succeeding Platonists, who,
+accepting in part the Pythagorean cosmology, made the study
+of geometry preliminary to that of philosophy. The many
+discoveries made by this school were facilitated in no small
+measure by the clarification of the axioms and definitions, the
+logical sequence of propositions which was adopted, and, more
+especially, by the formulation of the analytic method, <i>i.e.</i> of
+assuming the truth of a proposition and then reasoning to a
+<span class="pagenum"><a name="page676" id="page676"></a>676</span>
+known truth. The main strength of the Platonist geometers
+lies in stereometry or the geometry of solids. The Pythagoreans
+had dealt with the sphere and regular solids, but the pyramid,
+prism, cone and cylinder were but little known until the Platonists
+took them in hand. Eudoxus established their mensuration,
+proving the pyramid and cone to have one-third the content
+of a prism and cylinder on the same base and of the same height,
+and was probably the discoverer of a proof that the volumes of
+spheres are as the cubes of their radii. The discussion of sections
+of the cone and cylinder led to the discovery of the three curves
+named the parabola, ellipse and hyperbola (see <span class="sc"><a href="#artlinks">Conic Section</a></span>);
+it is difficult to over-estimate the importance of this discovery;
+its investigation marks the crowning achievement of Greek
+geometry, and led in later years to the fundamental theorems
+and methods of modern geometry.</p>
+
+<p>The presentation of the subject-matter of geometry as a connected
+and logical series of propositions, prefaced by <span class="grk" title="Horoi">&#8013;&#961;&#959;&#953;</span> or
+foundations, had been attempted by many; but it is to Euclid
+that we owe a complete exposition. Little indeed in the <i>Elements</i>
+is probably original except the arrangement; but in this Euclid
+surpassed such predecessors as Hippocrates, Leon, pupil of
+Neocleides, and Theudius of Magnesia, devising an apt logical
+model, although when scrutinized in the light of modern mathematical
+conceptions the proofs are riddled with fallacies. According
+to the commentator Proclus, the <i>Elements</i> were written with
+a twofold object, first, to introduce the novice to geometry, and
+secondly, to lead him to the regular solids; conic sections found
+no place therein. What Euclid did for the line and circle,
+Apollonius did for the conic sections, but there we have a discoverer
+as well as editor. These two works, which contain the greatest
+contributions to ancient geometry, are treated in detail in
+Section I. <i>Euclidean Geometry</i> and the articles <span class="sc"><a href="#artlinks">Euclid</a></span>; <span class="sc"><a href="#artlinks">Conic
+Section</a></span>; <span class="sc"><a href="#artlinks">Appolonius</a></span>. Between Euclid and Apollonius there
+flourished the illustrious Archimedes, whose geometrical discoveries
+are mainly concerned with the mensuration of the
+circle and conic sections, and of the sphere, cone and cylinder,
+and whose greatest contribution to geometrical method is the
+elevation of the method of exhaustion to the dignity of an instrument
+of research. Apollonius was followed by Nicomedes, the
+inventor of the conchoid; Diocles, the inventor of the cissoid;
+Zenodorus, the founder of the study of isoperimetrical figures;
+Hipparchus, the founder of trigonometry; and Heron the elder,
+who wrote after the manner of the Egyptians, and primarily
+directed attention to problems of practical surveying.</p>
+
+<p>Of the many isolated discoveries made by the later Alexandrian
+mathematicians, those of Menelaus are of importance. He
+showed how to treat spherical triangles, establishing their
+properties and determining their congruence; his theorem on
+the products of the segments in which the sides of a triangle
+are cut by a line was the foundation on which Carnot erected
+his theory of transversals. These propositions, and also those
+of Hipparchus, were utilized and developed by Ptolemy (<i>q.v.</i>),
+the expositor of trigonometry and discoverer of many isolated
+propositions. Mention may be made of the commentator Pappus,
+whose <i>Mathematical Collections</i> is valuable for its wealth of
+historical matter; of Theon, an editor of Euclid&rsquo;s <i>Elements</i> and
+commentator of Ptolemy&rsquo;s <i>Almagest</i>; of Proclus, a commentator
+of Euclid; and of Eutocius, a commentator of Apollonius and
+Archimedes.</p>
+
+<p>The Romans, essentially practical and having no inclination
+to study science <i>qua</i> science, only had a geometry which sufficed
+for surveying; and even here there were abundant inaccuracies,
+the empirical rules employed being akin to those of the Egyptians
+and Heron. The Hindus, likewise, gave more attention to
+computation, and their geometry was either of Greek origin or
+in the form presented in trigonometry, more particularly connected
+with arithmetic. It had no logical foundations; each
+proposition stood alone; and the results were empirical. The
+Arabs more closely followed the Greeks, a plan adopted as a
+sequel to the translation of the works of Euclid, Apollonius,
+Archimedes and many others into Arabic. Their chief contribution
+to geometry is exhibited in their solution of algebraic
+equations by intersecting conics, a step already taken by the
+Greeks in isolated cases, but only elevated into a <i>method</i> by Omar
+al Hayyami, who flourished in the 11th century. During the
+middle ages little was added to Greek and Arabic geometry.
+Leonardo of Pisa wrote a <i>Practica geometriae</i> (1220), wherein
+Euclidean methods are employed; but it was not until the 14th
+century that geometry, generally Euclid&rsquo;s <i>Elements</i>, became
+an essential item in university curricula. There was, however,
+no sign of original development, other branches of mathematics,
+mainly algebra and trigonometry, exercising a greater fascination
+until the 16th century, when the subject again came into favour.</p>
+
+<p>The extraordinary mathematical talent which came into being
+in the 16th and 17th centuries reacted on geometry and gave rise
+to all those characters which distinguish modern from ancient
+geometry. The first innovation of moment was the formulation
+of the principle of geometrical continuity by Kepler. The notion
+of infinity which it involved permitted generalizations and
+systematizations hitherto unthought of (see <span class="sc"><a href="#artlinks">Geometrical
+Continuity</a></span>); and the method of indefinite division applied to
+rectification, and quadrature and cubature problems avoided
+the cumbrous method of exhaustion and provided more accurate
+results. Further progress was made by Bonaventura Cavalieri,
+who, in his <i>Geometria indivisibilibus continuorum</i> (1620), devised
+a method intermediate between that of exhaustion and
+the infinitesimal calculus of Leibnitz and Newton. The logical
+basis of his system was corrected by Roberval and Pascal; and
+their discoveries, taken in conjunction with those of Leibnitz,
+Newton, and many others in the fluxional calculus, culminated
+in the branch of our subject known as differential geometry
+(see <span class="sc"><a href="#artlinks">Infinitesimal Calculus</a></span>; <span class="sc"><a href="#artlinks">Curve</a></span>; <span class="sc"><a href="#artlinks">Surface</a></span>).</p>
+
+<p>A second important advance followed the recognition that
+conics could be regarded as projections of a circle, a conception
+which led at the hands of Desargues and Pascal to modern
+<i>projective geometry</i> and <i>perspective</i>. A third, and perhaps the
+most important, advance attended the application of algebra to
+geometry by Descartes, who thereby founded <i>analytical geometry</i>.
+The new fields thus opened up were diligently explored, but the
+calculus exercised the greatest attraction and relatively little
+progress was made in geometry until the beginning of the 19th
+century, when a new era opened.</p>
+
+<p>Gaspard Monge was the first important contributor, stimulating
+analytical and differential geometry and founding <i>descriptive
+geometry</i> in a series of papers and especially in his lectures at the
+École polytechnique. Projective geometry, founded by Desargues,
+Pascal, Monge and L.N.M. Carnot, was crystallized by
+J.V. Poncelet, the creator of the modern methods. In his
+<i>Traité des propriétés des figures</i> (1822) the line and circular points
+at infinity, imaginaries, polar reciprocation, homology, cross-ratio
+and projection are systematically employed. In Germany,
+A.F. Möbius, J. Plücker and J. Steiner were making far-reaching
+contributions. Möbius, in his <i>Barycentrische Calcul</i> (1827),
+introduced homogeneous co-ordinates, and also the powerful
+notion of geometrical transformation, including the special
+cases of collineation and duality; Plücker, in his <i>Analytisch-geometrische
+Entwickelungen</i> (1828-1831), and his <i>System der
+analytischen Geometrie</i> (1835), introduced the abridged notation,
+line and plane co-ordinates, and the conception of generalized
+space elements; while Steiner, besides enriching geometry in
+numerous directions, was the first to systematically generate
+figures by projective pencils. We may also notice M. Chasles,
+whose <i>Aperēu historique</i> (1837) is a classic. Synthetic geometry,
+characterized by its fruitfulness and beauty, attracted most
+attention, and it so happened that its originally weak logical
+foundations became replaced by a more substantial set of axioms.
+These were found in the anharmonic ratio, a device leading to
+the liberation of synthetic geometry from metrical relations,
+and in involution, which yielded rigorous definitions of imaginaries.
+These innovations were made by K.J.C. von Staudt.
+Analytical geometry was stimulated by the algebra of invariants,
+a subject much developed by A. Cayley, G. Salmon, S.H. Aronhold,
+L.O. Hesse, and more particularly by R.F.A. Clebsch.</p>
+
+<p>The introduction of the line as a space element, initiated by
+<span class="pagenum"><a name="page677" id="page677"></a>677</span>
+H. Grassmann (1844) and Cayley (1859), yielded at the hands of
+Plücker a new geometry, termed <i>line geometry</i>, a subject
+developed more notably by F. Klein, Clebsch, C.T. Reye and
+F.O.R. Sturm (see Section V., <i>Line Geometry</i>).</p>
+
+<p><i>Non-euclidean geometries</i>, having primarily their origin in the
+discussion of Euclidean parallels, and treated by Wallis, Saccheri
+and Lambert, have been especially developed during the 19th
+century. Four lines of investigation may be distinguished:&mdash;the
+naļve-synthetic, associated with Lobatschewski, Bolyai,
+Gauss; the metric differential, studied by Riemann, Helmholtz,
+Beltrami; the projective, developed by Cayley, Klein, Clifford;
+and the critical-synthetic, promoted chiefly by the Italian
+mathematicians Peano, Veronese, Burali-Forte, Levi Civittą,
+and the Germans Pasch and Hilbert.</p>
+<div class="author">(C. E.*)</div>
+
+<p class="pt2 center sc">I. Euclidean Geometry</p>
+
+<p>This branch of the science of geometry is so named since its
+methods and arrangement are those laid down in Euclid&rsquo;s
+<i>Elements</i>.</p>
+
+<p>§ 1. <i>Axioms.</i>&mdash;The object of geometry is to investigate the
+properties of space. The first step must consist in establishing
+those fundamental properties from which all others follow by
+processes of deductive reasoning. They are laid down in the
+Axioms, and these ought to form such a system that nothing
+need be added to them in order fully to characterize space, and
+that nothing may be omitted without making the system incomplete.
+They must, in fact, completely &ldquo;define&rdquo; space.</p>
+
+<p>§ 2. <i>Definitions.</i>&mdash;The axioms of Euclidean Geometry are
+obtained from inspection of existent space and of solids in
+existent space,&mdash;hence from experience. The same source
+gives us the notions of the geometrical entities to which the
+axioms relate, viz. solids, surfaces, lines or curves, and points.
+A solid is directly given by experience; we have only to abstract
+all material from it in order to gain the notion of a geometrical
+solid. This has shape, size, position, and may be moved. Its
+boundary or boundaries are called surfaces. They separate one
+part of space from another, and are said to have no thickness.
+Their boundaries are curves or lines, and these have length
+only. Their boundaries, again, are points, which have no
+magnitude but only position. We thus come in three steps
+from solids to points which have no magnitude; in each step
+we lose one extension. Hence we say a solid has three dimensions,
+a surface two, a line one, and a point none. Space itself, of which
+a solid forms only a part, is also said to be of three dimensions.
+The same thing is intended to be expressed by saying that a
+solid has length, breadth and thickness, a surface length and
+breadth, a line length only, and a point no extension whatsoever.</p>
+
+<p>Euclid gives the essence of these statements as definitions:&mdash;</p>
+
+<div class="condensed list">
+<p>Def. 1, I. <i>A point is that which has no parts, or which has no magnitude.</i></p>
+
+<p>Def. 2, I. <i>A line is length without breadth.</i></p>
+
+<p>Def. 5, I. <i>A superficies is that which has only length and breadth.</i></p>
+
+<p>Def. 1, XI. <i>A solid is that which has length, breadth and thickness.</i></p>
+</div>
+
+<p>It is to be noted that the synthetic method is adopted by
+Euclid; the analytical derivation of the successive ideas of
+&ldquo;surface,&rdquo; &ldquo;line,&rdquo; and &ldquo;point&rdquo; from the experimental realization
+of a &ldquo;solid&rdquo; does not find a place in his system, although
+possessing more advantages.</p>
+
+<p>If we allow motion in geometry, we may generate these
+entities by moving a point, a line, or a surface, thus:&mdash;</p>
+
+<table class="reg f90" summary="poem"><tr><td> <div class="poemr">
+<p>The path of a moving point is a line.</p>
+
+<p>The path of a moving line is, in general, a surface.</p>
+
+<p>The path of a moving surface is, in general, a solid.</p>
+</div> </td></tr></table>
+
+<p>And we may then assume that the lines, surfaces and solids,
+as defined before, can all be generated in this manner. From
+this generation of the entities it follows again that the boundaries&mdash;the
+first and last position of the moving element&mdash;of a line are
+points, and so on; and thus we come back to the considerations
+with which we started.</p>
+
+<p>Euclid points this out in his definitions,&mdash;Def. 3, I., Def. 6, I.,
+and Def. 2, XI. He does not, however, show the connexion
+which these definitions have with those mentioned before.
+When points and lines have been defined, a statement like
+Def. 3, I., &ldquo;The extremities of a line are points,&rdquo; is a proposition
+which either has to be proved, and then it is a theorem, or which
+has to be taken for granted, in which case it is an axiom. And
+so with Def. 6, I., and Def. 2, XI.</p>
+
+<p>§ 3. Euclid&rsquo;s definitions mentioned above are attempts to
+describe, in a few words, notions which we have obtained by
+inspection of and abstraction from solids. A few more notions
+have to be added to these, principally those of the simplest
+line&mdash;the straight line, and of the simplest surface&mdash;the flat
+surface or plane. These notions we possess, but to define them
+accurately is difficult. Euclid&rsquo;s Definition 4, I., &ldquo;A straight
+line is that which lies evenly between its extreme points,&rdquo; must
+be meaningless to any one who has not the notion of straightness
+in his mind. Neither does it state a property of the straight
+line which can be used in any further investigation. Such a
+property is given in Axiom 10, I. It is really this axiom, together
+with Postulates 2 and 3, which characterizes the straight line.</p>
+
+<p>Whilst for the straight line the verbal definition and axiom
+are kept apart, Euclid mixes them up in the case of the plane.
+Here the Definition 7, I., includes an axiom. It defines a plane
+as a surface which has the property that every straight line
+which joins any two points in it lies altogether in the surface.
+But if we take a straight line and a point in such a surface, and
+draw all straight lines which join the latter to all points in the
+first line, the surface will be fully determined. This construction
+is therefore sufficient as a definition. That every other straight
+line which joins any two points in this surface lies altogether
+in it is a further property, and to assume it gives another axiom.</p>
+
+<p>Thus a number of Euclid&rsquo;s axioms are hidden among his first
+definitions. A still greater confusion exists in the present
+editions of Euclid between the postulates and axioms so called,
+but this is due to later editors and not to Euclid himself. The
+latter had the last three axioms put together with the postulates
+(<span class="grk" title="aitźmata">&#945;&#7984;&#964;&#942;&#956;&#945;&#964;&#945;</span>), so that these were meant to include all assumptions
+relating to space. The remaining assumptions, which relate to
+magnitudes in general, viz. the first eight &ldquo;axioms&rdquo; in modern
+editions, were called &ldquo;common notions&rdquo; (<span class="grk" title="koivai ennoiai">&#954;&#959;&#953;&#957;&#945;&#8054; &#7956;&#957;&#957;&#959;&#953;&#945;&#953;</span>).
+Of the latter a few may be said to be definitions. Thus the eighth
+might be taken as a definition of &ldquo;equal,&rdquo; and the seventh
+of &ldquo;halves.&rdquo; If we wish to collect the axioms used in Euclid&rsquo;s
+<i>Elements</i>, we have therefore to take the three postulates, the
+last three axioms as generally given, a few axioms hidden in the
+definitions, and an axiom used by Euclid in the proof of Prop.
+4, I, and on a few other occasions, viz. that figures may be
+moved in space without change of shape or size.</p>
+
+<p>§ 4. <i>Postulates.</i>&mdash;The assumptions actually made by Euclid
+may be stated as follows:&mdash;</p>
+
+<div class="condensed">
+<p>(1) Straight lines exist which have the property that any one of
+them may be produced both ways without limit, that through any
+two points in space such a line may be drawn, and that any two of
+them coincide throughout their indefinite extensions as soon as two
+points in the one coincide with two points in the other. (This
+gives the contents of Def. 4, part of Def. 35, the first two Postulates,
+and Axiom 10.)</p>
+
+<p>(2) Plane surfaces or planes exist having the property laid down
+in Def. 7, that every straight line joining any two points in such a
+surface lies altogether in it.</p>
+
+<p>(3) Right angles, as defined in Def. 10, are possible, and all right
+angles are equal; that is to say, wherever in space we take a plane,
+and wherever in that plane we construct a right angle, all angles
+thus constructed will be equal, so that any one of them may be made
+to coincide with any other. (Axiom 11.)</p>
+
+<p>(4) The 12th Axiom of Euclid. This we shall not state now, but
+only introduce it when we cannot proceed any further without it.</p>
+
+<p>(5) Figures maybe freely moved in space without change of shape
+or size. This is assumed by Euclid, but not stated as an axiom.</p>
+
+<p>(6) In any plane a circle may be described, having any point in
+that plane as centre, and its distance from any other point in that
+plane as radius. (Postulate 3.)</p>
+</div>
+
+<p>The definitions which have not been mentioned are all
+&ldquo;nominal definitions,&rdquo; that is to say, they fix a name for a
+thing described. Many of them overdetermine a figure.</p>
+
+<p>§ 5. Euclid&rsquo;s <i>Elements</i> (see <span class="sc"><a href="#artlinks">Euclid</a></span>) are contained in thirteen
+books. Of these the first four and the sixth are devoted to
+&ldquo;plane geometry,&rdquo; as the investigation of figures in a plane is
+generally called. The 5th book contains the theory of proportion
+<span class="pagenum"><a name="page678" id="page678"></a>678</span>
+which is used in Book VI. The 7th, 8th and 9th books are purely
+arithmetical, whilst the 10th contains a most ingenious treatment
+of geometrical irrational quantities. These four books will be
+excluded from our survey. The remaining three books relate to
+figures in space, or, as it is generally called, to &ldquo;solid geometry.&rdquo;
+The 7th, 8th, 9th, 10th, 13th and part of the 11th and 12th
+books are now generally omitted from the school editions of the
+<i>Elements</i>. In the first four and in the 6th book it is to be understood
+that all figures are drawn in a plane.</p>
+
+<div class="condensed">
+<p class="pt2 center sc">Book I. of Euclid&rsquo;s &ldquo;Elements.&rdquo;</p>
+
+<p>§ 6. According to the third postulate it is possible to draw in
+any plane a circle which has its centre at any given point, and its
+radius equal to the distance of this point from any other point
+given in the plane. This makes it possible (Prop. 1) to construct
+on a given line AB an equilateral triangle, by drawing first a circle
+with A as centre and AB as radius, and then a circle with B as
+centre and BA as radius. The point where these circles intersect&mdash;that
+they intersect Euclid quietly assumes&mdash;is the vertex of the
+required triangle. Euclid does not suppose, however, that a circle
+may be drawn which has its radius equal to the distance between
+any two points unless one of the points be the centre. This implies
+also that we are not supposed to be able to make any straight line
+equal to any other straight line, or to carry a distance about in space.
+Euclid therefore next solves the problem: It is required along a
+given straight line from a point in it to set off a distance equal to
+the length of another straight line given anywhere in the plane.
+This is done in two steps. It is shown in Prop. 2 how a straight line
+may be drawn from a given point equal in length to another given
+straight line not drawn from that point. And then the problem
+itself is solved in Prop. 3, by drawing first through the given point
+some straight line of the required length, and then about the same
+point as centre a circle having this length as radius. This circle
+will cut off from the given straight line a length equal to the required
+one. Nowadays, instead of going through this long process, we
+take a pair of compasses and set off the given length by its aid.
+This assumes that we may move a length about without changing it.
+But Euclid has not assumed it, and this proceeding would be fully
+justified by his desire not to take for granted more than was necessary,
+if he were not obliged at his very next step actually to make this
+assumption, though without stating it.</p>
+
+<p>§ 7. We now come (in Prop. 4) to the first theorem. It is the
+fundamental theorem of Euclid&rsquo;s whole system, there being only a
+very few propositions (like Props. 13, 14, 15, I.), except those in the
+5th book and the first half of the 11th, which do not depend upon
+it. It is stated very accurately, though somewhat clumsily, as
+follows:&mdash;</p>
+
+<p><i>If two triangles have two sides of the one equal to two sides of the
+other, each to each, and have also the angles contained by those sides
+equal to one another, they shall also have their bases or third sides
+equal; and the two triangles shall be equal; and their other angles
+shall be equal, each to each, namely, those to which the equal sides are
+opposite.</i></p>
+
+<p>That is to say, the triangles are &ldquo;identically&rdquo; equal, and one
+may be considered as a copy of the other. The proof is very simple.
+The first triangle is taken up and placed on the second, so that the
+parts of the triangles which are known to be equal fall upon each
+other. It is then easily seen that also the remaining parts of one
+coincide with those of the other, and that they are therefore equal.
+This process of applying one figure to another Euclid scarcely uses
+again, though many proofs would be simplified by doing so. The
+process introduces motion into geometry, and includes, as already
+stated, the axiom that figures may be moved without change of
+shape or size.</p>
+
+<p>If the last proposition be applied to an isosceles triangle, which
+has two sides equal, we obtain the theorem (Prop. 5), <i>if two sides
+of a triangle are equal, then the angles opposite these sides are equal</i>.</p>
+
+<p>Euclid&rsquo;s proof is somewhat complicated, and a stumbling-block
+to many schoolboys. The proof becomes much simpler if we consider
+the isosceles triangle ABC (AB = AC) twice over, once as a triangle
+BAC, and once as a triangle CAB; and now remember that AB, AC
+in the first are equal respectively to AC, AB in the second, and the
+angles included by these sides are equal. Hence the triangles are
+equal, and the angles in the one are equal to those in the other, viz.
+those which are opposite equal sides, <i>i.e.</i> angle ABC in the first
+equals angle ACB in the second, as they are opposite the equal
+sides AC and AB in the two triangles.</p>
+
+<p>There follows the converse theorem (Prop. 6). <i>If two angles in
+a triangle are equal, then the sides opposite them are equal</i>,&mdash;<i>i.e.</i> the
+triangle is isosceles. The proof given consists in what is called a
+<i>reductio ad absurdum</i>, a kind of proof often used by Euclid, and
+principally in proving the converse of a previous theorem. It
+assumes that the theorem to be proved is wrong, and then shows
+that this assumption leads to an absurdity, <i>i.e.</i> to a conclusion
+which is in contradiction to a proposition proved before&mdash;that
+therefore the assumption made cannot be true, and hence that
+the theorem is true. It is often stated that Euclid invented this
+kind of proof, but the method is most likely much older.</p>
+
+<p>§ 8. It is next proved that <i>two triangles which have the three sides
+of the one equal respectively to those of the other are identically equal,
+hence that the angles of the one are equal respectively to those of the
+other, those being equal which are opposite equal sides</i>. This is Prop. 8,
+Prop. 7 containing only a first step towards its proof.</p>
+
+<p>These theorems allow now of the solution of a number of problems,
+viz.:&mdash;</p>
+
+<p><i>To bisect a given angle</i> (Prop. 9).</p>
+
+<p><i>To bisect a given finite straight line</i> (Prop. 10).</p>
+
+<p><i>To draw a straight line perpendicularly to a given straight line
+through a given point in it</i> (Prop. 11), <i>and also through a given point
+not in it</i> (Prop. 12).</p>
+
+<p>The solutions all depend upon properties of isosceles triangles.</p>
+
+<p>§ 9. The next three theorems relate to angles only, and might have
+been proved before Prop. 4, or even at the very beginning. The
+first (Prop. 13) says, <i>The angles which one straight line makes with
+another straight line on one side of it either are two right angles or
+are together equal to two right angles</i>. This theorem would have
+been unnecessary if Euclid had admitted the notion of an angle
+such that its two limits are in the same straight line, and had besides
+defined the sum of two angles.</p>
+
+<p>Its converse (Prop. 14) is of great use, inasmuch as it enables us
+in many cases to prove that two straight lines drawn from the same
+point are one the continuation of the other. So also is</p>
+
+<p>Prop. 15. <i>If two straight lines cut one another, the vertical or opposite
+angles shall be equal.</i></p>
+
+<p>§ 10. Euclid returns now to properties of triangles. Of great
+importance for the next steps (though afterwards superseded by a
+more complete theorem) is</p>
+
+<p>Prop. 16. <i>If one side of a triangle be produced, the exterior angle
+shall be greater than either of the interior opposite angles.</i></p>
+
+<p>Prop. 17. <i>Any two angles of a triangle are together less than two
+right angles, is an immediate consequence of it.</i> By the aid of these
+two, the following fundamental properties of triangles are easily
+proved:&mdash;</p>
+
+<p>Prop. 18. <i>The greater side of every triangle has the greater angle
+opposite to it</i>;</p>
+
+<p>Its converse, Prop. 19. <i>The greater angle of every triangle is subtended
+by the greater side, or has the greater side opposite to it</i>;</p>
+
+<p>Prop. 20. <i>Any two sides of a triangle are together greater than the
+third side</i>;</p>
+
+<p>And also Prop. 21. <i>If from the ends of the side of a triangle there
+be drawn two straight lines to a point within the triangle, these shall
+be less than the other two sides of the triangle, but shall contain a greater
+angle.</i></p>
+
+<p>§ 11. Having solved two problems (Props. 22, 23), he returns to two
+triangles which have two sides of the one equal respectively to two
+sides of the other. It is known (Prop. 4) that if the included angles
+are equal then the third sides are equal; and conversely (Prop. 8),
+if the third sides are equal, then the angles included by the first
+sides are equal. From this it follows that if the included angles are
+not equal, the third sides are not equal; and conversely, that if the
+third sides are not equal, the included angles are not equal. Euclid
+now completes this knowledge by proving, that &ldquo;<i>if the included
+angles are not equal, then the third side in that triangle is the greater
+which contains the greater angle</i>&rdquo;; and conversely, that &ldquo;<i>if the third
+sides are unequal, that triangle contains the greater angle which contains
+the greater side</i>.&rdquo; These are Prop. 24 and Prop. 25.</p>
+
+<p>§ 12. The next theorem (Prop. 26) says that <i>if two triangles have
+one side and two angles of the one equal respectively to one side and
+two angles of the other, viz. in both triangles either the angles adjacent
+to the equal side, or one angle adjacent and one angle opposite it, then
+the two triangles are identically equal</i>.</p>
+
+<p>This theorem belongs to a group with Prop. 4 and Prop. 8. Its
+first case might have been given immediately after Prop. 4, but the
+second case requires Prop. 16 for its proof.</p>
+
+<p>§ 13. We come now to the investigation of parallel straight lines,
+<i>i.e.</i> of straight lines which lie in the same plane, and cannot be made
+to meet however far they be produced either way. The investigation
+which starts from Prop. 16, will become clearer if a few names be
+explained which are not all used by Euclid. If two straight lines
+be cut by a third, the latter is now generally called a &ldquo;transversal&rdquo;
+of the figure. It forms at the two points where it cuts the given lines
+four angles with each. Those of the angles which lie between the
+given lines are called interior angles, and of these, again, any two
+which lie on opposite sides of the transversal but one at each of the
+two points are called &ldquo;alternate angles.&rdquo;</p>
+
+<p>We may now state Prop. 16 thus:&mdash;<i>If two straight lines which
+meet are cut by a transversal, their alternate angles are unequal</i>. For
+the lines will form a triangle, and one of the alternate angles will
+be an exterior angle to the triangle, the other interior and opposite
+to it.</p>
+
+<p>From this follows at once the theorem contained in Prop. 27.
+<i>If two straight lines which are cut by a transversal make alternate
+angles equal, the lines cannot meet, however far they be produced,
+hence they are parallel.</i> This proves the existence of parallel
+lines.</p>
+
+<p>Prop. 28 states the same fact in different forms. <i>If a straight
+line, falling on two other straight lines, make the exterior angle equal
+to the interior and opposite angle on the same side of the line, or make</i>
+<span class="pagenum"><a name="page679" id="page679"></a>679</span>
+<i>the interior angles on the same side together equal to two right angles,
+the two straight lines shall be parallel to one another</i>.</p>
+
+<p>Hence we know that, &ldquo;if two straight lines which are cut by a
+transversal meet, their alternate angles are not equal&rdquo;; and hence
+that, &ldquo;if alternate angles are equal, then the lines are parallel.&rdquo;</p>
+
+<p>The question now arises, Are the propositions converse to these
+true or not? That is to say, &ldquo;If alternate angles are unequal, do
+the lines meet?&rdquo; And &ldquo;if the lines are parallel, are alternate
+angles necessarily equal?&rdquo;</p>
+
+<p>The answer to either of these two questions implies the answer
+to the other. But it has been found impossible to prove that the
+negation or the affirmation of either is true.</p>
+
+<p>The difficulty which thus arises is overcome by Euclid assuming
+that the first question has to be answered in the affirmative. This
+gives his last axiom (12), which we quote in his own words.</p>
+
+<p>Axiom 12.&mdash;<i>If a straight line meet two straight lines, so as to make
+the two interior angles on the same side of it taken together less than
+two right angles, these straight lines, being continually produced, shall
+at length meet on that side on which are the angles which are less than
+two right angles.</i></p>
+
+<p>The answer to the second of the above questions follows from this,
+and gives the theorem Prop. 29:&mdash;<i>If a straight line fall on two parallel
+straight lines, it makes the alternate angles equal to one another, and
+the exterior angle equal to the interior and opposite angle on the same
+side, and also the two interior angles on the same side together equal
+to two right angles</i>.</p>
+
+<p>§ 14. With this a new part of elementary geometry begins. The
+earlier propositions are independent of this axiom, and would be
+true even if a wrong assumption had been made in it. They all
+relate to figures in a plane. But a plane is only one among an infinite
+number of conceivable surfaces. We may draw figures on any one
+of them and study their properties. We may, for instance, take a
+sphere instead of the plane, and obtain &ldquo;spherical&rdquo; in the place of
+&ldquo;plane&rdquo; geometry. If on one of these surfaces lines and figures
+could be drawn, answering to all the definitions of our plane figures,
+and if the axioms with the exception of the last all hold, then all
+propositions up to the 28th will be true for these figures. This is
+the case in spherical geometry if we substitute &ldquo;shortest line&rdquo; or
+&ldquo;great circle&rdquo; for &ldquo;straight line,&rdquo; &ldquo;small circle&rdquo; for &ldquo;circle,&rdquo; and
+if, besides, we limit all figures to a part of the sphere which is less
+than a hemisphere, so that two points on it cannot be opposite ends
+of a diameter, and therefore determine always one and only one great
+circle.</p>
+
+<p>For spherical triangles, therefore, all the important propositions
+4, 8, 26; 5 and 6; and 18, 19 and 20 will hold good.</p>
+
+<p>This remark will be sufficient to show the impossibility of proving
+Euclid&rsquo;s last axiom, which would mean proving that this axiom is
+a consequence of the others, and hence that the theory of parallels
+would hold on a spherical surface, where the other axioms do hold,
+whilst parallels do not even exist.</p>
+
+<p>It follows that the axiom in question states an inherent difference
+between the plane and other surfaces, and that the plane is only
+fully characterized when this axiom is added to the other assumptions.</p>
+
+<p>§ 15. The introduction of the new axiom and of parallel lines leads
+to a new class of propositions.</p>
+
+<p>After proving (Prop. 30) that &ldquo;<i>two lines which are each parallel
+to a third are parallel to each other</i>,&rdquo; we obtain the new properties
+of triangles contained in Prop. 32. Of these the second part is the
+most important, viz. the theorem, <i>The three interior angles of every
+triangle are together equal to two right angles</i>.</p>
+
+<p>As easy deductions not given by Euclid but added by Simson
+follow the propositions about the angles in polygons, they are given
+in English editions as corollaries to Prop. 32.</p>
+
+<p>These theorems do not hold for spherical figures. The sum of the
+interior angles of a spherical triangle is always greater than two
+right angles, and increases with the area.</p>
+
+<p>§ 16. The theory of parallels as such may be said to be finished
+with Props. 33 and 34, which state properties of the parallelogram,
+<i>i.e.</i> of a quadrilateral formed by two pairs of parallels. They are&mdash;</p>
+
+<p>Prop. 33. <i>The straight lines which join the extremities of two equal
+and parallel straight lines towards the same parts are themselves equal
+and parallel</i>; and</p>
+
+<p>Prop. 34. <i>The opposite sides and angles of a parallelogram are
+equal to one another, and the diameter (diagonal) bisects the parallelogram,
+that is, divides it into two equal parts.</i></p>
+
+<p>§ 17. The rest of the first book relates to areas of figures.</p>
+
+<p>The theory is made to depend upon the theorems&mdash;</p>
+
+<p>Prop. 35. <i>Parallelograms on the same base and between the same
+parallels are equal to one another</i>; and</p>
+
+<p>Prop. 36. <i>Parallelograms on equal bases and between the same
+parallels are equal to one another</i>.</p>
+
+<p>As each parallelogram is bisected by a diagonal, the last theorems
+hold also if the word parallelogram be replaced by &ldquo;triangle,&rdquo; as is
+done in Props. 37 and 38.</p>
+
+<p>It is to be remarked that Euclid proves these propositions only
+in the case when the parallelograms or triangles have their bases in
+the same straight line.</p>
+
+<p>The theorems converse to the last form the contents of the next
+three propositions, viz.: Props, 40 and 41.&mdash;<i>Equal triangles, on
+the same or on equal bases, in the same straight line, and on the same
+side of it, are between the same parallels</i>.</p>
+
+<p>That the two cases here stated are given by Euclid in two separate
+propositions proved separately is characteristic of his method.</p>
+
+<p>§ 18. To compare areas of other figures, Euclid shows first, in
+Prop. 42, how <i>to draw a parallelogram which is equal in area to a
+given triangle, and has one of its angles equal to a given angle</i>. If the
+given angle is right, then the problem is solved <i>to draw a &ldquo;rectangle&rdquo;
+equal in area to a given triangle</i>.</p>
+
+<p>Next this parallelogram is transformed into another parallelogram,
+<i>which has one of its sides equal to a given straight line</i>, whilst its angles
+remain unaltered. This may be done by aid of the theorem in</p>
+
+<p>Prop. 43. <i>The complements of the parallelograms which are about
+the diameter of any parallelogram are equal to one another.</i></p>
+
+<p>Thus the problem (Prop. 44) is solved to <i>construct a parallelogram
+on a given line, which is equal in area to a given triangle, and which
+has one angle equal to a given angle</i> (generally a right angle).</p>
+
+<p>As every polygon can be divided into a number of triangles, we
+can now construct a parallelogram having a given angle, say a
+right angle, and being equal in area to a given polygon. For each
+of the triangles into which the polygon has been divided, a parallelogram
+may be constructed, having one side equal to a given straight
+line and one angle equal to a given angle. If these parallelograms
+be placed side by side, they may be added together to form a single
+parallelogram, having still one side of the given length. This is
+done in Prop. 45.</p>
+
+<p>Herewith a means is found to compare areas of different polygons.
+We need only construct two rectangles equal in area to the given
+polygons, and having each one side of given length. By comparing
+the unequal sides we are enabled to judge whether the areas are
+equal, or which is the greater. Euclid does not state this consequence,
+but the problem is taken up again at the end of the second book,
+where it is shown how to construct a square equal in area to a given
+polygon.</p>
+
+<p>Prop. 46 is: <i>To describe a square on a given straight line</i>.</p>
+
+<p>§ 19. The first book concludes with one of the most important
+theorems in the whole of geometry, and one which has been celebrated
+since the earliest times. It is stated, but on doubtful authority,
+that Pythagoras discovered it, and it has been called by his name.
+If we call that side in a right-angled triangle which is opposite the
+right angle the hypotenuse, we may state it as follows:&mdash;</p>
+
+<p>Theorem of Pythagoras (Prop. 47).&mdash;<i>In every right-angled triangle
+the square on the hypotenuse is equal to the sum of the squares of the
+other sides.</i></p>
+
+<p>And conversely&mdash;</p>
+
+<p>Prop. 48. <i>If the square described on one of the sides of a triangle be
+equal to the squares described on the other sides, then the angle contained
+by these two sides is a right angle.</i></p>
+
+<p>On this theorem (Prop. 47) almost all geometrical measurement
+depends, which cannot be directly obtained.</p>
+
+<p class="pt2 center sc">Book II.</p>
+
+<p>§ 20. The propositions in the second book are very different in
+character from those in the first; they all relate to areas of rectangles
+and squares. Their true significance is best seen by stating them in
+an algebraic form. This is often done by expressing the lengths of
+lines by aid of numbers, which tell how many times a chosen unit
+is contained in the lines. If there is a unit to be found which is contained
+an exact number of times in each side of a rectangle, it is
+easily seen, and generally shown in the teaching of arithmetic, that
+the rectangle contains a number of unit squares equal to the product
+of the numbers which measure the sides, a unit square being the
+square on the unit line. If, however, no such unit can be found,
+this process requires that connexion between lines and numbers
+which is only established by aid of ratios of lines, and which is therefore
+at this stage altogether inadmissible. But there exists another
+way of connecting these propositions with algebra, based on modern
+notions which seem destined greatly to change and to simplify
+mathematics. We shall introduce here as much of it as is required
+for our present purpose.</p>
+
+<p>At the beginning of the second book we find a definition according
+to which &ldquo;a rectangle is said to be &lsquo;contained&rsquo; by the two sides
+which contain one of its right angles&rdquo;; in the text this phraseology
+is extended by speaking of rectangles contained by any two straight
+lines, meaning the rectangle which has two adjacent sides equal to
+the two straight lines.</p>
+
+<p>We shall denote a finite straight line by a single small letter,
+a, b, c, ... x, and the area of the rectangle contained by two lines
+a and b by ab, and this we shall call the product of the two lines a
+and b. It will be understood that this definition has nothing to do
+with the definition of a product of numbers.</p>
+
+<p>We define as follows:&mdash;</p>
+
+<p>The <i>sum</i> of two straight lines a and b means a straight line c which
+may be divided in two parts equal respectively to a and b. This sum
+is denoted by a + b.</p>
+
+<p>The <i>difference</i> of two lines a and b (in symbols, a-b) means a line
+c which when added to b gives a; that is,</p>
+
+<p class="center">a &minus; b = c if b + c = a.</p>
+
+<p>The <i>product</i> of two lines a and b (in symbols, ab) means the area
+<span class="pagenum"><a name="page680" id="page680"></a>680</span>
+of the rectangle contained by the lines a and b. For aa, which
+means the square on the line a, we write a².</p>
+
+<p>§ 21. The first ten of the fourteen propositions of the second book
+may then be written in the form of formulae as follows:&mdash;</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcc">Prop.</td> <td class="tcr">1.</td> <td class="tcl">a (b + c + d + ... ) = ab + ac + ad + ...</td></tr>
+
+<tr><td class="tcc">&rdquo;</td> <td class="tcr">2.</td> <td class="tcl">ab + ac = a² if b + c = a.</td></tr>
+
+<tr><td class="tcc">&rdquo;</td> <td class="tcr">3.</td> <td class="tcl">a (a + b) = a² + ab.</td></tr>
+
+<tr><td class="tcc">&rdquo;</td> <td class="tcr">4.</td> <td class="tcl">(a + b)² = a² + 2ab + b².</td></tr>
+
+<tr><td class="tcc">&rdquo;</td> <td class="tcr">5.</td> <td class="tcl">(a + b)(a &minus; b) + b² = a².</td></tr>
+
+<tr><td class="tcc">&rdquo;</td> <td class="tcr">6.</td> <td class="tcl">(a + b)(a &minus; b) + b² = a².</td></tr>
+
+<tr><td class="tcc">&rdquo;</td> <td class="tcr">7.</td> <td class="tcl">a² + (a &minus; b)² = 2a (a &minus; b) + b².</td></tr>
+
+<tr><td class="tcc">&rdquo;</td> <td class="tcr">8.</td> <td class="tcl">4(a + b)a + b² = (2a + b)².</td></tr>
+
+<tr><td class="tcc">&rdquo;</td> <td class="tcr">9.</td> <td class="tcl">(a + b)² + (a &minus; b)² = 2a² + 2b².</td></tr>
+
+<tr><td class="tcc">&rdquo;</td> <td class="tcr">10.</td> <td class="tcl">(a + b)² + (a &minus; b)² = 2a² + 2b².</td></tr>
+</table>
+
+<p>It will be seen that 5 and 6, and also 9 and 10, are identical. In
+Euclid&rsquo;s statement they do not look the same, the figures being
+arranged differently.</p>
+
+<p>If the letters a, b, c, ... denoted numbers, it follows from algebra
+that each of these formulae is true. But this does not prove them in
+our case, where the letters denote lines, and their products areas
+without any reference to numbers. To prove them we have to
+discover the laws which rule the operations introduced, viz. addition
+and multiplication of segments. This we shall do now; and we shall
+find that these laws are the same with those which hold in algebraical
+addition and multiplication.</p>
+
+<p>§ 22. In a sum of numbers we may change the order in which
+the numbers are added, and we may also add the numbers together
+in groups and then add these groups. But this also holds for the
+sum of segments and for the sum of rectangles, as a little consideration
+shows. That the sum of rectangles has always a meaning
+follows from the Props. 43-45 in the first book. These laws about
+addition are reducible to the two&mdash;</p>
+
+<p class="center">a + b = b + a</p>
+<div class="author">(1),</div>
+
+<p class="center">a + (b + c) = a + b + c</p>
+<div class="author">(2);</div>
+
+<p class="noind">or, when expressed for rectangles,</p>
+
+<p class="center">ab + ed = ed + ab</p>
+<div class="author">(3),</div>
+
+<p class="center">ab + (cd + ef) = ab + cd + ef</p>
+<div class="author">(4).</div>
+
+<p class="noind">The brackets mean that the terms in the bracket have been added
+together before they are added to another term. The more general
+cases for more terms may be deduced from the above.</p>
+
+<p>For the product of two numbers we have the law that it remains
+unaltered if the factors be interchanged. This also holds for our
+geometrical product. For if ab denotes the area of the rectangle
+which has a as base and b as altitude, then ba will denote the area
+of the rectangle which has b as base and a as altitude. But in a
+rectangle we may take either of the two lines which contain it as
+base, and then the other will be the altitude. This gives</p>
+
+<p class="center">ab = ba</p>
+<div class="aut">(5).</div>
+
+<p>In order further to multiply a sum by a number, we have in algebra
+the rule:&mdash;Multiply each term of the sum, and add the products
+thus obtained. That this holds for our geometrical products is
+shown by Euclid in his first proposition of the second book, where
+he proves that the area of a rectangle whose base is the sum of a
+number of segments is equal to the sum of rectangles which have
+these segments separately as bases. In symbols this gives, in the
+simplest case,</p>
+
+<p class="center">a(b + c) = ab + ac</p>
+
+<p class="noind">and</p>
+
+<p class="center">(b + c)a = ba + ca</p>
+<div class="aut">(6).</div>
+
+<p class="noind">To these laws, which have been investigated by Sir William Hamilton
+and by Hermann Grassmann, the former has given special names.
+He calls the laws expressed in</p>
+
+<table class="reg" summary="poem"><tr><td> <div class="poemr">
+<p>(1) and (3) the commutative law for addition;</p>
+
+<p class="i3">(5) the commutative law for  multiplication;</p>
+
+<p>(2) and (4) the associative laws for addition;</p>
+
+<p class="i3">(6) the distributive law.</p>
+</div> </td></tr></table>
+
+<p>§ 23. Having proved that these six laws hold, we can at once
+prove every one of the above propositions in their algebraical form.</p>
+
+<p>The first is proved geometrically, it being one of the fundamental
+laws. The next two propositions are only special cases of the first.
+Of the others we shall prove one, viz. the fourth:&mdash;</p>
+
+<p class="center">(a + b)² = (a + b)(a + b) = (a + b)a + (a + b)b</p>
+<div class="aut">by (6).</div>
+
+<p class="noind">But</p>
+<p class="center">(a + b)a = aa + ba</p>
+<div class="aut">by (6),</div>
+
+<p class="center">= aa + ab</p>
+<div class="aut">by (5);</div>
+
+<p class="noind">and</p>
+
+<p class="center">(a + b)b = ab + bb</p>
+<div class="aut">by (6).</div>
+
+<p class="noind">Therefore</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcr">(a + b)²</td> <td class="tcl">= aa + ab + (ab + bb)</td></tr>
+<tr><td class="tcr">&nbsp;</td> <td class="tcl">= aa + (ab + ab) + bb</td></tr>
+<tr><td class="tcr">&nbsp;</td> <td class="tcl">= aa + 2ab + bb</td></tr>
+</table>
+
+<div class="aut">by (4).</div>
+
+<p>This gives the theorem in question.</p>
+
+<p>In the same manner every one of the first ten propositions is
+proved.</p>
+
+<p>It will be seen that the operations performed are exactly the same
+as if the letters denoted numbers.</p>
+
+<p>Props. 5 and 6 may also be written thus&mdash;</p>
+
+<p class="center">(a + b)(a &minus; b) = a² &minus; b².</p>
+
+<p>Prop. 7, which is an easy consequence of Prop. 4, may be transformed.
+If we denote by c the line a + b, so that</p>
+
+<p class="center">c = a + b, a = c &minus; b,</p>
+
+<p class="noind">we get</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcr">c² + (c &minus; b)²</td> <td class="tcl">= 2c(c &minus; b) + b²</td></tr>
+<tr><td class="tcr">&nbsp;</td> <td class="tcl">= 2c² &minus; 2bc + b².</td></tr>
+</table>
+
+<p class="noind">Subtracting c² from both sides, and writing a for c, we get</p>
+
+<p class="center">(a &minus; b)² = a² &minus; 2ab + b².</p>
+
+<p>In Euclid&rsquo;s <i>Elements</i> this form of the theorem does not appear,
+all propositions being so stated that the notion of subtraction does
+not enter into them.</p>
+
+<p>§ 24. The remaining two theorems (Props. 12 and 13) connect
+the square on one side of a triangle with the sum of the squares on
+the other sides, in case that the angle between the latter is acute or
+obtuse. They are important theorems in trigonometry, where it is
+possible to include them in a single theorem.</p>
+
+<p>§ 25. There are in the second book two problems, Props. 11 and 14.</p>
+
+<p>If written in the above symbolic language, the former requires to
+find a line x such that a(a &minus; x) = x². Prop. 11 contains, therefore,
+the solution of a quadratic equation, which we may write x² + ax = a².
+The solution is required later on in the construction of a regular
+decagon.</p>
+
+<p>More important is the problem in the last proposition (Prop. 14).
+It requires the construction of a square equal in area to a given
+rectangle, hence a solution of the equation</p>
+
+<p class="center">x² = ab.</p>
+
+<p>In Book I., 42-45, it has been shown how a rectangle may be constructed
+equal in area to a given figure bounded by straight lines.
+By aid of the new proposition we may therefore now determine a
+line such that the square on that line is equal in area to any given
+rectilinear figure, or we can <i>square</i> any such figure.</p>
+
+<p>As of two squares that is the greater which has the greater side,
+it follows that now the comparison of two areas has been reduced
+to the comparison of two lines.</p>
+
+<p>The problem of reducing other areas to squares is frequently met
+with among Greek mathematicians. We need only mention the
+problem of squaring the circle (see <span class="sc"><a href="#artlinks">Circle</a></span>).</p>
+
+<p>In the present day the comparison of areas is performed in a
+simpler way by reducing all areas to rectangles having a common
+base. Their altitudes give then a measure of their areas.</p>
+
+<p>The construction of a rectangle having the base u, and being equal
+in area to a given rectangle, depends upon Prop. 43, I. This therefore
+gives a solution of the equation</p>
+
+<p class="center">ab = ux,</p>
+
+<p class="noind">where x denotes the unknown altitude.</p>
+
+<p class="pt2 center sc">Book III.</p>
+
+<p>§ 26. The third book of the <i>Elements</i> relates exclusively to properties
+of the circle. A circle and its circumference have been defined
+in Book I., Def. 15. We restate it here in slightly different words:&mdash;</p>
+
+<p><i>Definition</i>.&mdash;The circumference of a circle is a plane curve such
+that all points in it have the same distance from a fixed point in
+the plane. This point is called the &ldquo;centre&rdquo; of the circle.</p>
+
+<p>Of the new definitions, of which eleven are given at the beginning
+of the third book, a few only require special mention. The first,
+which says that circles with equal radii are equal, is in part a theorem,
+but easily proved by applying the one circle to the other. Or it
+may be considered proved by aid of Prop. 24, equal circles not being
+used till after this theorem.</p>
+
+<p>In the second definition is explained what is meant by a line
+which &ldquo;touches&rdquo; a circle. Such a line is now generally called a
+tangent to the circle. The introduction of this name allows us to
+state many of Euclid&rsquo;s propositions in a much shorter form.</p>
+
+<p>For the same reason we shall call a straight line joining two points
+on the circumference of a circle a &ldquo;chord.&rdquo;</p>
+
+<p>Definitions 4 and 5 may be replaced with a slight generalization
+by the following:&mdash;</p>
+
+<p><i>Definition</i>.&mdash;By the distance of a point from a line is meant the
+length of the perpendicular drawn from the point to the line.</p>
+
+<p>§ 27. From the definition of a circle it follows that every circle
+has a centre. Prop. 1 requires to find it when the circle is given,
+<i>i.e.</i> when its circumference is drawn.</p>
+
+<p>To solve this problem a chord is drawn (that is, any two points in
+the circumference are joined), and through the point where this is
+bisected a perpendicular to it is erected. Euclid then proves, first,
+that no point off this perpendicular can be the centre, hence that the
+centre must lie in this line; and, secondly, that of the points on the
+perpendicular one only can be the centre, viz. the one which bisects
+the parts of the perpendicular bounded by the circle. In the second
+part Euclid silently assumes that the perpendicular there used does
+cut the circumference in two, and only in two points. The proof
+therefore is incomplete. The proof of the first part, however, is
+exact. By drawing two non-parallel chords, and the perpendiculars
+which bisect them, the centre will be found as the point where these
+perpendiculars intersect.</p>
+
+<p>§ 28. In Prop. 2 it is proved that a chord of a circle lies altogether
+within the circle.</p>
+
+<p><span class="pagenum"><a name="page681" id="page681"></a>681</span></p>
+
+<p>What we have called the first part of Euclid&rsquo;s solution of Prop. 1
+may be stated as a theorem:&mdash;</p>
+
+<p><i>Every straight line which bisects a chord, and is at right angles to it,
+passes through the centre of the circle.</i></p>
+
+<p>The converse to this gives Prop. 3, which may be stated thus:&mdash;</p>
+
+<p><i>If a straight line through the centre of a circle bisect a chord, then
+it is perpendicular to the chord, and if it be perpendicular to the chord
+it bisects it.</i></p>
+
+<p>An easy consequence of this is the following theorem, which is
+essentially the same as Prop. 4:&mdash;</p>
+
+<p><i>Two chords of a circle, of which neither passes through the centre,
+cannot bisect each other.</i></p>
+
+<p>These last three theorems are fundamental for the theory of the
+circle. It is to be remarked that Euclid never proves that a straight
+line cannot have more than two points in common with a circumference.</p>
+
+<p>§ 29. The next two propositions (5 and 6) might be replaced by
+a single and a simpler theorem, viz:&mdash;</p>
+
+<p><i>Two circles which have a common centre, and whose circumferences
+have one point in common, coincide.</i></p>
+
+<p>Or, more in agreement with Euclid&rsquo;s form:&mdash;</p>
+
+<p><i>Two different circles, whose circumferences have a point in common,
+cannot have the same centre.</i></p>
+
+<p>That Euclid treats of two cases is characteristic of Greek mathematics.</p>
+
+<p>The next two propositions (7 and 8) again belong together. They
+may be combined thus:&mdash;</p>
+
+<p><i>If from a point in a plane of a circle, which is not the centre, straight
+lines be drawn to the different points of the circumference, then of all
+these lines one is the shortest, and one the longest, and these lie both in
+that straight line which joins the given point to the centre. Of all the
+remaining lines each is equal to one and only one other, and these
+equal lines lie on opposite sides of the shortest or longest, and make
+equal angles with them.</i></p>
+
+<p>Euclid distinguishes the two cases where the given point lies within
+or without the circle, omitting the case where it lies in the circumference.</p>
+
+<p>From the last proposition it follows that if from a point more
+than two equal straight lines can be drawn to the circumference,
+this point must be the centre. This is Prop. 9.</p>
+
+<p>As a consequence of this we get</p>
+
+<p><i>If the circumferences of the two circles have three points in common
+they coincide.</i></p>
+
+<p>For in this case the two circles have a common centre, because
+from the centre of the one three equal lines can be drawn to points
+on the circumference of the other. But two circles which have a
+common centre, and whose circumferences have a point in common,
+coincide. (Compare above statement of Props. 5 and 6.)</p>
+
+<p>This theorem may also be stated thus:&mdash;</p>
+
+<p><i>Through three points only one circumference may be drawn; or,
+Three points determine a circle.</i></p>
+
+<p>Euclid does not give the theorem in this form. He proves, however,
+<i>that the two circles cannot cut another in more than two points</i>
+(Prop. 10), and <i>that two circles cannot touch one another in more points
+than one</i> (Prop. 13).</p>
+
+<p>§ 30. Propositions 11 and 12 assert that <i>if two circles touch, then
+the point of contact lies on the line joining their centres</i>. This gives
+two propositions, because the circles may touch either internally
+or externally.</p>
+
+<p>§ 31. Propositions 14 and 15 relate to the length of chords. The
+first says <i>that equal chords are equidistant from the centre, and that
+chords which are equidistant from the centre are equal</i>;</p>
+
+<p>Whilst Prop. 15 compares unequal chords, viz. <i>Of all chords the
+diameter is the greatest, and of other chords that is the greater which
+is nearer to the centre</i>; and conversely, <i>the greater chord is nearer to
+the centre</i>.</p>
+
+<p>§ 32. In Prop. 16 the tangent to a circle is for the first time introduced.
+The proposition is meant to show that the straight line
+at the end point of the diameter and at right angles to it is a tangent.
+The proposition itself does not state this. It runs thus:&mdash;</p>
+
+<p>Prop. 16. <i>The straight line drawn at right angles to the diameter
+of a circle, from the extremity of it, falls without the circle; and no
+straight line can be drawn from the extremity, between that straight
+line and the circumference, so as not to cut the circle.</i></p>
+
+<p><i>Corollary</i>.&mdash;The straight line at right angles to a diameter drawn
+through the end point of it touches the circle.</p>
+
+<p>The statement of the proposition and its whole treatment show
+the difficulties which the tangents presented to Euclid.</p>
+
+<p>Prop. 17 solves the problem <i>through a given point, either in the
+circumference or without it, to draw a tangent to a given circle</i>.</p>
+
+<p>Closely connected with Prop. 16 are Props. 18 and 19, which
+state (Prop. 18), <i>that the line joining the centre of a circle to the point
+of contact of a tangent is perpendicular to the tangent</i>; and conversely
+(Prop. 19), <i>that the straight line through the point of contact
+of, and perpendicular to, a tangent to a circle passes through the centre
+of the circle</i>.</p>
+
+<p>§ 33. The rest of the book relates to angles connected with a
+circle, viz. angles which have the vertex either at the centre or
+on the circumference, and which are called respectively angles
+at the centre and angles at the circumference. Between these
+two kinds of angles exists the important relation expressed as
+follows:&mdash;</p>
+
+<p>Prop. 20. <i>The angle at the centre of a circle is double of the angle
+at the circumference on the same base, that is, on the same arc.</i></p>
+
+<p>This is of great importance for its consequences, of which the
+two following are the principal:&mdash;</p>
+
+<p>Prop. 21. <i>The angles in the same segment of a circle are equal to
+one another</i>;</p>
+
+<p>Prop. 22. <i>The opposite angles of any quadrilateral figure inscribed
+in a circle are together equal to two right angles.</i></p>
+
+<p>Further consequences are:&mdash;</p>
+
+<p>Prop. 23. <i>On the same straight line, and on the same side of it, there
+cannot be two similar segments of circles, not coinciding with one
+another</i>;</p>
+
+<p>Prop. 24. <i>Similar segments of circles on equal straight lines are
+equal to one another.</i></p>
+
+<p>The problem Prop. 25. <i>A segment of a circle being given to describe
+the circle of which it is a segment</i>, may be solved much more easily
+by aid of the construction described in relation to Prop. 1, III.,
+in § 27.</p>
+
+<p>§ 34. There follow four theorems connecting the angles at the
+centre, the arcs into which they divide the circumference, and the
+chords subtending these arcs. They are expressed for angles, arcs
+and chords in equal circles, but they hold also for angles, arcs and
+chords in the same circle.</p>
+
+<p>The theorems are:&mdash;</p>
+
+<p>Prop. 26. <i>In equal circles equal angles stand on equal arcs, whether
+they be at the centres or circumferences</i>;</p>
+
+<p>Prop. 27. (converse to Prop. 26). <i>In equal circles the angles which
+stand on equal arcs are equal to one another, whether they be at the
+centres or the circumferences</i>;</p>
+
+<p>Prop. 28. <i>In equal circles equal straight lines</i> (equal chords) <i>cut
+off equal arcs, the greater equal to the greater, and the less equal to
+the less</i>;</p>
+
+<p>Prop. 29 (converse to Prop. 28). <i>In equal circles equal arcs are
+subtended by equal straight lines.</i></p>
+
+<p>§ 35. Other important consequences of Props. 20-22 are:&mdash;</p>
+
+<p>Prop. 31. <i>In a circle the angle in a semicircle is a right angle;
+but the angle in a segment greater than a semicircle is less than a right
+angle; and the angle in a segment less than a semicircle is greater than
+a right angle</i>;</p>
+
+<p>Prop. 32. <i>If a straight line touch a circle, and from the point of
+contact a straight line be drawn cutting the circle, the angles which
+this line makes with the line touching the circle shall be equal to the
+angles which are in the alternate segments of the circle.</i></p>
+
+<p>§ 36. Propositions 30, 33, 34, contain problems which are solved
+by aid of the propositions preceding them:&mdash;</p>
+
+<p>Prop. 30. <i>To bisect a given arc, that is, to divide it into two equal
+parts</i>;</p>
+
+<p>Prop. 33. <i>On a given straight line to describe a segment of a circle
+containing an angle equal to a given rectilineal angle</i>;</p>
+
+<p>Prop. 34. <i>From a given circle to cut off a segment containing an
+angle equal to a given rectilineal angle</i>.</p>
+
+<p>§ 37. If we draw chords through a point A within a circle, they
+will each be divided by A into two segments. Between these segments
+the law holds that the rectangle contained by them has the
+same area on whatever chord through A the segments are taken.
+The value of this rectangle changes, of course, with the position
+of A.</p>
+
+<p>A similar theorem holds if the point A be taken without the circle.
+On every straight line through A, which cuts the circle in two points
+B and C, we have two segments AB and AC, and the rectangles
+contained by them are again equal to one another, and equal to the
+square on a tangent drawn from A to the circle.</p>
+
+<p>The first of these theorems gives Prop. 35, and the second Prop.
+36, with its corollary, whilst Prop. 37, the last of Book III., gives
+the converse to Prop. 36. The first two theorems may be combined
+in one:&mdash;</p>
+
+<p><i>If through a point A in the plane of a circle a straight line be drawn
+cutting the circle in B and C, then the rectangle AB.AC has a constant
+value so long as the point A be fixed; and if from A a tangent AD can
+be drawn to the circle, touching at D, then the above rectangle equals the
+square on AD.</i></p>
+
+<p>Prop. 37 may be stated thus:&mdash;</p>
+
+<p><i>If from a point A without a circle a line be drawn cutting the circle
+in B and C, and another line to a point D on the circle, and AB.AC =
+AD², then the line AD touches the circle at D.</i></p>
+
+<p>It is not difficult to prove also the converse to the general proposition
+as above stated. This proposition and its converse may be
+expressed as follows:&mdash;</p>
+
+<p><i>If four points ABCD be taken on the circumference of a circle, and
+if the lines AB, CD, produced if necessary, meet at E, then</i></p>
+
+<p class="center">EA·EB = EC·ED;</p>
+
+<p class="noind"><i>and conversely, if this relation holds then the four points lie on a circle,
+that is, the circle drawn through three of them passes through the
+fourth.</i></p>
+
+<p>That a circle may always be drawn through three points, provided
+that they do not lie in a straight line, is proved only later on in
+Book IV.</p>
+
+<p><span class="pagenum"><a name="page682" id="page682"></a>682</span></p>
+
+<p class="pt2 center sc">Book IV.</p>
+
+<p>§ 38. The fourth book contains only problems, all relating to
+the construction of triangles and polygons inscribed in and circumscribed
+about circles, and of circles inscribed in or circumscribed
+about triangles and polygons. They are nearly all given for their
+own sake, and not for future use in the construction of figures, as
+are most of those in the former books. In seven definitions at the
+beginning of the book it is explained what is understood by figures
+inscribed in or described about other figures, with special reference
+to the case where one figure is a circle. Instead, however, of saying
+that one figure is described about another, it is now generally said
+that the one figure is circumscribed about the other. We may then
+state the definitions 3 or 4 thus:&mdash;</p>
+
+<p><i>Definition.</i>&mdash;A polygon is said to be inscribed in a circle, and the
+circle is said to be circumscribed about the polygon, if the vertices
+of the polygon lie in the circumference of the circle.</p>
+
+<p>And definitions 5 and 6 thus:&mdash;</p>
+
+<p><i>Definition.</i>&mdash;A polygon is said to be circumscribed about a circle,
+and a circle is said to be inscribed in a polygon, if the sides of the
+polygon are tangents to the circle.</p>
+
+<p>§ 39. The first problem is merely constructive. It requires to
+draw in a given circle a chord equal to a given straight line, which
+is not greater than the diameter of the circle. The problem is not
+a determinate one, inasmuch as the chord may be drawn from any
+point in the circumference. This may be said of almost all problems
+in this book, especially of the next two. They are:&mdash;</p>
+
+<p>Prop. 2. <i>In a given circle to inscribe a triangle equiangular to a
+given triangle;</i></p>
+
+<p>Prop. 3. <i>About a given circle to circumscribe a triangle equiangular
+to a given triangle.</i></p>
+
+<p>§ 40. Of somewhat greater interest are the next problems, where
+the triangles are given and the circles to be found.</p>
+
+<p>Prop. 4. <i>To inscribe a circle in a given triangle.</i></p>
+
+<p>The result is that the problem has always a solution, viz. the
+centre of the circle is the point where the bisectors of two of the
+interior angles of the triangle meet. The solution shows, though
+Euclid does not state this, that the problem has but one solution;
+and also,</p>
+
+<p><i>The three bisectors of the interior angles of any triangle meet in a
+point, and this is the centre of the circle inscribed in the triangle.</i></p>
+
+<p>The solutions of most of the other problems contain also theorems.
+Of these we shall state those which are of special interest; Euclid
+does not state any one of them.</p>
+
+<p>§ 41. Prop. 5. <i>To circumscribe a circle about a given triangle.</i></p>
+
+<p>The one solution which always exists contains the following:&mdash;</p>
+
+<p><i>The three straight lines which bisect the sides of a triangle at right
+angles meet in a point, and this point is the centre of the circle circumscribed
+about the triangle.</i></p>
+
+<p>Euclid adds in a corollary the following property:&mdash;</p>
+
+<p>The centre of the circle circumscribed about a triangle lies within,
+on a side of, or without the triangle, according as the triangle is
+acute-angled, right-angled or obtuse-angled.</p>
+
+<p>§ 42. Whilst it is always possible to draw a circle which is inscribed
+in or circumscribed about a given triangle, this is not the case with
+quadrilaterals or polygons of more sides. Of those for which this
+is possible the regular polygons, <i>i.e.</i> polygons which have all their
+sides and angles equal, are the most interesting. In each of them a
+circle may be inscribed, and another may be circumscribed about it.</p>
+
+<p>Euclid does not use the word regular, but he describes the polygons
+in question as <i>equiangular</i> and <i>equilateral</i>. We shall use the name
+regular polygon. The regular triangle is equilateral, the regular
+quadrilateral is the square.</p>
+
+<p>Euclid considers the regular polygons of 4, 5, 6 and 15 sides.
+For each of the first three he solves the problems&mdash;(1) to inscribe
+such a polygon in a given circle; (2) to circumscribe it about a
+given circle; (3) to inscribe a circle in, and (4) to circumscribe a
+circle about, such a polygon.</p>
+
+<p>For the regular triangle the problems are not repeated, because
+more general problems have been solved.</p>
+
+<p>Props. 6, 7, 8 and 9 solve these problems for the square.</p>
+
+<p>The general problem of inscribing in a given circle a regular
+polygon of n sides depends upon the problem of dividing the circumference
+of a circle into n equal parts, or what comes to the same
+thing, of drawing from the centre of the circle n radii such that the
+angles between consecutive radii are equal, that is, to divide the
+space about the centre into n equal angles. Thus, if it is required
+to inscribe a square in a circle, we have to draw four lines from the
+centre, making the four angles equal. This is done by drawing
+two diameters at right angles to one another. The ends of these
+diameters are the vertices of the required square. If, on the other
+hand, tangents be drawn at these ends, we obtain a square circumscribed
+about the circle.</p>
+
+<p>§ 43. To construct a <i>regular pentagon</i>, we find it convenient first
+to construct a <i>regular decagon</i>. This requires to divide the space
+about the centre into ten equal angles. Each will be <span class="spp">1</span>&frasl;<span class="suu">10</span>th of a right
+angle, or <span class="spp">1</span>&frasl;<span class="suu">5</span>th of two right angles. If we suppose the decagon constructed,
+and if we join the centre to the end of one side, we get an
+isosceles triangle, where the angle at the centre equals <span class="spp">1</span>&frasl;<span class="suu">5</span>th of two
+right angles; hence each of the angles at the base will be <span class="spp">2</span>&frasl;<span class="suu">5</span>ths of
+two right angles, as all three angles together equal two right angles.
+Thus we have to construct an isosceles triangle, having the angle at
+the vertex equal to half an angle at the base. This is solved in
+Prop. 10, by aid of the problem in Prop. 11 of the second book. If
+we make the sides of this triangle equal to the radius of the given
+circle, then the base will be the side of the regular decagon inscribed
+in the circle. This side being known the decagon can be constructed,
+and if the vertices are joined alternately, leaving out half their
+number, we obtain the regular pentagon. (Prop. 11.)</p>
+
+<p>Euclid does not proceed thus. He wants the pentagon before
+the decagon. This, however, does not change the real nature of
+his solution, nor does his solution become simpler by not mentioning
+the decagon.</p>
+
+<p>Once the regular pentagon is inscribed, it is easy to circumscribe
+another by drawing tangents at the vertices of the inscribed pentagon.
+This is shown in Prop. 12.</p>
+
+<p>Props. 13 and 14 teach how a circle may be inscribed in or circumscribed
+about any given regular pentagon.</p>
+
+<p>§ 44. The <i>regular hexagon</i> is more easily constructed, as shown
+in Prop. 15. The result is that the side of the regular hexagon
+inscribed in a circle is equal to the radius of the circle.</p>
+
+<p>For this polygon the other three problems mentioned are not
+solved.</p>
+
+<p>§ 45. The book closes with Prop. 16. To inscribe a regular
+quindecagon in a given circle. If we inscribe a regular pentagon
+and a regular hexagon in the circle, having one vertex in common,
+then the arc from the common vertex to the next vertex of the
+pentagon is <span class="spp">1</span>&frasl;<span class="suu">5</span>th of the circumference, and to the next vertex of the
+hexagon is <span class="spp">1</span>&frasl;<span class="suu">6</span>th of the circumference. The difference between these
+arcs is, therefore, <span class="spp">1</span>&frasl;<span class="suu">5</span> &minus; <span class="spp">1</span>&frasl;<span class="suu">6</span> = <span class="spp">1</span>&frasl;<span class="suu">30</span>th of the circumference. The latter may,
+therefore, be divided into thirty, and hence also in fifteen equal parts,
+and the regular quindecagon be described.</p>
+
+<p>§ 46. We conclude with a few theorems about regular polygons
+which are not given by Euclid.</p>
+
+<p><i>The straight lines perpendicular to and bisecting the sides of any
+regular polygon meet in a point. The straight lines bisecting the angles
+in the regular polygon meet in the same point. This point is the centre
+of the circles circumscribed about and inscribed in the regular polygon.</i></p>
+
+<p>We can bisect any given arc (Prop. 30, III.). Hence we can divide
+a circumference into 2n equal parts as soon as it has been divided
+into n equal parts, or as soon as a regular polygon of n sides has been
+constructed. Hence&mdash;</p>
+
+<p><i>If a regular polygon of n sides has been constructed, then a regular
+polygon of 2n sides, of 4n, of 8n sides, &amp;c., may also be constructed.</i>
+Euclid shows how to construct regular polygons of 3, 4, 5 and 15
+sides. It follows that we can construct regular polygons of</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcr">3,</td> <td class="tcr">6,</td> <td class="tcr">12,</td> <td class="tcr">24</td> <td class="tcc">sides</td></tr>
+<tr><td class="tcr">4,</td> <td class="tcr">8,</td> <td class="tcr">16,</td> <td class="tcr">32</td> <td class="tcc">&rdquo;</td></tr>
+<tr><td class="tcr">5,</td> <td class="tcr">10,</td> <td class="tcr">20,</td> <td class="tcr">40</td> <td class="tcc">&rdquo;</td></tr>
+<tr><td class="tcr">15,</td> <td class="tcr">30,</td> <td class="tcr">60,</td> <td class="tcr">120</td> <td class="tcc">&rdquo;</td></tr>
+</table>
+
+<p>The construction of any new regular polygon not included in one
+of these series will give rise to a new series. Till the beginning of the
+19th century nothing was added to the knowledge of regular polygons
+as given by Euclid. Then Gauss, in his celebrated <i>Arithmetic</i>,
+proved that every regular polygon of 2<span class="sp">n</span> + 1 sides may be constructed
+if this number 2<span class="sp">n</span> + 1 be prime, and that no others except those
+with 2<span class="sp">m</span> (2<span class="sp">n</span> + 1) sides can be constructed by elementary methods.
+This shows that regular polygons of 7, 9, 13 sides cannot thus be
+constructed, but that a regular polygon of 17 sides is possible; for
+17 = 2<span class="sp">4</span> + 1. The next polygon is one of 257 sides. The construction
+becomes already rather complicated for 17 sides.</p>
+
+<p class="pt2 center sc">Book V.</p>
+
+<p>§ 47. The fifth book of the <i>Elements</i> is not exclusively geometrical.
+It contains the theory of ratios and proportion of quantities in
+general. The treatment, as here given, is admirable, and in every
+respect superior to the algebraical method by which Euclid&rsquo;s theory
+is now generally replaced. We shall treat the subject in order to
+show why the usual algebraical treatment of proportion is not really
+sound. We begin by quoting those definitions at the beginning of
+Book V. which are most important. These definitions have given
+rise to much discussion.</p>
+
+<p>The only definitions which are essential for the fifth book are
+Defs. 1, 2, 4, 5, 6 and 7. Of the remainder 3, 8 and 9 are more
+than useless, and probably not Euclid&rsquo;s, but additions of later editors,
+of whom Theon of Alexandria was the most prominent. Defs. 10
+and 11 belong rather to the sixth book, whilst all the others are
+merely nominal. The really important ones are 4, 5, 6 and 7.</p>
+
+<p>§ 48. To define a magnitude is not attempted by Euclid. The
+first two definitions state what is meant by a &ldquo;part,&rdquo; that is, a
+submultiple or measure, and by a &ldquo;multiple&rdquo; of a given magnitude.
+The meaning of Def. 4 is that two given quantities can have
+a ratio to one another only in case that they are comparable as to
+their magnitude, that is, if they are of the same kind.</p>
+
+<p>Def. 3, which is probably due to Theon, professes to define a ratio,
+but is as meaningless as it is uncalled for, for all that is wanted is
+given in Defs. 5 and 7.</p>
+
+<p>In Def. 5 it is explained what is meant by saying that two magnitudes
+have the same ratio to one another as two other magnitudes,
+<span class="pagenum"><a name="page683" id="page683"></a>683</span>
+and in Def. 7 what we have to understand by a greater or a less ratio.
+The 6th definition is only nominal, explaining the meaning of the
+word <i>proportional</i>.</p>
+
+<p>Euclid represents magnitudes by lines, and often denotes them
+either by single letters or, like lines, by two letters. We shall use
+only single letters for the purpose. If a and b denote two magnitudes
+of the same kind, their ratio will be denoted by a : b; if c and d are
+two other magnitudes of the same kind, but possibly of a different
+kind from a and b, then if c and d have the same ratio to one another
+as a and b, this will be expressed by writing&mdash;</p>
+
+<p class="center">a : b :: c : d.</p>
+
+<p>Further, if m is a (whole) number, ma shall denote the multiple
+of a which is obtained by taking it m times.</p>
+
+<p>§ 49. The whole theory of ratios is based on Def. 5.</p>
+
+<p>Def. 5. <i>The first of four magnitudes is said to have the same ratio
+to the second that the third has to the fourth when, any equimultiples
+whatever of the first and the third being taken, and any equimultiples
+whatever of the second and the fourth, if the multiple of the first be less
+than that of the second, the multiple of the third is also less than that of
+the fourth; and if the multiple of the first is equal to that of the second,
+the multiple of the third is also equal to that of the fourth; and if the
+multiple of the first is greater than that of the second, the multiple of
+the third is also greater than that of the fourth.</i></p>
+
+<p>It will be well to show at once in an example how this definition
+can be used, by proving the first part of the first proposition in the
+sixth book. <i>Triangles of the same altitude are to one another as
+their bases</i>, or if a and b are the bases, and &alpha; and &beta; the areas, of two
+triangles which have the same altitude, then a : b :: &alpha; : &beta;.</p>
+
+<p>To prove this, we have, according to Definition 5, to show&mdash;</p>
+
+<table class="reg" summary="poem"><tr><td> <div class="poemr">
+  <p>if ma &gt; nb, then m&alpha; &gt; n&beta;,</p>
+  <p>if ma = nb, then m&alpha; = n&beta;,</p>
+  <p>if ma &lt; nb, then m&alpha; &lt; n&beta;.</p>
+</div> </td></tr></table>
+
+<p class="noind">That this is true is in our case easily seen. We may suppose that
+the triangles have a common vertex, and their bases in the same
+line. We set off the base a along the line containing the bases
+m times; we then join the different parts of division to the vertex,
+and get m triangles all equal to &alpha;. The triangle on ma as base equals,
+therefore, m&alpha;. If we proceed in the same manner with the base b,
+setting it off n times, we find that the area of the triangle on the
+base nb equals n&beta;, the vertex of all triangles being the same. But
+if two triangles have the same altitude, then their areas are equal
+if the bases are equal; hence m&alpha; = n&beta; if ma = nb, and if their bases
+are unequal, then that has the greater area which is on the greater
+base; in other words, m&alpha; is greater than, equal to, or less than
+n&beta;, according as ma is greater than, equal to, or less than nb, which
+was to be proved.</p>
+
+<p>§ 50. It will be seen that even in this example it does not become
+evident what a ratio really is. It is still an open question whether
+ratios are magnitudes which we can compare. We do not know
+whether the ratio of two lines is a magnitude of the same kind as the
+ratio of two areas. Though we might say that Def. 5 defines <i>equal
+</i>ratios, still we do not know whether they are equal in the sense of
+the axiom, that two things which are equal to a third are equal to
+one another. That this is the case requires a proof, and until this
+proof is given we shall use the :: instead of the sign = , which, however,
+we shall afterwards introduce.</p>
+
+<p>As soon as it has been established that all ratios are like magnitudes,
+it becomes easy to show that, in some cases at least, they
+are numbers. This step was never made by Greek mathematicians.
+They distinguished always most carefully between continuous
+magnitudes and the discrete series of numbers. In modern times
+it has become the custom to ignore this difference.</p>
+
+<p>If, in determining the ratio of two lines, a common measure can
+be found, which is contained m times in the first, and n times in
+the second, then the ratio of the two lines equals the ratio of the
+two numbers m : n. This is shown by Euclid in Prop. 5, X. But the
+ratio of two numbers is, as a rule, a fraction, and the Greeks did
+not, as we do, consider fractions as numbers. Far less had they
+any notion of introducing irrational numbers, which are neither
+whole nor fractional, as we are obliged to do if we wish to say that
+all ratios are numbers. The incommensurable numbers which are
+thus introduced as ratios of incommensurable quantities are nowadays
+as familiar to us as fractions; but a proof is generally omitted
+that we may apply to them the rules which have been established
+for rational numbers only. Euclid&rsquo;s treatment of ratios avoids this
+difficulty. His definitions hold for commensurable as well as for
+incommensurable quantities. Even the notion of incommensurable
+quantities is avoided in Book V. But he proves that the more
+elementary rules of algebra hold for ratios. We shall state all
+his propositions in that algebraical form to which we are now accustomed.
+This may, of course, be done without changing the character
+of Euclid&rsquo;s method.</p>
+
+<p>§. 51. Using the notation explained above we express the first
+propositions as follows:&mdash;</p>
+
+<p>Prop. 1. If</p>
+
+<p class="center">a = ma&prime;, b = mb&prime;, c = mc&prime;,</p>
+
+<p class="noind">then</p>
+
+<p class="center">a + b + c = m(a&prime; + b&prime; + c&prime;).</p>
+
+<p>Prop. 2. If</p>
+
+<table class="reg" summary="poem"><tr><td> <div class="poemr">
+<p>a = mb, and c = md,</p>
+<p>e = nb, and f = nd,</p>
+</div> </td></tr></table>
+
+<p class="noind">then a + e is the same multiple of b as c + f is of d, viz.:&mdash;</p>
+
+<p class="center">a + e = (m + n)b, and c + f = (m + n)d.</p>
+
+<p>Prop. 3. If a = mb, c = md, then is na the same multiple of b
+that nc is of d, viz. na = nmb, nc = nmd.</p>
+
+<p>Prop. 4. If</p>
+
+<p class="center">a : b :: c : d,</p>
+
+<p class="noind">then</p>
+
+<p class="center">ma : nb :: mc : nd.</p>
+
+<p>Prop. 5. If</p>
+
+<p class="center">a = mb, and c = md,</p>
+
+<p class="noind">then</p>
+
+<p class="center">a &minus; c = m(b &minus; d).</p>
+
+<p>Prop. 6. If</p>
+
+<p class="center">a = mb, c = md,</p>
+
+<p class="noind">then are a &minus; nb and c &minus; nd either equal to, or equimultiples of, b
+and d, viz. a &minus; nb = (m &minus; n)b and c &minus; nd = (m &minus; n)d, where m &minus; n may
+be unity.</p>
+
+<p>All these propositions relate to <i>equimultiples</i>. Now follow propositions
+about ratios which are compared as to their magnitude.</p>
+
+<p>§ 52. Prop. 7. If a = b, then a : c :: b : c and c : a :: c : b.</p>
+
+<p>The proof is simply this. As a = b we know that ma = mb; therefore
+if</p>
+
+<p class="center">ma &gt; nc, then mb &gt; nc,</p>
+
+<p class="noind">if</p>
+
+<p class="center">ma = nc, then mb = nc,</p>
+
+<p class="noind">if</p>
+
+<p class="center">ma &lt; nc, then mb &lt; nc,</p>
+
+<p class="noind">therefore the first proportion holds by Definition 5.</p>
+
+<p>Prop. 8. If</p>
+
+<p class="center">a &gt; b, then a : c &gt; b : c,</p>
+
+<p class="noind">and</p>
+
+<p class="center">c : a &lt; c : b.</p>
+
+<p class="noind">The proof depends on Definition 7.</p>
+
+<p>Prop. 9 (converse to Prop. 7). If</p>
+
+<p class="center">a : c :: b : c,</p>
+
+<p class="noind">or if</p>
+
+<p class="center">c : a :: c : b, then a = b.</p>
+
+<p>Prop. 10 (converse to Prop. 8). If</p>
+
+<p class="center">a : c &gt; b : c, then a &gt; b,</p>
+
+<p class="noind">and if</p>
+
+<p class="center">c : a &lt; c : b, then a &lt; b.</p>
+
+<p>Prop. 11. If</p>
+
+<p class="center">a : b :: c : d,</p>
+
+<p class="noind">and</p>
+
+<p class="center">a : b :: e : f,</p>
+
+<p class="noind">then</p>
+
+<p class="center">c : d :: e : f.</p>
+
+<p>In words, <i>if too ratios are equal to a third, they are equal to one
+another</i>. After these propositions have been proved, we have a
+right to consider a ratio as a <i>magnitude</i>, for only now can we consider
+a ratio as something for which the axiom about magnitudes
+holds: things which are equal to a third are equal to one another.</p>
+
+<p>We shall indicate this by writing in future the sign = instead
+of ::. The remaining propositions, which explain themselves, may
+then be stated as follows:</p>
+
+<p>§ 53. Prop. 12. If</p>
+
+<p class="center">a : b = c : d = e : f,</p>
+
+<p class="noind">then</p>
+
+<p class="center">a + c + e : b + d + f = a : b.</p>
+
+<p>Prop. 13. If</p>
+
+<p class="center">a : b = c : d and c : d &gt; e : f,</p>
+
+<p class="noind">then</p>
+
+<p class="center">a : b &gt; e : f.</p>
+
+<p>Prop. 14. If</p>
+
+<p class="center">a : b = c : d, and a &gt; c, then b &gt; d.</p>
+
+<p>Prop. 15. Magnitudes have the same ratio to one another that
+their equimultiples have&mdash;</p>
+
+<p class="center">ma : mb = a : b.</p>
+
+<p>Prop. 16. If a, b, c, d are magnitudes of the same kind, and if</p>
+
+<p class="center">a : b = c : d,</p>
+
+<p class="noind">then</p>
+
+<p class="center">a : c = b : d.</p>
+
+<p>Prop. 17. If</p>
+
+<p class="center">a + b : b = c + d : d,</p>
+
+<p class="noind">then</p>
+
+<p class="center">a : b = c : d.</p>
+
+<p>Prop. 18 (converse to 17). If</p>
+
+<p class="center">a : b = c : d</p>
+
+<p class="noind">then</p>
+
+<p class="center">a + b : b = c + d : d.</p>
+
+<p>Prop. 19. If a, b, c, d are quantities of the same kind, and if</p>
+
+<p class="center">a : b = c : d,</p>
+
+<p class="noind">then</p>
+
+<p class="center">a &minus; c : b &minus; d = a : b.</p>
+
+<p>§ 54. Prop. 20. <i>If there be three magnitudes, and another three,
+which have the same ratio, taken two and two, then if the first be greater
+than the third, the fourth shall be greater than the sixth: and if equal,
+equal; and if less, less.</i></p>
+
+<p>If we understand by</p>
+
+<p class="center">a : b : c : d : e : ... = a&prime; : b&prime; : c&prime; : d&prime; : e&prime; : ...</p>
+
+<p class="noind">that the ratio of any two consecutive magnitudes on the first side
+equals that of the corresponding magnitudes on the second side,
+we may write this theorem in symbols, thus:&mdash;</p>
+
+<p>If a, b, c be quantities of one, and d, e, f magnitudes of the same
+or any other kind, such that</p>
+
+<p class="center">a : b : c = d : e : f,</p>
+
+<p class="noind">and if</p>
+
+<p class="center">a &gt; c, then d &gt; f,</p>
+
+<p class="noind">but if</p>
+
+<p class="center">a = c, then d = f,</p>
+
+<p class="noind">and if</p>
+
+<p class="center">a &lt; c, then d &lt; f.</p>
+
+<p>Prop. 21. If</p>
+
+<p class="center">a : b = e : f and b : c = d : e,</p>
+
+<p class="noind">or if</p>
+
+<p class="center">a : b : c = 1/f : 1/e : 1/d,</p>
+
+<span class="pagenum"><a name="page684" id="page684"></a>684</span>
+
+<p class="noind">and if</p>
+
+<p class="center">a &gt; c, then d &gt; f,</p>
+
+<p class="noind">but if</p>
+
+<p class="center">a = c, then d = f,</p>
+
+<p class="noind">and if</p>
+
+<p class="center">a &lt; c, then d &lt; f.</p>
+
+<p>By aid of these two propositions the following two are proved.</p>
+
+<p>§ 55. Prop. 22. <i>If there be any number of magnitudes, and as
+many others, which have the same ratio, taken two and two in order,
+the first shall have to the last of the first magnitudes the same ratio
+which the first of the others has to the last.</i></p>
+
+<p>We may state it more generally, thus:</p>
+
+<p class="noind">If</p>
+
+<p class="center">a : b : c : d : e: ... = a&prime; : b&prime; : c&prime; : d&prime; : e&prime; : ... ,</p>
+
+<p class="noind">then not only have two consecutive, but any two magnitudes on
+the first side, the same ratio as the corresponding magnitudes on
+the other. For instance&mdash;</p>
+
+<p class="center">a : c = a&prime; : c&prime;; b : e = b&prime; : e&prime;, &amp;c.</p>
+
+<p>Prop. 23 we state only in symbols, viz.:&mdash;</p>
+
+<p class="center">a : b : c : d : e : ... = 1/a&prime; : 1/b&prime; : 1/c&prime; : 1/d&prime; : 1/e&prime; ...,</p>
+
+<p class="noind">then</p>
+
+<p class="center">a : c = c&prime; : a&prime;,<br />
+b : e = e&prime; : b&prime;,</p>
+
+<p class="noind">and so on.</p>
+
+<p>Prop. 24 comes to this: If a : b = c : d and e : b = f : d, then</p>
+
+<p class="center">a + e : b = c + f : d.</p>
+
+<p>Some of the proportions which are considered in the above propositions
+have special names. These we have omitted, as being of
+no use, since algebra has enabled us to bring the different operations
+contained in the propositions under a common point of view.</p>
+
+<p>§ 56. The last proposition in the fifth book is of a different
+character.</p>
+
+<p>Prop. 25. <i>If four magnitudes of the same kind be proportional,
+the greatest and least of them together shall be greater than the other
+two together.</i> In symbols&mdash;</p>
+
+<p>If a, b, c, d be magnitudes of the same kind, and if a : b = c : d,
+and if a is the greatest, hence d the least, then a + d &gt; b + c.</p>
+
+<p>§ 57. We return once again to the question. What is a ratio?
+We have seen that we may treat ratios as magnitudes, and that all
+ratios are magnitudes of the same kind, for we may compare any
+two as to their magnitude. It will presently be shown that ratios
+of lines may be considered as <i>quotients</i> of lines, so that a ratio appears
+as answer to the question, How often is one line contained in another?
+But the answer to this question is given by a number, at least in
+some cases, and in all cases if we admit incommensurable numbers.
+Considered from this point of view, we may say the fifth book of the
+<i>Elements</i> shows that some of the simpler algebraical operations
+hold for incommensurable numbers. In the ordinary algebraical
+treatment of numbers this proof is altogether omitted, or given by
+a process of limits which does not seem to be natural to the subject.</p>
+
+<p class="pt2 center sc">Book VI.</p>
+
+<p>§ 58. The sixth book contains the theory of similar figures.
+After a few definitions explaining terms, the first proposition gives
+the first application of the theory of proportion.</p>
+
+<p>Prop. 1. <i>Triangles and parallelograms of the same altitude are to
+one another as their bases.</i></p>
+
+<p>The proof has already been considered in § 49.</p>
+
+<p>From this follows easily the important theorem</p>
+
+<p>Prop. 2. <i>If a straight line be drawn parallel to one of the sides
+of a triangle it shall cut the other sides, or those sides produced, proportionally;
+and if the sides or the sides produced be cut proportionally,
+the straight line which joins the points of section shall be parallel to
+the remaining side of the triangle.</i></p>
+
+<p>§ 59. The next proposition, together with one added by Simson
+as Prop. A, may be expressed more conveniently if we introduce a
+modern phraseology, viz. if in a line AB we assume a point C between
+A and B, we shall say that C divides AB internally in the ratio
+AC : CB; but if C be taken in the line AB produced, we shall say
+that AB is divided externally in the ratio AC : CB.</p>
+
+<p>The two propositions then come to this:</p>
+
+<p>Prop. 3. <i>The bisector of an angle in a triangle divides the opposite
+side internally in a ratio equal to the ratio of the two sides including
+that angle;</i> and conversely, <i>if a line through the vertex of a triangle
+divide the base internally in the ratio of the two other sides, then that
+line bisects the angle at the vertex</i>.</p>
+
+<p>Simson&rsquo;s Prop. A. <i>The line which bisects an exterior angle of a
+triangle divides the opposite side externally in the ratio of the other
+sides;</i> and conversely, <i>if a line through the vertex of a triangle divide
+the base externally in the ratio of the sides, then it bisects an exterior
+angle at the vertex of the triangle</i>.</p>
+
+<p>If we combine both we have&mdash;</p>
+
+<p><i>The two lines which bisect the interior and exterior angles at one
+vertex of a triangle divide the opposite side internally and externally
+in the same ratio, viz. in the ratio of the other two sides.</i></p>
+
+<p>§ 60. The next four propositions contain the theory of similar
+triangles, of which four cases are considered. They may be stated
+together.</p>
+
+<p><i>Two triangles are similar</i>,&mdash;</p>
+
+<p>1. (Prop. 4). <i>If the triangles are equiangular:</i></p>
+
+<p>2. (Prop. 5). <i>If the sides of the one are proportional to those of
+the other</i>;</p>
+
+<p>3. (Prop. 6). <i>If two sides in one are proportional to two sides in
+the other, and if the angles contained by these sides are equal</i>;</p>
+
+<p>4. (Prop. 7). <i>If two sides in one are proportional to two sides in
+the other, if the angles opposite homologous sides are equal, and if
+the angles opposite the other homologous sides are both acute, both right
+or both obtuse; homologous sides being in each case those which are
+opposite equal angles</i>.</p>
+
+<p>An important application of these theorems is at once made to
+a right-angled triangle, viz.:&mdash;</p>
+
+<p>Prop. 8. <i>In a right-angled triangle, if a perpendicular be drawn
+from the right angle to the base, the triangles on each side of it are
+similar to the whole triangle, and to one another</i>.</p>
+
+<p><i>Corollary.</i>&mdash;From this it is manifest that the perpendicular
+drawn from the right angle of a right-angled triangle to the base
+is a mean proportional between the segments of the base, and also
+that each of the sides is a mean proportional between the base and
+the segment of the base adjacent to that side.</p>
+
+<p>§ 61. There follow four propositions containing problems, in
+language slightly different from Euclid&rsquo;s, viz.:&mdash;</p>
+
+<p>Prop. 9. <i>To divide a straight line into a given number of equal
+parts</i>.</p>
+
+<p>Prop. 10. <i>To divide a straight line in a given ratio</i>.</p>
+
+<p>Prop. 11. <i>To find a third proportional to two given straight lines</i>.</p>
+
+<p>Prop. 12. <i>To find a fourth proportional to three given straight
+lines</i>.</p>
+
+<p>Prop. 13. <i>To find a mean proportional between two given straight
+lines</i>.</p>
+
+<p>The last three may be written as equations with one unknown
+quantity&mdash;viz. if we call the given straight lines a, b, c, and the
+required line x, we have to find a line x so that</p>
+
+<p>Prop. 11.</p>
+
+<p class="center">a : b = b : x;</p>
+
+<p>Prop. 12.</p>
+
+<p class="center">a : b = c : x;</p>
+
+<p>Prop. 13.</p>
+
+<p class="center">a : x = x : b.</p>
+
+<p>We shall see presently how these may be written without the
+signs of ratios.</p>
+
+<p>§ 62. Euclid considers next proportions connected with parallelograms
+and triangles which are equal in area.</p>
+
+<p>Prop. 14. <i>Equal parallelograms which have one angle of the one
+equal to one angle of the other have their sides about the equal angles
+reciprocally proportional; and parallelograms which have one angle
+of the one equal to one angle of the other, and their sides about the equal
+angles reciprocally proportional, are equal to one another</i>.</p>
+
+<p>Prop. 15. <i>Equal triangles which have one angle of the one equal
+to one angle of the other, have their sides about the equal angles reciprocally
+proportional; and triangles which have one angle of the one equal
+to one angle of the other, and their sides about the equal angles reciprocally
+proportional, are equal to one another</i>.</p>
+
+<table class="flt" style="float: right; width: 320px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:269px; height:167px" src="images/img684.jpg" alt="" /></td></tr></table>
+
+<p>The latter proposition is really the same as the former, for if, as
+in the accompanying diagram,
+in the figure belonging to the
+former the two equal parallelograms
+AB and BC be bisected
+by the lines DF and EG, and
+if EF be drawn, we get the
+figure belonging to the latter.</p>
+
+<p>It is worth noticing that
+the lines FE and DG are
+parallel. We may state therefore
+the theorem&mdash;</p>
+
+<p><i>If two triangles are equal in
+area, and have one angle in the one vertically opposite to one angle
+in the other, then the two straight lines which join the remaining two
+vertices of the one to those of the other triangle are parallel</i>.</p>
+
+<p>§ 63. A most important theorem is</p>
+
+<p><i>Prop. 16. If four straight lines be proportionals, the rectangle
+contained by the extremes is equal to the rectangle contained by the
+means; and if the rectangle contained by the extremes be equal to the
+rectangle contained by the means, the four straight lines are proportionals</i>.</p>
+
+<p>In symbols, if a, b, c, d are the four lines, and</p>
+
+<p class="noind">if</p>
+
+<p class="center">a : b = c : d,</p>
+
+<p class="noind">then</p>
+
+<p class="center">ad = bc;</p>
+
+<p class="noind">and conversely, if</p>
+
+<p class="center">ad = bc,</p>
+
+<p class="noind">then</p>
+
+<p class="center">a : b = c : d,</p>
+
+<p class="noind">where ad and bc denote (as in § 20), the areas of the rectangles
+contained by a and d and by b and c respectively.</p>
+
+<p>This allows us to transform every proportion between four lines
+into an equation between two products.</p>
+
+<p>It shows further that the operation of forming a product of two
+lines, and the operation of forming their ratio are each the inverse
+of the other.</p>
+
+<p>If we now define a quotient a/b of two lines as the <i>number</i> which
+multiplied into b gives a, so that</p>
+
+<table class="math0" summary="math">
+<tr><td>a</td>
+<td rowspan="2">b = a,</td></tr>
+<tr><td class="denom">b</td></tr></table>
+
+<p><span class="pagenum"><a name="page685" id="page685"></a>685</span></p>
+
+<p class="noind">we see that from the equality of two quotients</p>
+
+<table class="math0" summary="math">
+<tr><td>a</td>
+<td rowspan="2">=</td> <td>c</td></tr>
+<tr><td class="denom">b</td> <td class="denom">d</td></tr></table>
+
+<p class="noind">follows, if we multiply both sides by bd,</p>
+
+<table class="math0" summary="math">
+<tr><td>a</td>
+<td rowspan="2">b·d =</td> <td>c</td>
+<td rowspan="2">d·b,</td></tr>
+<tr><td class="denom">b</td> <td class="denom">d</td></tr></table>
+
+<p class="center">ad = cb.</p>
+
+<p>But from this it follows, according to the last theorem, that</p>
+
+<p class="center">a : b = c : d.</p>
+
+<p>Hence we conclude that the quotient a/b and the ratio a : b are
+different forms of the same magnitude, only with this important
+difference that the quotient a/b would have a meaning only if a and
+b have a common measure, until we introduce incommensurable
+numbers, while the ratio a : b has always a meaning, and thus gives
+rise to the introduction of incommensurable numbers.</p>
+
+<p>Thus it is really the theory of ratios in the fifth book which enables
+us to extend the geometrical calculus given before in connexion
+with Book II. It will also be seen that if we write the ratios in
+Book V. as quotients, or rather as fractions, then most of the theorems
+state properties of quotients or of fractions.</p>
+
+<p>§ 64. Prop. 17. <i>If three straight lines are proportional the rectangle
+contained by the extremes is equal to the square on the mean;</i> and
+conversely, is only a special case of 16. After the problem, Prop.
+18, <i>On a given straight line to describe a rectilineal figure similar
+and similarly situated to a given rectilineal figure</i>, there follows another
+fundamental theorem:</p>
+
+<p>Prop. 19. <i>Similar triangles are to one another in the duplicate
+ratio of their homologous sides.</i> In other words, the areas of similar
+triangles are to one another as the squares on homologous sides.
+This is generalized in:</p>
+
+<p>Prop. 20. <i>Similar polygons may be divided into the same number
+of similar triangles, having the same ratio to one another that the
+polygons have; and the polygons are to one another in the duplicate
+ratio of their homologous sides.</i></p>
+
+<p>§ 65. Prop. 21. <i>Rectilineal figures which are similar to the same
+rectilineal figure are also similar to each other</i>, is an immediate consequence
+of the definition of similar figures. As similar figures
+may be said to be equal in &ldquo;shape&rdquo; but not in &ldquo;size,&rdquo; we may state
+it also thus:</p>
+
+<p>&ldquo;Figures which are equal in shape to a third are equal in shape
+to each other.&rdquo;</p>
+
+<p>Prop. 22. <i>If four straight lines be proportionals, the similar
+rectilineal figures similarly described on them shall also be proportionals;
+and if the similar rectilineal figures similarly described on four
+straight lines be proportionals, those straight lines shall be proportionals.</i></p>
+
+<p>This is essentially the same as the following:&mdash;</p>
+
+<p class="noind"><i>If</i></p>
+
+<p class="center">a : b = c : d,</p>
+
+<p class="noind"><i>then</i></p>
+
+<p class="center">a² : b² = c² : d².</p>
+
+<p>§ 66. Now follows a proposition which has been much discussed
+with regard to Euclid&rsquo;s exact meaning in saying that a ratio is
+<i>compounded</i> of two other ratios, viz.:</p>
+
+<p>Prop. 23. <i>Parallelograms which are equiangular to one another,
+have to one another the ratio which is compounded of the ratios of their
+sides.</i></p>
+
+<p>The proof of the proposition makes its meaning clear. In symbols
+the ratio a : c is compounded of the two ratios a : b and b : c, and if
+a : b = a&prime; : b&prime;, b : c = b&Prime; : c&Prime;, then a : c is compounded of a&prime; : b&prime; and
+b&Prime; : c&Prime;.</p>
+
+<p>If we consider the ratios as numbers, we may say that the one
+ratio is the product of those of which it is compounded, or in symbols,</p>
+
+<table class="math0" summary="math">
+<tr><td>a</td>
+<td rowspan="2">=</td> <td>a</td>
+<td rowspan="2">·</td> <td>b</td>
+<td rowspan="2">=</td> <td>a&prime;</td>
+<td rowspan="2">·</td> <td>b&Prime;</td>
+<td rowspan="2">, if</td> <td>a</td>
+<td rowspan="2">=</td> <td>a&prime;</td>
+<td rowspan="2">and</td> <td>b</td>
+<td rowspan="2">=</td> <td>b&Prime;</td>
+<td rowspan="2">.</td></tr>
+<tr><td class="denom">c</td> <td class="denom">b</td>
+<td class="denom">c</td> <td class="denom">b&prime;</td>
+<td class="denom">c&Prime;</td> <td class="denom">b</td>
+<td class="denom">b&prime;</td> <td class="denom">c</td>
+<td class="denom">c&Prime;</td></tr></table>
+
+<p>The theorem in Prop. 23 is the foundation of all mensuration of
+areas. From it we see at once that two rectangles have the ratio
+of their areas compounded of the ratios of their sides.</p>
+
+<p>If A is the area of a rectangle contained by a and b, and B that
+of a rectangle contained by c and d, so that A = ab, B = cd, then
+A : B = ab : cd, and this is, the theorem says, compounded of the
+ratios a : c and b : d. In forms of quotients,</p>
+
+<table class="math0" summary="math">
+<tr><td>a</td>
+<td rowspan="2">·</td> <td>b</td>
+<td rowspan="2">=</td> <td>ab</td>
+<td rowspan="2">.</td></tr>
+<tr><td class="denom">c</td> <td class="denom">d</td>
+<td class="denom">cd</td></tr></table>
+
+<p>This shows how to multiply quotients in our geometrical calculus.</p>
+
+<p>Further, <i>Two triangles have the ratios of their areas compounded
+of the ratios of their bases and their altitude.</i> For a triangle is equal
+in area to half a parallelogram which has the same base and the
+same altitude.</p>
+
+<p>§ 67. To bring these theorems to the form in which they are usually
+given, we assume a straight line u as our unit of length (generally
+an inch, a foot, a mile, &amp;c.), and determine the number &alpha; which
+expresses how often u is contained in a line a, so that &alpha; denotes the
+ratio a : u whether commensurable or not, and that a = &alpha;u. We
+call this number &alpha; the numerical value of a. If in the same manner
+&beta; be the numerical value of a line b we have</p>
+
+<p class="center">a : b = &alpha; : &beta;;</p>
+
+<p class="noind">in words: <i>The ratio of two lines (and of two like quantities in general)
+is equal to that of their numerical values.</i></p>
+
+<p>This is easily proved by observing that a = &alpha;u, b = &beta;u, therefore
+a : b = &alpha;u : &beta;u, and this may without difficulty be shown to equal &alpha; : &beta;.</p>
+
+<p>If now a, b be base and altitude of one, a&prime;, b&prime; those of another
+parallelogram, &alpha;, &beta; and &alpha;&prime;, &beta;&prime; their numerical values respectively,
+and A, A&prime; their areas, then</p>
+
+<table class="math0" summary="math">
+<tr><td>A</td>
+<td rowspan="2">=</td> <td>a</td>
+<td rowspan="2">·</td> <td>b</td>
+<td rowspan="2">=</td> <td>&alpha;</td>
+<td rowspan="2">·</td> <td>&beta;</td>
+<td rowspan="2">=</td> <td>&alpha;&beta;</td>
+<td rowspan="2">.</td></tr>
+<tr><td class="denom">A&prime;</td> <td class="denom">a&prime;</td>
+<td class="denom">b&prime;</td> <td class="denom">&alpha;&prime;</td>
+<td class="denom">&beta;&prime;</td> <td class="denom">&alpha;&prime;&beta;&prime;</td></tr></table>
+
+<p class="noind">In words: <i>The areas of two parallelograms are to each other as the
+products of the numerical values of their bases and altitudes.</i></p>
+
+<p>If especially the second parallelogram is the unit square, <i>i.e.</i> a
+square on the unit of length, then &alpha;&prime; = &beta;&prime; = 1, A&prime; = u², and we have</p>
+
+<table class="math0" summary="math">
+<tr><td>A</td>
+<td rowspan="2">= &alpha;&beta; or A = &alpha;&beta;·u².</td></tr>
+<tr><td class="denom">A&prime;</td></tr></table>
+
+<p>This gives the theorem: The number of unit squares contained in
+a parallelogram equals the product of the numerical values of base
+and altitude, and similarly the number of unit squares contained in
+a triangle equals half the product of the numerical values of base
+and altitude.</p>
+
+<p>This is often stated by saying that the area of a parallelogram is
+equal to the product of the base and the altitude, meaning by this
+product the product of the numerical values, and not the product as
+defined above in § 20.</p>
+
+<p>§ 68. Propositions 24 and 26 relate to parallelograms about
+diagonals, such as are considered in Book I., 43. They are&mdash;</p>
+
+<p>Prop. 24. <i>Parallelograms about the diameter of any parallelogram
+are similar to the whole parallelogram and to one another</i>; and its
+converse (Prop. 26), <i>If two similar parallelograms have a common
+angle, and be similarly situated, they are about the same diameter.</i></p>
+
+<p>Between these is inserted a problem.</p>
+
+<p>Prop. 25. <i>To describe a rectilineal figure which shall be similar to
+one given rectilinear figure, and equal to another given rectilineal
+figure</i>.</p>
+
+<p>§ 69. Prop. 27 contains a theorem relating to the theory of
+maxima and minima. We may state it thus:</p>
+
+<p>Prop. 27. <i>If a parallelogram be divided into two by a straight line
+cutting the base, and if on half the base another parallelogram be constructed
+similar to one of those parts, then this third parallelogram is
+greater than the other part.</i></p>
+
+<p>Of far greater interest than this general theorem is a special case
+of it, where the parallelograms are changed into rectangles, and
+where one of the parts into which the parallelogram is divided is
+made a square; for then the theorem changes into one which is
+easily recognized to be identical with the following:&mdash;</p>
+
+<p><i>Of all rectangles which have the same perimeter the square has the
+greatest area.</i></p>
+
+<p>This may also be stated thus:&mdash;</p>
+
+<p><i>Of all rectangles which have the same area the square has the least
+perimeter.</i></p>
+
+<p>§ 70. The next three propositions contain problems which may
+be said to be solutions of quadratic equations. The first two are,
+like the last, involved in somewhat obscure language. We transcribe
+them as follows:</p>
+
+<p><i>Problem</i>.&mdash;To describe on a given base a parallelogram, and to
+divide it either internally (Prop. 28) or externally (Prop. 29) from
+a point on the base into two parallelograms, of which the one has
+a given size (is equal in area to a given figure), whilst the other
+has a given shape (is similar to a given parallelogram).</p>
+
+<p>If we express this again in symbols, calling the given base a, the
+one part x, and the altitude y, we have to determine x and y in the
+first case from the equations</p>
+
+<p class="center">(a &minus; x)y = k²,</p>
+
+<table class="math0" summary="math">
+<tr><td>x</td>
+<td rowspan="2">=</td> <td>p</td>
+<td rowspan="2">,</td></tr>
+<tr><td class="denom">y</td> <td class="denom">q</td></tr></table>
+
+<p class="noind">k² being the given size of the first, and p and q the base and altitude
+of the parallelogram which determine the shape of the second of the
+required parallelograms.</p>
+
+<p>If we substitute the value of y, we get</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">(a &minus; x)x =</td> <td>pk²</td>
+<td rowspan="2">,</td></tr>
+<tr><td class="denom">q</td></tr></table>
+
+<p class="noind">or,</p>
+
+<p class="center">ax &minus; x² = b²,</p>
+
+<p class="noind">where a and b are known quantities, taking b² = pk²/q.</p>
+
+<p>The second case (Prop. 29) gives rise, in the same manner, to the
+quadratic</p>
+
+<p class="center">ax + x² = b².</p>
+
+<p>The next problem&mdash;</p>
+
+<p>Prop. 30. <i>To cut a given straight line in extreme and mean ratio</i>,
+leads to the equation</p>
+
+<p class="center">ax + x² = a².</p>
+
+<p><span class="pagenum"><a name="page686" id="page686"></a>686</span></p>
+
+<p>This is, therefore, only a special case of the last, and is, besides,
+an old acquaintance, being essentially the same problem as that
+proposed in II. 11.</p>
+
+<p>Prop. 30 may therefore be solved in two ways, either by aid of
+Prop. 29 or by aid of II. 11. Euclid gives both solutions.</p>
+
+<p>§ 71. Prop. 31 (Theorem). <i>In any right-angled triangle, any
+rectilineal figure described on the side subtending the right angle is
+equal to the similar and similarly-described figures on the sides containing
+the right angle</i>,&mdash;is a pretty generalization of the theorem of
+Pythagoras (I. 47).</p>
+
+<p>Leaving out the next proposition, which is of little interest, we
+come to the last in this book.</p>
+
+<p>Prop. 33. <i>In equal circles angles, whether at the centres or the
+circumferences, have the same ratio which the arcs on which they stand
+have to one another; so also have the sectors</i>.</p>
+
+<p>Of this, the part relating to angles at the centre is of special
+importance; it enables us to measure angles by arcs.</p>
+
+<p>With this closes that part of the <i>Elements</i> which is devoted to
+the study of figures in a plane.</p>
+
+<p class="pt2 center sc">Book XI.</p>
+
+<p>§ 72. In this book figures are considered which are not confined
+to a plane, viz. first relations between lines and planes in space,
+and afterwards properties of solids.</p>
+
+<p>Of new definitions we mention those which relate to the perpendicularity
+and the inclination of lines and planes.</p>
+
+<p>Def. 3. <i>A straight line is perpendicular, or at right angles, to a
+plane when it makes right angles with every straight line meeting it
+in that plane</i>.</p>
+
+<p>The definition of perpendicular planes (Def. 4) offers no difficulty.
+Euclid defines the inclination of lines to planes and of planes to
+planes (Defs. 5 and 6) by aid of plane angles, included by straight
+lines, with which we have been made familiar in the first books.</p>
+
+<p>The other important definitions are those of parallel planes,
+which never meet (Def. 8), and of solid angles formed by three or
+more planes meeting in a point (Def. 9).</p>
+
+<p>To these we add the definition of a line parallel to a plane as a
+line which does not meet the plane.</p>
+
+<p>§ 73. Before we investigate the contents of Book XI., it will be
+well to recapitulate shortly what we know of planes and lines from
+the definitions and axioms of the first book. There a plane has
+been defined as a surface which has the property that every straight
+line which joins two points in it lies altogether in it. This is equivalent
+to saying that a straight line which has two points in a plane
+has all points in the plane. Hence, a straight line which does not
+lie in the plane cannot have more than one point in common with
+the plane. This is virtually the same as Euclid&rsquo;s Prop. 1, viz.:&mdash;</p>
+
+<p>Prop. 1. <i>One part of a straight line cannot be in a plane and another
+part without it</i>.</p>
+
+<p>It also follows, as was pointed out in § 3, in discussing the definitions
+of Book I., that a plane is determined already by one straight
+line and a point without it, viz. if all lines be drawn through the
+point, and cutting the line, they will form a plane.</p>
+
+<p>This may be stated thus:&mdash;</p>
+
+<p><i>A plane is determined</i>&mdash;</p>
+
+<p>1st, <i>By a straight line and a point which does not lie on it;</i></p>
+
+<p>2nd, <i>By three points which do not lie in a straight line</i>; for if two
+of these points be joined by a straight line we have case 1;</p>
+
+<p>3rd, <i>By two intersecting straight lines</i>; for the point of intersection
+and two other points, one in each line, give case 2;</p>
+
+<p>4th, <i>By two parallel lines</i> (Def. 35, I.).</p>
+
+<p>The third case of this theorem is Euclid&rsquo;s</p>
+
+<p>Prop. 2. <i>Two straight lines which cut one another are in one plane,
+and three straight lines which meet one another are in one plane</i>.</p>
+
+<p>And the fourth is Euclid&rsquo;s</p>
+
+<p>Prop. 7. <i>If two straight lines be parallel, the straight line drawn
+from any point in one to any point in the other is in the same plane
+with the parallels</i>. From the definition of a plane further follows</p>
+
+<p>Prop. 3. <i>If two planes cut one another, their common section is a
+straight line</i>.</p>
+
+<p>§ 74. Whilst these propositions are virtually contained in the
+definition of a plane, the next gives us a new and fundamental
+property of space, showing at the same time that it is possible to
+have a straight line perpendicular to a plane, according to Def. 3.
+It states&mdash;</p>
+
+<p>Prop. 4. <i>If a straight line is perpendicular to two straight lines
+in a plane which it meets, then it is perpendicular to all lines in the plane
+which it meets, and hence it is perpendicular to the plane</i>.</p>
+
+<p>Def. 3 may be stated thus: If a straight line is perpendicular
+to a plane, then it is perpendicular to every line in the plane which
+it meets. The converse to this would be</p>
+
+<p><i>All straight lines which meet a given straight line in the same point,
+and are perpendicular to it, lie in a plane which is perpendicular to
+that line</i>.</p>
+
+<p>This Euclid states thus:</p>
+
+<p>Prop. 5. <i>If three straight lines meet all at one point, and a straight
+line stands at right angles to each of them at that point, the three straight
+lines shall be in one and the same plane</i>.</p>
+
+<p>§ 75. There follow theorems relating to the theory of parallel
+lines in space, viz.:&mdash;</p>
+
+<p>Prop. 6. <i>Any two lines which are perpendicular to the same plane
+are parallel to each other;</i> and conversely</p>
+
+<p>Prop. 8. <i>If of two parallel straight lines one is perpendicular to a
+plane, the other is so also.</i></p>
+
+<p>Prop. 7. <i>If two straight lines are parallel, the straight line which
+joins any point in one to any point in the other is in the same plane as
+the parallels.</i> (See above, § 73.)</p>
+
+<p>Prop. 9. <i>Two straight lines which are each of them parallel to the
+same straight line, and not in the same plane with it, are parallel to
+one another</i>; where the words, &ldquo;and not in the same plane with
+it,&rdquo; may be omitted, for they exclude the case of three parallels
+in a plane, which has been proved before; and</p>
+
+<p>Prop. 10. <i>If two angles in different planes have the two limits of
+the one parallel to those of the other, then the angles are equal.</i> That
+their planes are parallel is shown later on in Prop. 15.</p>
+
+<p>This theorem is not necessarily true, for the angles in question
+may be supplementary; but then the one angle will be equal to
+that which is adjacent and supplementary to the other, and this
+latter angle will also have its limits parallel to those of the first.</p>
+
+<p>From this theorem it follows that if we take any two straight
+lines in space which do not meet, and if we draw through any point
+P in space two lines parallel to them, then the angle included by
+these lines will always be the same, whatever the position of the
+point P may be. This angle has in modern times been called the
+angle between the given lines:&mdash;</p>
+
+<p><i>By the angles between two not intersecting lines we understand the
+angles which two intersecting lines include that are parallel respectively
+to the two given lines.</i></p>
+
+<p>§ 76. It is now possible to solve the following two problems:&mdash;</p>
+
+<p><i>To draw a straight line perpendicular to a given plane from a given
+point which lies</i></p>
+
+<p>1. <i>Not in the plane</i> (Prop. 11).</p>
+
+<p>2. <i>In the plane</i> (Prop. 12).</p>
+
+<p>The second case is easily reduced to the first&mdash;viz. if by aid of
+the first we have drawn any perpendicular to the plane from some
+point without it, we need only draw through the given point in the
+plane a line parallel to it, in order to have the required perpendicular
+given. The solution of the first part is of interest in itself. It depends
+upon a construction which may be expressed as a theorem.</p>
+
+<p><i>If from a point A without a plane a perpendicular AB be drawn to the
+plane, and if from the foot B of this perpendicular another perpendicular
+BC be drawn to any straight line in the plane, then the straight line
+joining A to the foot C of this second perpendicular will also be perpendicular
+to the line in the plane.</i></p>
+
+<p>The theory of perpendiculars to a plane is concluded by the
+theorem&mdash;</p>
+
+<p>Prop. 13. <i>Through any point in space, whether in or without a
+plane, only one straight line can be drawn perpendicular to the plane.</i></p>
+
+<p>§ 77. The next four propositions treat of parallel planes. It is
+shown <i>that planes which have a common perpendicular are parallel</i>
+(Prop. 14); <i>that two planes are parallel if two intersecting straight
+lines in the one are parallel respectively to two straight lines in the
+other plane</i> (Prop. 15); <i>that parallel planes are cut by any plane in
+parallel straight lines</i> (Prop. 16); and lastly, <i>that any two straight
+lines are cut proportionally by a series of parallel planes</i> (Prop. 17).</p>
+
+<p>This theory is made more complete by adding the following
+theorems, which are easy deductions from the last: <i>Two parallel
+planes have common perpendiculars</i> (converse to 14); and <i>Two
+planes which are parallel to a third plane are parallel to each other.</i></p>
+
+<p>It will be noted that Prop. 15 at once allows of the solution of
+the problem: &ldquo;Through a given point to draw a plane parallel to
+a given plane.&rdquo; And it is also easily proved that this problem
+allows always of one, and only of one, solution.</p>
+
+<p>§ 78. We come now to planes which are perpendicular to one
+another. Two theorems relate to them.</p>
+
+<p>Prop. 18. <i>If a straight line be at right angles to a plane, every
+plane which passes through it shall be at right angles to that plane.</i></p>
+
+<p>Prop. 19. <i>If two planes which cut one another be each of them
+perpendicular to a third plane, their common section shall be perpendicular
+to the same plane.</i></p>
+
+<p>§ 79. If three planes pass through a common point, and if they
+bound each other, a solid angle of three faces, or a <i>trihedral</i> angle,
+is formed, and similarly by more planes a solid angle of more faces,
+or a <i>polyhedral</i> angle. These have many properties which are quite
+analogous to those of triangles and polygons in a plane. Euclid
+states some, viz.:&mdash;</p>
+
+<p>Prop. 20. <i>If a solid angle be contained by three plane angles, any
+two of them are together greater than the third.</i></p>
+
+<p>But the next&mdash;</p>
+
+<p>Prop. 21. <i>Every solid angle is contained by plane angles, which
+are together less than four right angles</i>&mdash;has no analogous theorem
+in the plane.</p>
+
+<p>We may mention, however, that the theorems about triangles
+contained in the propositions of Book I., which do not depend
+upon the theory of parallels (that is all up to Prop. 27), have their
+corresponding theorems about trihedral angles. The latter are
+formed, if for &ldquo;side of a triangle&rdquo; we write &ldquo;plane angle&rdquo; or
+&ldquo;face&rdquo; of trihedral angle, and for &ldquo;angle of triangle&rdquo; we substitute
+&ldquo;angle between two faces&rdquo; where the planes containing the
+solid angle are called its <i>faces</i>. We get, for instance, from I. 4, the
+<span class="pagenum"><a name="page687" id="page687"></a>687</span>
+theorem, <i>If two trihedral angles have the angles of two faces in the one
+equal to the angles of two faces in the other, and have likewise the angles
+included by these faces equal, then the angles in the remaining faces are
+equal, and the angles between the other faces are equal each to each, viz.
+those which are opposite equal faces.</i> The solid angles themselves are
+not necessarily equal, for they may be only symmetrical like the
+right hand and the left.</p>
+
+<p>The connexion indicated between triangles and trihedral angles
+will also be recognized in</p>
+
+<p>Prop. 22. <i>If every two of three plane angles be greater than the
+third, and if the straight lines which contain them be all equal, a triangle
+may be made of the straight lines that join the extremities of those equal
+straight lines.</i></p>
+
+<p>And Prop. 23 solves the problem, <i>To construct a trihedral angle
+having the angles of its faces equal to three given plane angles, any two
+of them being greater than the third.</i> It is, of course, analogous to the
+problem of constructing a triangle having its sides of given length.</p>
+
+<p>Two other theorems of this kind are added by Simson in his
+edition of Euclid&rsquo;s <i>Elements</i>.</p>
+
+<p>§ 80. These are the principal properties of lines and planes in
+space, but before we go on to their applications it will be well to
+define the word <i>distance</i>. In geometry distance means always
+&ldquo;shortest distance&rdquo;; viz. the distance of a point from a straight
+line, or from a plane, is the length of the perpendicular from the
+point to the line or plane. The distance between two non-intersecting
+lines is the length of their common perpendicular, there being
+but one. The distance between two parallel lines or between two
+parallel planes is the length of the common perpendicular between
+the lines or the planes.</p>
+
+<p>§ 81. <i>Parallelepipeds</i>.&mdash;The rest of the book is devoted to the
+study of the parallelepiped. In Prop. 24 the possibility of such
+a solid is proved, viz.:&mdash;</p>
+
+<p>Prop. 24. <i>If a solid be contained by six planes two and two of
+which are parallel, the opposite planes are similar and equal parallelograms.</i></p>
+
+<p>Euclid calls this solid henceforth a parallelepiped, though he
+never defines the word. Either face of it may be taken as <i>base</i>,
+and its distance from the opposite face as <i>altitude</i>.</p>
+
+<p>Prop. 25. <i>If a solid parallelepiped be cut by a plane parallel to
+two of its opposite planes, it divides the whole into two solids, the base
+of one of which shall be to the base of the other as the one solid is to the
+other.</i></p>
+
+<p>This theorem corresponds to the theorem (VI. 1) that parallelograms
+between the same parallels are to one another as their bases.
+A similar analogy is to be observed among a number of the remaining
+propositions.</p>
+
+<p>§ 82. After solving a few problems we come to</p>
+
+<p>Prop. 28. <i>If a solid parallelepiped be cut by a plane passing
+through the diagonals of two of the opposite planes, it shall be cut in
+two equal parts.</i></p>
+
+<p>In the proof of this, as of several other propositions, Euclid
+neglects the difference between solids which are symmetrical like
+the right hand and the left.</p>
+
+<p>Prop. 31. <i>Solid parallelepipeds, which are upon equal bases, and
+of the same altitude, are equal to one another.</i></p>
+
+<p>Props. 29 and 30 contain special cases of this theorem leading up
+to the proof of the general theorem.</p>
+
+<p>As consequences of this fundamental theorem we get</p>
+
+<p>Prop. 32. <i>Solid parallelepipeds, which have the same altitude, are
+to one another as their bases;</i> and</p>
+
+<p>Prop. 33. <i>Similar solid parallelepipeds are to one another in the
+triplicate ratio of their homologous sides.</i></p>
+
+<p>If we consider, as in § 67, the ratios of lines as numbers, we may
+also say&mdash;</p>
+
+<p><i>The ratio of the volumes of similar parallelepipeds is equal to the
+ratio of the third powers of homologous sides.</i></p>
+
+<p>Parallelepipeds which are not similar but equal are compared by
+aid of the theorem</p>
+
+<p>Prop. 34. <i>The bases and altitudes of equal solid parallelepipeds
+<span class="correction" title="amended from and">are</span> reciprocally proportional; and if the bases and altitudes be reciprocally
+proportional, the solid parallelepipeds are equal.</i></p>
+
+<p>§ 83. Of the following propositions the 37th and 40th are of
+special interest.</p>
+
+<p>Prop. 37. <i>If four straight lines be proportionals, the similar solid
+parallelepipeds, similarly described from them, shall also be proportionals;
+and if the similar parallelepipeds similarly described
+from four straight lines be proportionals, the straight lines shall be
+proportionals.</i></p>
+
+<p>In symbols it says&mdash;</p>
+
+<p class="center">If a : b = c : d, then a³ : b³ = c³: d³.</p>
+
+<p>Prop. 40 teaches how to compare the volumes of triangular
+prisms with those of parallelepipeds, by proving <i>that a triangular
+prism is equal in volume to a parallelepiped, which has its altitude
+and its base equal to the altitude and the base of the triangular
+prism.</i></p>
+
+<p>§ 84. From these propositions follow all results relating to the
+mensuration of volumes. We shall state these as we did in the case
+of areas. The starting-point is the &ldquo;rectangular&rdquo; parallelepiped,
+which has every edge perpendicular to the planes it meets, and
+which takes the place of the rectangle in the plane. If this has all
+its edges equal we obtain the &ldquo;cube.&rdquo;</p>
+
+<p>If we take a certain line u as unit length, then the square on u is
+the unit of area, and the cube on u the unit of volume, that is to
+say, if we wish to measure a volume we have to determine how
+many unit cubes it contains.</p>
+
+<p>A rectangular parallelepiped has, as a rule, the three edges unequal,
+which meet at a point. Every other edge is equal to one
+of them. If a, b, c be the three edges meeting at a point, then we
+may take the rectangle contained by two of them, say by b and c,
+as base and the third as altitude. Let V be its volume, V&prime; that of
+another rectangular parallelepiped which has the edges a&prime;, b, c,
+hence the same base as the first. It follows then easily, from Prop.
+25 or 32, that V : V&prime; = a : a&prime;; or in words,</p>
+
+<p><i>Rectangular parallelepipeds on equal bases are proportional to their
+altitudes.</i></p>
+
+<p>If we have two rectangular parallelepipeds, of which the first has
+the volume V and the edges a, b, c, and the second, the volume V&prime;
+and the edges a&prime;, b&prime;, c&prime;, we may compare them by aid of two new
+ones which have respectively the edges a&prime;, b, c and a&prime;, b&prime;, c, and the
+volumes V<span class="su">1</span> and V<span class="su">2</span>. We then have</p>
+
+<p class="center">V : V<span class="su">1</span> = a : a&prime;; V<span class="su">1</span> : V<span class="su">2</span> = b : b&prime;, V<span class="su">2</span> : V&prime; = c : c&prime;.</p>
+
+<p>Compounding these, we have</p>
+
+<p class="center">V : V&prime; = (a : a&prime;) (b : b&prime;) (c : c&prime;),</p>
+
+<p class="noind">or</p>
+
+<table class="math0" summary="math">
+<tr><td>V</td>
+<td rowspan="2">=</td> <td>a</td>
+<td rowspan="2">·</td> <td>b</td>
+<td rowspan="2">·</td> <td>c</td>
+<td rowspan="2">.</td></tr>
+<tr><td class="denom">V&prime;</td> <td class="denom">a&prime;</td>
+<td class="denom">b&prime;</td> <td class="denom">c&prime;</td></tr></table>
+
+<p class="noind">Hence, as a special case, making V&prime; equal to the unit cube U on u
+we get</p>
+
+<table class="math0" summary="math">
+<tr><td>V</td>
+<td rowspan="2">=</td> <td>a</td>
+<td rowspan="2">·</td> <td>b</td>
+<td rowspan="2">·</td> <td>c</td>
+<td rowspan="2">= &alpha;·&beta;·&gamma;,</td></tr>
+<tr><td class="denom">U</td> <td class="denom">u</td>
+<td class="denom">u</td> <td class="denom">u</td></tr></table>
+
+<p class="noind">where &alpha;, &beta;, &gamma; are the numerical values of a, b, c; that is, <i>The number
+of unit cubes in a rectangular parallelepiped</i> is equal to the product
+of the numerical values of its three edges. This is generally expressed
+by saying the volume of a rectangular parallelepiped is
+measured by the product of its sides, or by the product of its base
+into its altitude, which in this case is the same.</p>
+
+<p>Prop. 31 allows us to extend this to any parallelepipeds, and Props.
+28 or 40, to triangular prisms.</p>
+
+<p><i>The volume of any parallelepiped, or of any triangular prism, is
+measured by the product of base and altitude.</i></p>
+
+<p>The consideration that any polygonal prism may be divided into
+a number of triangular prisms, which have the same altitude and
+the sum of their bases equal to the base of the polygonal prism,
+shows further that the same holds for any prism whatever.</p>
+
+<p class="pt2 center sc">Book XII.</p>
+
+<p>§ 85. In the last part of Book XI. we have learnt how to compare
+the volumes of parallelepipeds and of prisms. In order to determine
+the volume of any solid bounded by plane faces we must determine
+the volume of pyramids, for every such solid may be decomposed
+into a number of pyramids.</p>
+
+<p>As every pyramid may again be decomposed into triangular
+pyramids, it becomes only necessary to determine their volume.
+This is done by the</p>
+
+<p><i>Theorem.</i>&mdash;Every triangular pyramid is equal in volume to one
+third of a triangular prism having the same base and the same
+altitude as the pyramid.</p>
+
+<p>This is an immediate consequence of Euclid&rsquo;s</p>
+
+<p>Prop. 7. <i>Every prism having a triangular base may be divided
+into three pyramids that have triangular bases, and are equal to one
+another.</i></p>
+
+<p>The proof of this theorem is difficult, because the three triangular
+pyramids into which the prism is divided are by no means equal in
+shape, and cannot be made to coincide. It has first to be proved
+that two triangular pyramids have equal volumes, if they have
+equal bases and equal altitudes. This Euclid does in the following
+manner. He first shows (Prop. 3) that a triangular pyramid may
+be divided into four parts, of which two are equal triangular pyramids
+similar to the whole pyramid, whilst the other two are equal triangular
+prisms, and further, that these two prisms together are
+greater than the two pyramids, hence more than half the given
+pyramid. He next shows (Prop. 4) that if two triangular pyramids
+are given, having equal bases and equal altitudes, and if each be
+divided as above, then the two triangular prisms in the one are
+equal to those in the other, and each of the remaining pyramids in
+the one has its base and altitude equal to the base and altitude of
+the remaining pyramids in the other. Hence to these pyramids the
+same process is again applicable. We are thus enabled to cut out
+of the two given pyramids equal parts, each greater than half the
+original pyramid. Of the remainder we can again cut out equal
+parts greater than half these remainders, and so on as far as we like.
+This process may be continued till the last remainder is smaller
+than any assignable quantity, however small. It follows, so we
+should conclude at present, that the two volumes must be equal, for
+they cannot differ by any assignable quantity.</p>
+
+<p>To Greek mathematicians this conclusion offers far greater
+<span class="pagenum"><a name="page688" id="page688"></a>688</span>
+difficulties. They prove elaborately, by a <i>reductio ad absurdum</i>,
+that the volumes cannot be unequal. This proof must be read in
+the <i>Elements.</i> We must, however, state that we have in the above
+not proved Euclid&rsquo;s Prop. 5, but only a special case of it. Euclid
+does not suppose that the bases of the two pyramids to be compared
+are equal, and hence he proves that the volumes are as the bases.
+The reasoning of the proof becomes clearer in the special case, from
+which the general one may be easily deduced.</p>
+
+<p>§ 86. Prop. 6 extends the result to pyramids with polygonal
+bases. From these results follow again the rules at present given
+for the mensuration of solids, viz. a pyramid is the third part of a
+triangular prism having the same base and the same altitude. But
+a triangular prism is equal in volume to a parallelepiped which
+has the same base and altitude. Hence if B is the base and h the
+altitude, we have</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcl">Volume of prism</td> <td class="tcl">= Bh,</td></tr>
+<tr><td class="tcl">Volume of pyramid</td> <td class="tcl">= <span class="spp">1</span>&frasl;<span class="suu">3</span>Bh,</td></tr>
+</table>
+
+<p class="noind">statements which have to be taken in the sense that B means the
+number of square units in the base, h the number of units of length
+in the altitude, or that B and h denote the numerical values of base
+and altitude.</p>
+
+<p>§ 87. A method similar to that used in proving Prop. 5 leads to
+the following results relating to solids bounded by simple curved
+surfaces:&mdash;</p>
+
+<p>Prop. 10. <i>Every cone is the third part of a cylinder which has the
+same base, and is of an equal altitude with it.</i></p>
+
+<p>Prop. 11. <i>Cones or cylinders of the same altitude are to one another
+as their bases.</i></p>
+
+<p>Prop. 12. <i>Similar cones or cylinders have to one another the triplicate
+ratio of that which the diameters of their bases have.</i></p>
+
+<p>Prop. 13. <i>If a cylinder be cut by a plane parallel to its opposite
+planes or bases, it divides the cylinder into two cylinders, one of which
+is to the other as the axis of the first to the axis of the other;</i> which
+may also be stated thus:&mdash;</p>
+
+<p><i>Cylinders on the same base are proportional to their altitudes.</i></p>
+
+<p>Prop. 14. <i>Cones or cylinders upon equal bases are to one another
+as their altitudes.</i></p>
+
+<p>Prop. 15. <i>The bases and altitudes of equal cones or cylinders are
+reciprocally proportional, and if the bases and altitudes be reciprocally
+proportional, the cones or cylinders are equal to one another.</i></p>
+
+<p>These theorems again lead to formulae in mensuration, if we
+compare a cylinder with a prism having its base and altitude equal to
+the base and altitude of the cylinder. This may be done by the
+method of exhaustion. We get, then, the result that their bases are
+equal, and have, if B denotes the numerical value of the base, and
+h that of the altitude,</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcl">Volume of cylinder</td> <td class="tcl">= Bh,</td></tr>
+<tr><td class="tcl">Volume of cone</td> <td class="tcl">= <span class="spp">1</span>&frasl;<span class="suu">3</span>Bh.</td></tr>
+</table>
+
+<p>§ 88. The remaining propositions relate to circles and spheres.
+Of the sphere only one property is proved, viz.:&mdash;</p>
+
+<p>Prop. 18. <i>Spheres have to one another the triplicate ratio of that
+which their diameters have.</i> The mensuration of the sphere, like
+that of the circle, the cylinder and the cone, had not been settled
+in the time of Euclid. It was done by Archimedes.</p>
+
+<p class="pt2 center sc">Book XIII.</p>
+
+<p>§ 89. The 13th and last book of Euclid&rsquo;s <i>Elements</i> is devoted to
+the regular solids (see <span class="sc"><a href="#artlinks">Polyhedron</a></span>). It is shown that there are
+five of them, viz.:&mdash;</p>
+
+<p>1. The regular <i>tetrahedron</i>, with 4 triangular faces and 4 vertices;</p>
+
+<p>2. The <i>cube</i>, with 8 vertices and 6 square faces;</p>
+
+<p>3. The <i>octahedron</i>, with 6 vertices and 8 triangular faces;</p>
+
+<p>4. The <i>dodecahedron</i>, with 12 pentagonal faces, 3 at each of the
+20 vertices;</p>
+
+<p>5. The <i>icosahedron</i>, with 20 triangular faces, 5 at each of the
+12 vertices.</p>
+
+<p>It is shown how to inscribe these solids in a given sphere, and
+how to determine the lengths of their edges.</p>
+
+<p>§ 90. The 13th book, and therefore the <i>Elements</i>, conclude with
+the scholium, &ldquo;that no other regular solid exists besides the five
+ones enumerated.&rdquo;</p>
+
+<p>The proof is very simple. Each face is a regular polygon, hence
+the angles of the faces at any vertex must be angles in equal regular
+polygons, must be together less than four right angles (XI. 21), and
+must be three or more in number. Each angle in a regular triangle
+equals two-thirds of one right angle. Hence it is possible to form
+a solid angle with three, four or five regular triangles or faces.
+These give the solid angles of the tetrahedron, the octahedron and
+the icosahedron. The angle in a square (the regular quadrilateral)
+equals one right angle. Hence three will form a solid angle, that
+of the cube, and four will not. The angle in the regular pentagon
+equals <span class="spp">6</span>&frasl;<span class="suu">5</span> of a right angle. Hence three of them equal <span class="spp">18</span>&frasl;<span class="suu">5</span> (<i>i.e.</i> less
+than 4) right angles, and form the solid angle of the dodecahedron.
+Three regular polygons of six or more sides cannot form a solid
+angle. Therefore no other regular solids are possible.</p>
+</div>
+<div class="author">(O. H.)</div>
+
+<p class="pt2 center sc">II. Projective Geometry</p>
+
+<p>It is difficult, at the outset, to characterize projective geometry
+as compared with Euclidean. But a few examples will at least
+indicate the practical differences between the two.</p>
+
+<p>In Euclid&rsquo;s <i>Elements</i> almost all propositions refer to the <i>magnitude</i>
+of lines, angles, areas or volumes, and therefore to measurement.
+The statement that an angle is right, or that two straight
+lines are parallel, refers to measurement. On the other hand,
+the fact that a straight line does or does not cut a circle is independent
+of measurement, it being dependent only upon the
+mutual &ldquo;position&rdquo; of the line and the circle. This difference
+becomes clearer if we project any figure from one plane to another
+(see <span class="sc"><a href="#artlinks">Projection</a></span>). By this the length of lines, the magnitude
+of angles and areas, is altered, so that the projection, or shadow,
+of a square on a plane will not be a square; it will, however,
+be some quadrilateral. Again, the projection of a circle will not
+be a circle, but some other curve more or less resembling a circle.
+But one property may be stated at once&mdash;no straight line can cut
+the projection of a circle in more than two points, because no
+straight line can cut a circle in more than two points. There
+are, then, some properties of figures which do not alter by
+projection, whilst others do. To the latter belong nearly all
+properties relating to measurement, at least in the form in which
+they are generally given. The others are said to be projective
+properties, and their investigation forms the subject of projective
+geometry.</p>
+
+<p>Different as are the kinds of properties investigated in the old
+and the new sciences, the methods followed differ in a still
+greater degree. In Euclid each proposition stands by itself;
+its connexion with others is never indicated; the leading ideas
+contained in its proof are not stated; general principles do not
+exist. In the modern methods, on the other hand, the greatest
+importance is attached to the leading thoughts which pervade
+the whole; and general principles, which bring whole groups of
+theorems under one aspect, are given rather than separate propositions.
+The whole tendency is towards generalization.
+A straight line is considered as given in its entirety, extending
+both ways to infinity, while Euclid never admits anything but
+finite quantities. The treatment of the infinite is in fact another
+fundamental difference between the two methods: Euclid avoids
+it; in modern geometry it is systematically introduced.</p>
+
+<p>Of the different modern methods of geometry, we shall treat
+principally of the methods of projection and correspondence which
+have proved to be the most powerful. These have become independent
+of Euclidean Geometry, especially through the <i>Geometrie
+der Lage</i> of V. Staudt and the <i>Ausdehnungslehre</i> of Grassmann.</p>
+
+<p>For the sake of brevity we shall presuppose a knowledge of
+Euclid&rsquo;s <i>Elements</i>, although we shall use only a few of his propositions.</p>
+
+<div class="condensed">
+<p>§ 1. <i>Geometrical Elements.</i> We consider space as filled with points,
+lines and planes, and these we call the elements out of which our
+figures are to be formed, calling any combination of these elements a
+&ldquo;figure.&rdquo;</p>
+
+<p>By a line we mean a straight line in its entirety, extending both
+ways to infinity; and by a plane, a plane surface, extending in all
+directions to infinity.</p>
+
+<p>We accept the three-dimensional space of experience&mdash;the space
+assumed by Euclid&mdash;which has for its properties (among others):&mdash;</p>
+
+<p>Through any two points in space one and only one line may be
+drawn;</p>
+
+<p>Through any three points which are not in a line, one and only one
+plane may be placed;</p>
+
+<p>The intersection of two planes is a line;</p>
+
+<p>A line which has two points in common with a plane lies in the
+plane, hence the intersection of a line and a plane is a single point; and</p>
+
+<p>Three planes which do not meet in a line have one single point in
+common.</p>
+
+<p>These results may be stated differently in the following form:&mdash;</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p>I. A plane is determined&mdash;</p></td>
+<td class="tcl" style="width: 50%;"><p>A point is determined&mdash;</p></td></tr>
+
+<tr><td class="tcl1 rb3"><p>1. By three points which do not lie in a line;</p>
+<p>2. By two intersecting lines;</p>
+<p>3. By a line and a point which does not lie in it.</p></td>
+
+<td class="tcl1"><p>1. By three planes which do not pass through a line;</p>
+<p>2. By two intersecting lines</p>
+<p>3. By a plane and a line which does not lie in it.</p></td></tr>
+
+<tr><td class="tcl rb3"><p>A line is determined&mdash;</p></td>
+<td class="tcl">&nbsp;</td></tr>
+
+<tr><td class="tcl1 rb3"><p>1. By two points;</p></td>
+<td class="tcl1"><p>2. By two planes.</p></td></tr></table>
+
+<p><span class="pagenum"><a name="page689" id="page689"></a>689</span></p>
+
+<p>It will be observed that not only are planes determined by points,
+but also points by planes; that therefore the planes may be considered
+as elements, like points; and also that in any one of the
+above statements we may interchange the words point and plane,
+and we obtain again a correct statement, provided that these
+statements themselves are true. As they stand, we ought, in
+several cases, to add &ldquo;if they are not parallel,&rdquo; or some such words,
+parallel lines and planes being evidently left altogether out of
+consideration. To correct this we have to reconsider the theory of
+parallels.</p>
+
+<table class="flt" style="float: right; width: 350px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:294px; height:205px" src="images/img689.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 1.</span></td></tr></table>
+
+<p>§ 2. <i>Parallels. Point at Infinity.</i>&mdash;Let us take in a plane a line p
+(fig. 1), a point S not in this line, and a line q drawn through S.
+Then this line q will meet
+the line p in a point A. If
+we turn the line q about S
+towards q&rsquo;, its point of
+intersection with p will
+move along p towards B,
+passing, on continued turning,
+to a greater and greater
+distance, until it is moved
+out of our reach. If we
+turn q still farther, its continuation
+will meet p, but
+now at the other side of
+A. The point of intersection
+has disappeared to
+the right and reappeared
+to the left. There is one intermediate position where q is parallel
+to p&mdash;that is where it does not cut p. In every other position it
+cuts p in some finite point. If, on the other hand, we move the point
+A to an infinite distance in p, then the line q which passes through
+A will be a line which does not cut p at any finite point. Thus we
+are led to say: <i>Every</i> line through S which joins it to any point
+at an infinite distance in p is parallel to p. But by Euclid&rsquo;s 12th
+axiom there is but one line parallel to p through S. The difficulty in
+which we are thus involved is due to the fact that we try to reason
+about infinity as if we, with our finite capabilities, could comprehend
+the infinite. To overcome this difficulty, we may say that all points
+at infinity in a line <i>appear</i> to us as one, and may be replaced by a
+single &ldquo;ideal&rdquo; point.</p>
+
+<p>We may therefore now give the following definitions and axiom:&mdash;</p>
+
+<p><i>Definition.</i>&mdash;Lines which meet at infinity are called parallel.</p>
+
+<p><i>Axiom.</i>&mdash;All points at an infinite distance in a line may be considered
+as one single point.</p>
+
+<p><i>Definition.</i>&mdash;This ideal point is called the <i>point at infinity</i> in the
+line.</p>
+
+<p>The axiom is equivalent to Euclid&rsquo;s Axiom 12, for it follows from
+either that through any point only one line may be drawn parallel
+to a given line.</p>
+
+<p>This point at infinity in a line is reached whether we move a
+point in the one or in the opposite direction of a line to infinity.
+A line thus appears closed by this point, and we speak as if we
+could move a point along the line from one position A to another
+B in two ways, either through the point at infinity or through finite
+points only.</p>
+
+<p>It must never be forgotten that this point at infinity is ideal;
+in fact, the whole notion of &ldquo;infinity&rdquo; is only a mathematical
+conception, and owes its introduction (as a method of research) to
+the working generalizations which it permits.</p>
+
+<p>§ 3. <i>Line and Plane at Infinity.</i>&mdash;Having arrived at the notion of
+replacing all points at infinity in a line by one ideal point, there is no
+difficulty in replacing all points at infinity in a plane by one ideal
+line.</p>
+
+<p>To make this clear, let us suppose that a line p, which cuts two
+fixed lines a and b in the points A and B, moves parallel to itself
+to a greater and greater distance. It will at last cut both a and
+b at their points at infinity, so that a line which joins the two points
+at infinity in two intersecting lines lies altogether at infinity. Every
+other line in the plane will meet it therefore at infinity, and thus it
+contains all points at infinity in the plane.</p>
+
+<p><i>All points at infinity in a plane lie in a line, which is called the</i> line
+at infinity <i>in the plane.</i></p>
+
+<p>It follows that parallel planes must be considered as planes
+having a common line at infinity, for any other plane cuts them in
+parallel lines which have a point at infinity in common.</p>
+
+<p>If we next take two intersecting planes, then the point at infinity
+in their line of intersection lies in both planes, so that their lines
+at infinity meet. Hence every line at infinity meets every other
+line at infinity, and they are therefore all in one plane.</p>
+
+<p><i>All points at infinity in space may be considered as lying in one
+ideal plane, which is called the</i> plane at infinity.</p>
+
+<p>§ 4. <i>Parallelism.</i>&mdash;We have now the following definitions:&mdash;</p>
+
+<p>Parallel lines are lines which meet at infinity;</p>
+
+<p>Parallel planes are planes which meet at infinity;</p>
+
+<p>A line is parallel to a plane if it meets it at infinity.</p>
+
+<p>Theorems like this&mdash;Lines (or planes) which are parallel to a third
+are parallel to each other&mdash;follow at once.</p>
+
+<p>This view of parallels leads therefore to no contradiction of
+Euclid&rsquo;s <i>Elements.</i></p>
+
+<p>As immediate consequences we get the propositions:&mdash;</p>
+
+<p>Every line meets a plane in one point, or it lies in it;</p>
+
+<p>Every plane meets every other plane in a line;</p>
+
+<p>Any two lines in the same plane meet.</p>
+
+<p>§ 5. <i>Aggregates of Geometrical Elements.</i>&mdash;We have called points,
+lines and planes the elements of geometrical figures. We also say
+that an element of one kind contains one of the other if it lies in it
+or passes through it.</p>
+
+<p>All the elements of one kind which are contained in one or two
+elements of a different kind form aggregates which have to be
+enumerated. They are the following:&mdash;</p>
+
+<p>I. Of one dimension.</p>
+
+<div class="list1">
+<p>1. The <i>row</i>, or range, <i>of points</i> formed by all points in a line,
+          which is called its base.</p>
+
+<p>2. The <i>flat pencil</i> formed by all the lines through a point in
+          a plane. Its base is the point in the plane.</p>
+
+<p>3. The <i>axial pencil</i> formed by all planes through a line
+          which is called its base or axis.</p>
+</div>
+
+<p>II. Of two dimensions.</p>
+
+<div class="list1">
+<p>1. The field of points and lines&mdash;that is, a plane with all its
+          points and all its lines.</p>
+
+<p>2. The pencil of lines and planes&mdash;that is, a point in space
+          with all lines and all planes through it.</p>
+</div>
+
+<p>III. Of three dimensions.</p>
+
+<div class="list1">
+<p>The space of points&mdash;that is, all points in space.</p>
+
+<p>The space of planes&mdash;that is, all planes in space.</p>
+</div>
+
+<p>IV. Of four dimensions.</p>
+
+<div class="list1">
+<p>The space of lines, or all lines in space.</p>
+</div>
+
+<p>§ 6. <i>Meaning of &ldquo;Dimensions.&rdquo;</i>&mdash;The word dimension in the above
+needs explanation. If in a plane we take a row p and a pencil with
+centre Q, then through every point in p one line in the pencil will
+pass, and every ray in Q will cut p in one point, so that we are
+entitled to say a row contains as many points as a flat pencil lines,
+and, we may add, as an axial pencil planes, because an axial pencil
+is cut by a plane in a flat pencil.</p>
+
+<p>The number of elements in the row, in the flat pencil, and in the
+axial pencil is, of course, infinite and indefinite too, but the same in
+all. This number may be denoted by &infin;. Then a plane contains
+&infin;² points and as many lines. To see this, take a flat pencil in a
+plane. It contains &infin; lines, and each line contains &infin; points, whilst
+each point in the plane lies on one of these lines. Similarly, in a
+plane each line cuts a fixed line in a point. But this line is cut at
+each point by &infin; lines and contains &infin; points; hence there are &infin;²
+lines in a plane.</p>
+
+<p>A pencil in space contains as many lines as a plane contains
+points and as many planes as a plane contains lines, for any plane
+cuts the pencil in a field of points and lines. Hence a pencil contains
+&infin;² lines and &infin;² planes. <i>The field and the pencil are of two
+dimensions.</i></p>
+
+<p>To count the number of points in space we observe that each
+point lies on some line in a pencil. But the pencil contains &infin;²
+lines, and each line &infin; points; hence space contains &infin;³ points.
+Each plane cuts any fixed plane in a line. But a plane contains
+&infin;² lines, and through each pass &infin; planes; therefore space contains
+&infin;³ planes.</p>
+
+<p>Hence space contains as many planes as points, but it contains
+an infinite number of times more lines than points or planes. To
+count them, notice that every line cuts a fixed plane in one point.
+But &infin;² lines pass through each point, and there are &infin;² points in the
+plane. Hence there are &infin;<span class="sp">4</span> lines in space. <i>The space of points
+and planes is of three dimensions, but the space of lines is of four
+dimensions.</i></p>
+
+<p>A field of points or lines contains an infinite number of rows and
+flat pencils; a pencil contains an infinite number of flat pencils
+and of axial pencils; space contains a triple infinite number of
+pencils and of fields, &infin;<span class="sp">4</span> rows and axial pencils and &infin;<span class="sp">5</span> flat pencils&mdash;or,
+in other words, each point is a centre of &infin;² flat pencils.</p>
+
+<p>§ 7. The above enumeration allows a classification of figures.
+Figures in a row consist of groups of points only, and figures in
+the flat or axial pencil consist of groups of lines or planes. In the
+plane we may draw polygons; and in the pencil or in the point,
+solid angles, and so on.</p>
+
+<p>We may also distinguish the different measurements We have&mdash;</p>
+
+<table class="reg" summary="poem"><tr><td> <div class="poemr">
+<p>In the row, length of segment;</p>
+<p>In the flat pencil, angles;</p>
+<p>In the axial pencil, dihedral angles between two planes;</p>
+<p>In the plane, areas;</p>
+<p>In the pencil, solid angles;</p>
+<p>In the space of points or planes, volumes.</p>
+</div> </td></tr></table>
+
+<p class="pt2 center sc">Segments of a Line</p>
+
+<p>§ 8. Any two points A and B in space determine on the line through
+them a finite part, which may be considered as being described by
+a point moving from A to B. This we shall denote by AB, and
+distinguish it from BA, which is supposed as being described by a
+point moving from B to A, and hence in a direction or in a &ldquo;sense&rdquo;
+opposite to AB. Such a finite line, which has a definite sense, we
+shall call a &ldquo;segment,&rdquo; so that AB and BA denote different segments,
+which are said to be equal in length but of opposite sense. The one
+sense is often called positive and the other negative.</p>
+
+<p><span class="pagenum"><a name="page690" id="page690"></a>690</span></p>
+
+<p>In introducing the word &ldquo;sense&rdquo; for direction in a line, we have
+the word direction reserved for direction of the line itself, so that
+different lines have different directions, unless they be parallel,
+whilst in each line we have a positive and negative sense.</p>
+
+<p>We may also say, with Clifford, that AB denotes the &ldquo;step&rdquo; of
+going from A to B.</p>
+
+<table class="flt" style="float: right; width: 230px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:184px; height:129px" src="images/img690a.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 2.</span></td></tr></table>
+
+<p>§ 9. If we have three points A, B, C in a line (fig. 2), the step AB
+will bring us from A to B, and the step
+BC from B to C. Hence both steps are
+equivalent to the one step AC. This is
+expressed by saying that AC is the
+&ldquo;sum&rdquo; of AB and BC; in symbols&mdash;</p>
+
+<p class="center">AB + BC = AC,</p>
+
+<p class="noind">where account is to be taken of the
+sense.</p>
+
+<p>This equation is true whatever be the
+position of the three points on the line.
+As a special case we have</p>
+
+<p class="center">AB + BA = 0,</p>
+<div class="aut">(1)</div>
+
+<p class="noind">and similarly</p>
+
+<p class="center">AB + BC + CA = 0,</p>
+<div class="aut">(2)</div>
+
+<p class="noind">which again is true for any three points in a line.</p>
+
+<p>We further write</p>
+
+<p class="center">AB = &minus;BA.</p>
+
+<p class="noind">where &minus; denotes negative sense.</p>
+
+<p>We can then, just as in algebra, change subtraction of segments
+into addition by changing the sense, so that AB &minus; CB is the same
+as AB + (&minus;CB) or AB + BC. A figure will at once show the truth
+of this. The sense is, in fact, in every respect equivalent to the
+&ldquo;sign&rdquo; of a number in algebra.</p>
+
+<p>§ 10. Of the many formulae which exist between points in a line
+we shall have to use only one more, which connects the segments
+between any four points A, B, C, D in a line. We have</p>
+
+<p class="center">BC = BD + DC, CA = CD + DA, AB = AD + DB;</p>
+
+<p class="noind">or multiplying these by AD, BD, CD respectively, we get</p>
+
+<table class="reg" summary="poem"><tr><td> <div class="poemr">
+<p>BC · AD = BD · AD + DC · AD = BD · AD &minus; CD · AD</p>
+
+<p>CA · BD = CD · BD + DA · BD = CD · BD &minus; AD · BD</p>
+
+<p>AB · CD = AD · CD + DB · CD = AD · CD &minus; BD · CD.</p>
+</div> </td></tr></table>
+
+<p class="noind">It will be seen that the sum of the right-hand sides vanishes, hence
+that</p>
+
+<p class="center">BC · AD + CA · BD + AB · CD = 0</p>
+<div class="aut">(3)</div>
+
+<p class="noind">for any four points on a line.</p>
+
+<table class="flt" style="float: right; width: 390px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:335px; height:30px" src="images/img690b.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 3.</span></td></tr></table>
+
+<p>§ 11. If C is any point in the line AB, then we say that C divides
+the segment AB in the ratio AC/CB, account being taken of the
+sense of the two segments AC and CB. If C lies between A and B
+the ratio is positive, as AC and CB have the same sense. But if
+C lies without the segment AB, <i>i.e.</i> if C divides AB externally, then
+the ratio is negative.
+To see how the value of
+this ratio changes with
+C, we will move C along
+the whole line (fig. 3),
+whilst A and B remain fixed. If C lies at the point A, then AC = 0,
+hence the ratio AC : CB vanishes. As C moves towards B, AC
+increases and CB decreases, so that our ratio increases. At the
+middle point M of AB it assumes the value +1, and then increases
+till it reaches an infinitely large value, when C arrives at B. On
+passing beyond B the ratio becomes negative. If C is at P we have
+AC = AP = AB + BP, hence</p>
+
+<table class="math0" summary="math">
+<tr><td>AC</td>
+<td rowspan="2">=</td> <td>AB</td>
+<td rowspan="2">+</td> <td>BP</td>
+<td rowspan="2">= &minus;</td> <td>AB</td>
+<td rowspan="2">&minus; 1.</td></tr>
+<tr><td class="denom">CB</td> <td class="denom">PB</td>
+<td class="denom">PB</td> <td class="denom">BP</td></tr></table>
+
+<p class="noind">In the last expression the ratio AB : BP is positive, has its greatest
+value &infin; when C coincides with B, and vanishes when BC becomes
+infinite. Hence, as C moves from B to the right to the point at
+infinity, the ratio AC : CB varies from &minus;&infin; to &minus;1.</p>
+
+<p>If, on the other hand, C is to the left of A, say at Q, we have
+AC = AQ = AB + BQ = AB &minus; QB, hence AC/CB = AB/QB &minus; 1.</p>
+
+<p>Here AB &lt; QB, hence the ratio AB : QB is positive and always
+less than one, so that the whole is negative and &lt; 1. If C is at
+the point at infinity it is &minus;1, and then increases as C moves to the
+right, till for C at A we get the ratio = 0. Hence&mdash;</p>
+
+<p>&ldquo;As C moves along the line from an infinite distance to the left to
+an infinite distance at the right, the ratio always increases; it starts
+with the value &minus;1, reaches 0 at A, +1 at M, &infin; at B, now changes
+sign to &minus;&infin;, and increases till at an infinite distance it reaches
+again the value &minus;1. <i>It assumes therefore all possible values from
+-&infin; to +&infin;, and each value only once, so that not only does every
+position of</i> C <i>determine a definite value of the ratio</i> AC : CB, <i>but also,
+conversely, to every positive or negative value of this ratio belongs one
+single point in the line</i> AB.</p>
+
+<p>[Relations between segments of lines are interesting as showing an
+application of algebra to geometry. The genesis of such relations
+from algebraic identities is very simple. For example, if a, b, c, x
+be any four quantities, then</p>
+
+<table class="math0" summary="math">
+<tr><td>a</td>
+<td rowspan="2">+</td> <td>b</td>
+<td rowspan="2">+</td> <td>c</td>
+<td rowspan="2">=</td> <td>x</td>
+<td rowspan="2">;</td></tr>
+<tr><td class="denom">(a &minus; b)(a &minus; c)(x &minus; a)</td> <td class="denom">(b &minus; c)(b &minus; a)(x &minus; b)</td>
+<td class="denom">(c &minus; a)(c &minus; b)(x &minus; c)</td> <td class="denom">(x &minus; a)(x &minus; b)(x &minus; c)</td></tr></table>
+
+<p class="noind">this may be proved, cumbrously, by multiplying up, or, simply, by
+decomposing the right-hand member of the identity into partial
+fractions. Now take a line ABCDX, and let AB = a, AC = b, AD = c,
+AX = x. Then obviously (a &minus; b) = AB &minus; AC = &minus;BC, paying regard
+to signs; (a &minus; c) = AB &minus; AD = DB, and so on. Substituting these
+values in the identity we obtain the following relation connecting
+the segments formed by five points on a line:&mdash;</p>
+
+<table class="math0" summary="math">
+<tr><td>AB</td>
+<td rowspan="2">+</td> <td>AC</td>
+<td rowspan="2">+</td> <td>AD</td>
+<td rowspan="2">=</td> <td>AX</td>
+<td rowspan="2">.</td></tr>
+<tr><td class="denom">BC · BD · BX</td> <td class="denom">CD · CB · CX</td>
+<td class="denom">DB · DC · DX</td> <td class="denom">BX · CX · DX</td></tr></table>
+
+<p>Conversely, if a metrical relation be given, its validity may be
+tested by reducing to an algebraic equation, which is an identity
+if the relation be true. For example, if ABCDX be five collinear
+points, prove</p>
+
+<table class="math0" summary="math">
+<tr><td>AD · AX</td>
+<td rowspan="2">+</td> <td>BD · BX</td>
+<td rowspan="2">+</td> <td>CD · CX</td>
+<td rowspan="2">= 1.</td></tr>
+<tr><td class="denom">AB · AC</td> <td class="denom">BC · BA</td>
+<td class="denom">CA · CB</td></tr></table>
+
+<p class="noind">Clearing of fractions by multiplying throughout by AB · BC · CA,
+we have to prove</p>
+
+<p class="center">&minus;AD · AX · BC &minus; BD · BX · CA &minus; CD · CX · AB = AB · BC · CA.</p>
+
+<p class="noind">Take A as origin and let AB = a, AC = b, AD = c, AX = x. Substituting
+for the segments in terms of a, b, c, x, we obtain on simplification</p>
+
+<p class="center">a²b &minus; ab² = &minus;ab² + a²b, an obvious identity.</p>
+
+<p>An alternative method of testing a relation is illustrated in the
+<span class="correction" title="amended from example: following">following example:&mdash;</span> If A, B, C, D, E, F be six collinear points,
+then</p>
+
+<table class="math0" summary="math">
+<tr><td>AE · AF</td>
+<td rowspan="2">+</td> <td>BE · BF</td>
+<td rowspan="2">+</td> <td>CE · CF</td>
+<td rowspan="2">+</td> <td>DE · DF</td>
+<td rowspan="2">= 0.</td></tr>
+<tr><td class="denom">AB · AC · AD</td> <td class="denom">BC · BD · BA</td>
+<td class="denom">CD · CA · CB</td> <td class="denom">DA · DB · DC</td></tr></table>
+
+<p class="noind">Clearing of fractions by multiplying throughout by AB · BC · CD · DA,
+and reducing to a common origin O (calling OA = a, OB = b, &amp;c.),
+an equation containing the second and lower powers of OA ( = a),
+&amp;c., is obtained. Calling OA = x, it is found that x = b, x = c, x = d
+are solutions. Hence the quadratic has three roots; consequently
+it is an identity.</p>
+
+<p>The relations connecting five points which we have instanced above
+may be readily deduced from the six-point relation; the first by
+taking D at infinity, and the second by taking F at infinity, and then
+making the obvious permutations of the points.]</p>
+
+<p class="pt2 center sc">Projection and Cross-ratios</p>
+
+<p>§ 12. If we join a point A to a point S, then the point where the
+line SA cuts a fixed plane &pi; is called the projection of A on the
+plane &pi; from S as centre of projection. If we have two planes &pi;
+and &pi;&prime; and a point S, we may project every point A in &pi; to the
+other plane. If A&prime; is the projection of A, then A is also the projection
+of A&prime;, so that the relations are reciprocal. To every figure
+in &pi; we get as its projection a corresponding figure in &pi;&prime;.</p>
+
+<p>We shall determine such properties of figures as remain true for
+the projection, and which are called projective properties. For this
+purpose it will be sufficient to consider at first only constructions in
+one plane.</p>
+
+<table class="nobctr" style="clear: both;" summary="Illustration">
+<tr><td class="figcenter" colspan="2"><img style="width:472px; height:377px" src="images/img690c.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 4.</span></td>
+<td class="caption"><span class="sc">Fig. 5.</span></td></tr></table>
+
+<p>Let us suppose we have given in a plane two lines p and p&prime; and a
+centre S (fig. 4); we may then project the points in p from S to p&prime;.
+Let A&prime;, B&prime; ... be the projections of A, B ..., the point at infinity in
+p which we shall denote by I will be projected into a finite point
+<span class="pagenum"><a name="page691" id="page691"></a>691</span>
+I&prime; in p&prime;, viz. into the point where the parallel to p through S cuts
+p&prime;. Similarly one point J in p will be projected into the point
+J&prime; at infinity in p&prime;. This point J is of course the point where the
+parallel to p&prime; through S cuts p. We thus see that every point in p
+is projected into a single point in p&prime;.</p>
+
+<p>Fig. 5 shows that a segment AB will be projected into a segment
+A&prime;B&prime; which is not equal to it, at least not as a rule; and
+also that the ratio AC : CB is not equal to the ratio
+A&prime;C&prime; : C&prime;B&prime; formed by the projections. These ratios
+will become equal only if p and p&prime; are parallel, for
+in this case the triangle SAB is similar to the triangle
+SA&prime;B&prime;. Between three points in a line and their projections
+there exists therefore in general no relation.
+But between four points a relation does exist.</p>
+
+<p>§ 13. Let A, B, C, D be four points in p, A&prime;, B&prime;,
+C, D&prime; their projections in p&prime;, then the ratio of the two
+ratios AC : CB and AD : DB into which C and D
+divide the segment AB is equal to the corresponding
+expression between A&prime;, B&prime;, C&prime;, D&prime;. In symbols we have</p>
+
+<table class="math0" summary="math">
+<tr><td>AC</td>
+<td rowspan="2">:</td> <td>AD</td>
+<td rowspan="2">=</td> <td>A&prime;C&prime;</td>
+<td rowspan="2">:</td> <td>A&prime;D&prime;</td>
+<td rowspan="2">.</td></tr>
+<tr><td class="denom">CB</td> <td class="denom">DB</td>
+<td class="denom">C&prime;B&prime;</td> <td class="denom">D&prime;B&prime;</td></tr></table>
+
+<p>This is easily proved by aid of similar triangles.</p>
+
+<table class="flt" style="float: right; width: 350px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:304px; height:237px" src="images/img691.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 6.</span></td></tr></table>
+
+<p>Through the points A and B on p draw parallels to p&prime;, which cut
+the projecting rays in
+C<span class="su">2</span>, D<span class="su">2</span>, B<span class="su">2</span> and A<span class="su">1</span>, C<span class="su">1</span>,
+D<span class="su">1</span>, as indicated in
+fig. 6. The two triangles
+ACC<span class="su">2</span> and BCC<span class="su">1</span> will be
+similar, as will also be
+the triangles ADD<span class="su">2</span> and
+BDD<span class="su">1</span>.</p>
+
+<p>The proof is left to
+the reader.</p>
+
+<p>This result is of fundamental
+importance.</p>
+
+<p>The expression
+AC/CB : AD/DB has been
+called by Chasles the
+&ldquo;anharmonic ratio of the
+four points A, B, C, D.&rdquo;
+Professor Clifford proposed
+the shorter name of &ldquo;cross-ratio.&rdquo; We shall adopt the
+latter. We have then the</p>
+
+<p><span class="sc">Fundamental Theorem.</span>&mdash;<i>The cross-ratio of four points in a
+line is equal to the cross-ratio of their projections on any other line
+which lies in the same plane with it.</i></p>
+
+<p>§ 14. Before we draw conclusions from this result, we must investigate
+the meaning of a cross-ratio somewhat more fully.</p>
+
+<p>If four points A, B, C, D are given, and we wish to form their
+cross-ratio, we have first to divide them into two groups of two,
+the points in each group being taken in a definite order. Thus,
+let A, B be the first, C, D the second pair, A and C being the first
+points in each pair. The cross-ratio is then the ratio AC : CB
+divided by AD : DB. This will be denoted by (AB, CD), so that</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">(AB, CD) =</td> <td>AC</td>
+<td rowspan="2">:</td> <td>AD</td>
+<td rowspan="2">.</td></tr>
+<tr><td class="denom">CB</td> <td class="denom">DB</td></tr></table>
+
+<p>This is easily remembered. In order to write it out, make first
+the two lines for the fractions, and put above and below these
+the letters A and B in their places, thus, A/*B : A/*B; and then fill
+up, crosswise, the first by C and the other by D.</p>
+
+<p>§ 15. If we take the points in a different order, the value of the
+cross-ratio will change. We can do this in twenty-four different
+ways by forming all permutations of the letters. But of these
+twenty-four cross-ratios groups of four are equal, so that there are
+really only six different ones, and these six are reciprocals in pairs.</p>
+
+<p>We have the following rules:&mdash;</p>
+
+<p>I. If in a cross-ratio the two groups be interchanged, its value
+remains unaltered, <i>i.e.</i></p>
+
+<p class="center">(AB, CD) = (CD, AB) = (BA, DC) = (DC, BA).</p>
+
+<p>II. If in a cross-ratio the two points belonging to one of the two
+groups be interchanged, the cross-ratio changes into its reciprocal, <i>i.e.</i></p>
+
+<p class="center">(AB, CD) = 1/(AB, DC) = 1/(BA, CD) = 1/(CD, BA) = 1/(DC, AB).</p>
+
+<p>From I. and II. we see that eight cross-ratios are associated with
+(AB, CD).</p>
+
+<p>III. If in a cross-ratio the two middle letters be interchanged,
+the cross-ratio &alpha; changes into its complement 1 &minus; &alpha;, <i>i.e.</i> (AB, CD) =
+1 &minus; (AC, BD).</p>
+
+<p>[§ 16. If &lambda; = (AB, CD), &mu; = (AC, DB), &nu; = (AD, BC), then &lambda;, &mu;, &nu;
+and their reciprocals 1/&lambda;, 1/&mu;, 1/&nu; are the values of the total number
+of twenty-four cross-ratios. Moreover, &lambda;, &mu;, &nu; are connected by the
+relations</p>
+
+<p class="center">&lambda; + 1/&mu; = &mu; + 1/&nu; = &nu; + 1/&lambda; =
+&minus;&lambda;&mu;&nu; = 1;</p>
+
+<p class="noind">this proposition may be proved by substituting for &lambda;, &mu;, &nu; and
+reducing to a common origin. There are therefore four equations
+between three unknowns; hence if one cross-ratio be given, the
+remaining twenty-three are determinate. Moreover, two of the
+quantities &lambda;, &mu;, &nu; are positive, and the remaining one negative.</p>
+
+<p>The following scheme shows the twenty-four cross-ratios expressed
+in terms of &lambda;, &mu;, &nu;.]</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcl lb rb cl tb">(AB, CD)<br />(BA, DC)<br />(CD, AB)<br />(DC, BA)</td> <td class="tccm rb cl tb">&lambda;</td> <td class="tccm rb cl tb">1 &minus; &mu;</td> <td class="tccm rb2 cl tb">1/(1 &minus; &nu;)</td>
+  <td class="tcl rb cl tb">(AD, BC)<br />(BC, AD)<br />(CB, DA)<br />(DA, CB)</td> <td class="tccm rb cl tb">(&lambda; &minus; 1)/&lambda;</td> <td class="tccm rb cl tb">&mu;/(&mu; &minus; 1)</td> <td class="tccm rb cl tb">&nu;</td></tr>
+
+
+<tr><td class="tcl lb rb">(AC, DB)<br />(BD, CA)<br />(CA, BD)<br />(DB, AC)</td> <td class="tccm rb">1/(1 &minus; &lambda;)</td> <td class="tccm rb">1/&mu;</td> <td class="tccm rb2">(&nu; &minus; 1)/&nu;</td>
+  <td class="tcl rb">(AC, BD)<br />(BD, AC)<br />(CA, DB)<br />(DB, CA)</td> <td class="tccm rb">1 &minus; &lambda;</td> <td class="tccm rb">&mu;</td> <td class="tccm rb">&nu;/(&nu; &minus; 1)</td></tr>
+
+<tr><td class="tcl lb rb cl bb">(AB, DC)<br />(BA, CD)<br />(CD, BA)<br />(DC, AB)</td> <td class="tccm rb cl bb">1/&lambda;</td> <td class="tccm rb cl bb">1/(1 &minus; &mu;)</td> <td class="tccm rb2 cl bb">1 &minus; &nu;</td>
+  <td class="tcl rb cl bb">(AD, CB)<br />(BC, DA)<br />(CB, AD)<br />(DA, BC)</td> <td class="tccm rb cl bb">&lambda;/(&lambda; &minus; 1)</td> <td class="tccm rb cl bb">(&mu; &minus; 1)/&mu;</td> <td class="tccm rb cl bb">1/&nu;</td></tr>
+</table>
+
+<p>§ 17. If one of the points of which a cross-ratio is formed is the
+point at infinity in the line, the cross-ratio changes into a simple
+ratio. It is convenient to let the point at infinity occupy the last
+place in the symbolic expression for the cross-ratio. Thus if I is a
+point at infinity, we have (AB, CI) = &minus;AC/CB, because AI : IB = &minus;1.</p>
+
+<p>Every common ratio of three points in a line may thus be expressed
+as a cross-ratio, by adding the point at infinity to the group
+of points.</p>
+
+<p class="pt2 center sc">Harmonic Ranges</p>
+
+<p>§ 18. If the points have special positions, the cross-ratios may
+have such a value that, of the six different ones, two and two become
+equal. If the first two shall be equal, we get &lambda; = 1/&lambda;, or &lambda;² = 1,
+&lambda; = ±1.</p>
+
+<p>If we take &lambda; = +1, we have (AB, CD) = 1, or AC/CB = AD/DB;
+that is, the points C and D coincide, provided that A and B are
+different.</p>
+
+<p>If we take &lambda; = &minus;1, so that (AB, CD) = &minus;1, we have AC/CB =
+&minus;AD/DB. <i>Hence C and D divide AB internally and externally in the
+same ratio.</i></p>
+
+<p>The four points are in this case said to be <i>harmonic points</i>, and
+C <i>and</i> D <i>are said to be harmonic conjugates with regard to</i> A <i>and</i> B.</p>
+
+<p>But we have also (CD, AB) = &minus;1, so that A and B are harmonic
+conjugates with regard to C and D.</p>
+
+<p>The principal property of harmonic points is that their cross-ratio
+remains unaltered if we interchange the two points belonging to one
+pair, viz.</p>
+
+<p class="center">(AB, CD) = (AB, DC) = (BA, CD).</p>
+
+<p>For four harmonic points the six cross-ratios become equal two
+and two:</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&lambda; = &minus;1, 1 &minus; &lambda; = 2,</td> <td>&lambda;</td>
+<td rowspan="2">= ½,</td> <td>1</td>
+<td rowspan="2">= &minus;1,</td> <td>1</td>
+<td rowspan="2">= ½,</td> <td>&lambda; &minus; 1</td>
+<td rowspan="2">= 2.</td></tr>
+<tr><td class="denom">&lambda; &minus; 1</td> <td class="denom">&lambda;</td>
+<td class="denom">1 &minus; &lambda;</td> <td class="denom">&lambda;</td></tr></table>
+
+<p>Hence if we get four points whose cross-ratio is 2 or ½, then they
+are harmonic, but not arranged so that conjugates are paired. If
+this is the case the cross-ratio = &minus;1.</p>
+
+<p>§ 19. If we equate any two of the above six values of the cross-ratios,
+we get either &lambda; = 1, 0, &infin;, or &lambda; = &minus;1, 2, ½, or else &lambda; becomes
+a root of the equation &lambda;² &minus; &lambda; + 1 = 0, that is, an imaginary cube root of
+&minus;1. In this case the six values become three and three equal, so
+that only two different values remain. This case, though important
+in the theory of cubic curves, is for our purposes of no interest,
+whilst harmonic points are all-important.</p>
+
+<p>§ 20. From the definition of harmonic points, and by aid of § 11,
+the following properties are easily deduced.</p>
+
+<p>If C and D are harmonic conjugates with regard to A and B,
+then one of them lies in, the other without AB; it is impossible
+to move from A to B without passing either through C or through
+D; the one blocks the finite way, the other the way through infinity.
+This is expressed by saying A and B are &ldquo;separated&rdquo; by
+C and D.</p>
+
+<p>For every position of C there will be one and only one point
+D which is its harmonic conjugate with regard to any point pair
+A, B.</p>
+
+<p>If A and B are different points, and if C coincides with A or B,
+D does. But if A and B coincide, one of the points C or D, lying
+between them, coincides with them, and the other may be anywhere
+in the line. It follows that, &ldquo;<i>if of four harmonic conjugates two
+coincide, then a third coincides with them, and the fourth may be any
+point in the line</i>.&rdquo;</p>
+
+<p>If C is the middle point between A and B, then D is the point at
+infinity; for AC : CB = +1, hence AD : DB must be equal to &minus;1.
+<i>The harmonic conjugate of the point at infinity in a line with regard
+to two points</i> A, B <i>is the middle point of</i> AB.</p>
+
+<p>This important property gives a first example how metric properties
+are connected with projective ones.</p>
+
+<p>[§ 21. <i>Harmonic properties of the complete quadrilateral and quadrangle.</i></p>
+
+<p><span class="pagenum"><a name="page692" id="page692"></a>692</span></p>
+
+<table class="nobctr" style="clear: both;" summary="Illustration">
+<tr><td class="figcenter" colspan="2"><img style="width:500px; height:212px" src="images/img692.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 7.</span></td>
+<td class="caption"><span class="sc">Fig. 8.</span></td></tr></table>
+
+<p>A figure formed by four lines in a plane is called a <i>complete quadrilateral</i>,
+or, shorter, a <i>four-side</i>. The four sides meet in six points,
+named the &ldquo;vertices,&rdquo; which may be joined by three lines (other
+than the sides), named the &ldquo;diagonals&rdquo; or &ldquo;harmonic lines.&rdquo; The
+diagonals enclose the &ldquo;harmonic triangle of the quadrilateral.&rdquo; In
+fig. 7, A&prime;B&prime;C&prime;, B&prime;AC, C&prime;AB, CBA&prime; are the sides, A, A&prime;, B, B&prime;, C, C&prime;
+the vertices, AA&prime;, BB&prime;, CC&prime; the harmonic lines, and &alpha;&beta;&gamma; the harmonic
+triangle of the quadrilateral. A figure formed by four coplanar
+points is named a <i>complete quadrangle</i>, or, shorter, a <i>four-point</i>.
+The four points may be joined by six lines, named the &ldquo;sides,&rdquo;
+which intersect in three other points, termed the &ldquo;diagonal or
+harmonic points.&rdquo; The harmonic points are the vertices of the
+&ldquo;harmonic triangle of the complete quadrangle.&rdquo; In fig. 8, AA&prime;,
+BB&prime; are the points, AA&prime;, BB&prime;, A&prime;B&prime;, B&prime;A, AB, BA&prime; are the sides,
+L, M, N are the diagonal points, and LMN is the harmonic triangle
+of the quadrangle.</p>
+
+<p>The harmonic property of the complete quadrilateral is: Any
+diagonal or harmonic line is harmonically divided by the other
+two; and of a complete quadrangle: The angle at any harmonic
+point is divided harmonically by the joins to the other harmonic
+points. To prove the first theorem, we have to prove (AA&prime;, &beta;&gamma;),
+(BB&prime;, &gamma;&alpha;), (CC&prime;, &beta;&alpha;) are harmonic. Consider the cross-ratio (CC&prime;, &alpha;&beta;).
+Then projecting from A on BB&prime; we have A(CC&prime;, &alpha;&beta;) = A(B&prime;B, &alpha;&gamma;).
+Projecting from A&prime; on BB&prime;, A&prime;(CC&prime;, &alpha;&beta;) = A&prime;(BB&prime;, &alpha;&gamma;). Hence
+(B&prime;B, &alpha;&gamma;) = (BB&prime;, &alpha;&gamma;), <i>i.e.</i> the cross-ratio (BB&prime;, &alpha;&gamma;) equals that of its
+reciprocal; hence the range is harmonic.</p>
+
+<p>The second theorem states that the pencils L(BA, NM), M(B&prime;A, LN),
+N(BA, LM) are harmonic. Deferring the subject of harmonic pencils
+to the next section, it will suffice to state here that any transversal
+intersects an harmonic pencil in an harmonic range. Consider the
+pencil L(BA, NM), then it is sufficient to prove (BA&prime;, NM&prime;) is harmonic.
+This follows from the previous theorem by considering A&prime;B
+as a diagonal of the quadrilateral ALB&prime;M.]</p>
+
+<p>This property of the complete quadrilateral allows the solution
+of the problem:</p>
+
+<p><i>To construct the harmonic conjugate</i> D <i>to a point</i> C <i>with regard to two
+given points</i> A <i>and</i> B.</p>
+
+<p>Through A draw any two lines, and through C one cutting the
+former two in G and H. Join these points to B, cutting the former
+two lines in E and F. The point D where EF cuts AB will be the
+harmonic conjugate required.</p>
+
+<p>This remarkable construction requires nothing but the drawing
+of lines, and is therefore independent of measurement. In a similar
+manner the harmonic conjugate of the line VA for two lines VC,
+VD is constructed with the aid of the property of the complete
+quadrangle.</p>
+
+<p>§ 22. <i>Harmonic Pencils.</i>&mdash;The theory of cross-ratios may be extended
+from points in a row to lines in a flat pencil and to planes in
+an axial pencil. We have seen (§ 13) that if the lines which join four
+points A, B, C, D to any point S be cut by any other line in A&prime;, B&prime;, C&prime;,
+D&prime;, then (AB, CD) = (A&prime;B&prime;, C&prime;D&prime;). In other words, four lines in a
+flat pencil are cut by every other line in four points whose cross-ratio
+is constant.</p>
+
+<p><i>Definition.</i>&mdash;By the cross-ratio of four rays in a flat pencil is
+meant the cross-ratio of the four points in which the rays are cut
+by any line. If a, b, c, d be the lines, then this cross-ratio is denoted
+by (ab, cd).</p>
+
+<p><i>Definition.</i>&mdash;By the cross-ratio of four planes in an axial pencil
+is understood the cross-ratio of the four points in which any line
+cuts the planes, or, what is the same thing, the cross-ratio of the
+four rays in which any plane cuts the four planes.</p>
+
+<p>In order that this definition may have a meaning, it has to be
+proved that all lines cut the pencil in points which have the same
+cross-ratio. This is seen at once for two intersecting lines, as their
+plane cuts the axial pencil in a flat pencil, which is itself cut by
+the two lines. The cross-ratio of the four points on one line is
+therefore equal to that on the other, and equal to that of the four
+rays in the flat pencil.</p>
+
+<p>If two non-intersecting lines p and q cut the four planes in A, B,
+C, D and A&prime;, B&prime;, C&prime;, D&prime;, draw a line r to meet both p and q, and
+let this line cut the planes in A&Prime;, B&Prime;, C&Prime;, D&Prime;. Then (AB, CD) =
+(A&prime;B&prime;, C&prime;D&prime;), for each is equal to (A&Prime;B&Prime;, C&Prime;D&Prime;).</p>
+
+<p>§ 23. We may now also extend the notion of harmonic elements,
+viz.</p>
+
+<p><i>Definition.</i>&mdash;Four rays in a flat pencil and four planes in an axial
+pencil are said to be harmonic if their cross-ratio equals -1, that is,
+if they are cut by a line in four harmonic points.</p>
+
+<p>If we understand by a &ldquo;median line&rdquo; of a triangle a line which
+joins a vertex to the middle point of the opposite side, and by a
+&ldquo;median line&rdquo; of a parallelogram a line joining middle points of
+opposite sides, we get as special cases of the last theorem:</p>
+
+<p><i>The diagonals and median lines of a parallelogram form an harmonic
+pencil</i>; and</p>
+
+<p><i>At a vertex of any triangle, the two sides, the median line, and the
+line parallel to the base form an harmonic pencil.</i></p>
+
+<p>Taking the parallelogram a rectangle, or the triangle isosceles,
+we get:</p>
+
+<p><i>Any two lines and the bisections of their angles form an harmonic
+pencil.</i> Or:</p>
+
+<p><i>In an harmonic pencil, if two conjugate rays are perpendicular,
+then the other two are equally inclined to them</i>; and, conversely, <i>if
+one ray bisects the angle between conjugate rays, it is perpendicular to
+its conjugate</i>.</p>
+
+<p>This connects perpendicularity and bisection of angles with
+projective properties.</p>
+
+<p>§ 24. We add a few theorems and problems which are easily proved
+or solved by aid of harmonics.</p>
+
+<p>An harmonic pencil is cut by a line parallel to one of its rays in
+three equidistant points.</p>
+
+<p>Through a given point to draw a line such that the segment
+determined on it by a given angle is bisected at that point.</p>
+
+<p>Having given two parallel lines, to bisect on either any given
+segment without using a pair of compasses.</p>
+
+<p>Having given in a line a segment and its middle point, to draw
+through any given point in the plane a line parallel to the given line.</p>
+
+<p>To draw a line which joins a given point to the intersection of two
+given lines which meet off the drawing paper (by aid of § 21).</p>
+
+<p class="pt2 center sc">Correspondence. Homographic and Perspective Ranges</p>
+
+<p>§ 25. Two rows, p and p&prime;, which are one the projection of the
+other (as in fig. 5), stand in a definite relation to each other, characterized
+by the following properties.</p>
+
+<p>1. <i>To each point in either corresponds one point in the other</i>; that
+is, those points are said to correspond which are projections of one
+another.</p>
+
+<p>2. <i>The cross-ratio of any four points in one equals that of the corresponding
+points in the other.</i></p>
+
+<p>3. <i>The lines joining corresponding points all pass through the same
+point.</i></p>
+
+<p>If we suppose corresponding points marked, and the rows brought
+into any other position, then the lines joining corresponding points
+will no longer meet in a common point, and hence the third of the
+above properties will not hold any longer; but we have still a
+correspondence between the points in the two rows possessing the first
+two properties. Such a correspondence has been called a <i>one-one
+correspondence</i>, whilst the two rows between which such correspondence
+has been established are said to be <i>projective</i> or <i>homographic</i>.
+Two rows which are each the projection of the other are therefore
+<i>projective</i>. We shall presently see, also, that any two projective
+rows may always be placed in such a position that one appears as
+the projection of the other. If they are in such a position the rows
+are said to be in <i>perspective position</i>, or simply to be in <i>perspective</i>.</p>
+
+<p>§ 26. The notion of a one-one correspondence between rows may
+be extended to flat and axial pencils, viz. a flat pencil will be said
+to be projective to a flat pencil if to each ray in the first corresponds
+one ray in the second, and if the cross-ratio of four rays in one equals
+that of the corresponding rays in the second.</p>
+
+<p>Similarly an axial pencil may be projective to an axial pencil.
+But a flat pencil may also be projective to an axial pencil, or either
+pencil may be projective to a row. The definition is the same in each
+case: there is a one-one correspondence between the elements, and
+four elements have the same cross-ratio as the corresponding ones.</p>
+
+<p>§ 27. There is also in each case a special position which is called
+<i>perspective</i>, viz.</p>
+
+<p>1. Two projective rows are perspective if they lie in the same
+plane, and if the one row is a projection of the other.</p>
+
+<p>2. Two projective flat pencils are perspective&mdash;(1) if they lie in
+the same plane, and have a row as a common section; (2) if they
+lie in the same pencil (in space), and are both sections of the same
+axial pencil; (3) if they are in space and have a row as common
+section, or are both sections of the same axial pencil, one of the
+conditions involving the other.</p>
+
+<p>3. Two projective axial pencils, if their axes meet, and if they
+have a flat pencil as a common section.</p>
+
+<p>4. A row and a projective flat pencil, if the row is a section of the
+pencil, each point lying in its corresponding line.</p>
+
+<p>5. A row and a projective axial pencil, if the row is a section of the
+pencil, each point lying in its corresponding line.</p>
+
+<p>6. A flat and a projective axial pencil, if the former is a section
+of the other, each ray lying in its corresponding plane.</p>
+
+<p>That in each case the correspondence established by the position
+indicated is such as has been called projective follows at once from
+the definition. It is not so evident that the perspective position may
+always be obtained. We shall show in § 30 this for the first three
+<span class="pagenum"><a name="page693" id="page693"></a>693</span>
+cases. First, however, we shall give a few theorems which relate to
+the general correspondence, not to the perspective position.</p>
+
+<p>§ 28. <i>Two rows or pencils, flat or axial, which are projective to a
+third are projective to each other</i>; this follows at once from the
+definitions.</p>
+
+<p>§ 29. <i>If two rows, or two pencils, either flat or axial, or a row and a
+pencil, be projective, we may assume to any three elements in the one
+the three corresponding elements in the other, and then the correspondence
+is uniquely determined.</i></p>
+
+<p>For if in two projective rows we assume that the points A, B, C
+in the first correspond to the given points A&prime;, B&prime;, C&prime; in the second,
+then to any fourth point D in the first will correspond a point D&prime;
+in the second, so that</p>
+
+<p class="center">(AB, CD) = (A&prime;B&prime;, C&prime;D&prime;).</p>
+
+<p class="noind">But there is only one point, D&prime;, which makes the cross-ratio
+(A&prime;B&prime;, C&prime;D&prime;) equal to the given number (AB, CD).</p>
+
+<p>The same reasoning holds in the other cases.</p>
+
+<p>§ 30. If two rows are perspective, then the lines joining corresponding
+points all meet in a point, the centre of projection; and
+the point in which the two bases of the rows intersect as a point
+in the first row coincides with its corresponding point in the
+second.</p>
+
+<p>This follows from the definition. The converse also holds,
+viz.</p>
+
+<p><i>If two projective rows have such a position that one point in the one
+coincides with its corresponding point in the other, then they are perspective,
+that is, the lines joining corresponding points all pass through
+a common point, and form a flat pencil.</i></p>
+
+<p>For let A, B, C, D ... be points in the one, and A&prime;, B&prime;, C&prime;,
+D&prime; ... the corresponding points in the other row, and let A be made
+to coincide with its corresponding point A&prime;. Let S be the point where
+the lines BB&prime; and CC&prime; meet, and let us join S to the point D in the
+first row. This line will cut the second row in a point D&Prime;, so that
+A, B, C, D are projected from S into the points A, B&prime;, C&prime;, D&Prime;. The
+cross-ratio (AB, CD) is therefore equal to (AB&prime;, C&prime;D&Prime;), and by hypothesis
+it is equal to (A&prime;B&prime;, C&prime;D&prime;). Hence (A&prime;B&prime;, C&prime;D&Prime;) = (A&prime;B&prime;, C&prime;D&prime;),
+that is, D&Prime; is the same point as D&prime;.</p>
+
+<p>§ 31. If two projected flat pencils in the same plane are in perspective,
+then the intersections of corresponding lines form a row,
+and the line joining the two centres as a line in the first pencil
+corresponds to the same line as a line in the second. And conversely,</p>
+
+<p><i>If two projective pencils in the same plane, but with different centres,
+have one line in the one coincident with its corresponding line in the
+other, then the two pencils are perspective, that is, the intersection of
+corresponding lines lie in a line.</i></p>
+
+<p>The proof is the same as in § 30.</p>
+
+<p>§ 32. If two projective flat pencils in the same point (pencil in
+space), but not in the same plane, are perspective, then the planes
+joining corresponding rays all pass through a line (they form an
+axial pencil), and the line common to the two pencils (in which
+their planes intersect) corresponds to itself. And conversely:&mdash;</p>
+
+<p>If two flat pencils which have a common centre, but do not lie
+in a common plane, are placed so that one ray in the one coincides
+with its corresponding ray in the other, then they are perspective,
+that is, the planes joining corresponding lines all pass through a
+line.</p>
+
+<p>§ 33. If two projective axial pencils are perspective, then the intersection
+of corresponding planes lie in a plane, and the plane common
+to the two pencils (in which the two axes lie) corresponds to itself.
+And conversely:&mdash;</p>
+
+<p>If two projective axial pencils are placed in such a position that a
+plane in the one coincides with its corresponding plane, then the two
+pencils are perspective, that is, corresponding planes meet in lines
+which lie in a plane.</p>
+
+<p>The proof again is the same as in § 30.</p>
+
+<p>§ 34. These theorems relating to perspective position become
+illusory if the projective rows of pencils have a common base. We
+then have:&mdash;</p>
+
+<p>In two projective rows on the same line&mdash;and also in two projective
+and concentric flat pencils in the same plane, or in two
+projective axial pencils with a common axis&mdash;every element in the
+one coincides with its corresponding element in the other as soon
+as three elements in the one coincide with their corresponding
+elements in the other.</p>
+
+<p><i>Proof</i> (in case of two rows).&mdash;Between four elements A, B, C, D
+and their corresponding elements A&prime;, B&prime;, C&prime;, D&prime; exists the relation
+(ABCD) = (A&prime;B&prime;C&prime;D&prime;). If now A&prime;, B&prime;, C&prime; coincide respectively with
+A, B, C, we get (AB, CD) = (AB, CD&prime;), hence D and D&prime; coincide.</p>
+
+<p>The last theorem may also be stated thus:&mdash;</p>
+
+<p>In two projective rows or pencils, which have a common base
+but are not identical, not more than two elements in the one can
+coincide with their corresponding elements in the other.</p>
+
+<p>Thus two projective rows on the same line cannot have more
+than two pairs of coincident points unless every point coincides
+with its corresponding point.</p>
+
+<table class="flt" style="float: right; width: 370px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:309px; height:275px" src="images/img693a.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 9.</span></td></tr>
+<tr><td class="figright1"><img style="width:301px; height:266px" src="images/img693b.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 10.</span></td></tr>
+<tr><td class="figright1"><img style="width:316px; height:361px" src="images/img693c.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 11.</span></td></tr></table>
+
+<p>It is easy to construct two projective rows on the same line,
+which have two pairs of corresponding points coincident. Let the
+points A, B, C as points belonging to the one row correspond to A,
+B, and C&prime; as points in the second. Then A and B coincide with their
+corresponding points, but C does not. It is, however, not necessary
+that two such rows
+have twice a point
+coincident with its corresponding
+point; it is
+possible that this happens
+only once or not
+at all. Of this we shall
+see examples later.</p>
+
+<p>§ 35. If two projective
+rows or pencils are in
+perspective position, we
+know at once which
+element in one corresponds
+to any given
+element in the other.
+If p and q (fig. 9) are
+two projective rows, so
+that K corresponds to
+itself, and if we know
+that to A and B in p
+correspond A&prime; and B&prime; in q, then the point S, where AA&prime; meets BB&prime;,
+is the centre of projection, and hence, in order to find the point C&prime;
+corresponding to C, we have only to join C to S; the point C&prime;,
+where this line cuts q, is the point required.</p>
+
+<p>If two flat pencils, S<span class="su">1</span> and S<span class="su">2</span>, in a plane are perspective (fig. 10),
+we need only to know two pairs, a, a&prime; and b, b&prime;, of corresponding
+rays in order to find the
+axis s of projection. This
+being known, a ray c&prime; in
+S<span class="su">2</span>, corresponding to a given
+ray c in S<span class="su">1</span>, is found by
+joining S<span class="su">2</span> to the point
+where c cuts the axis s.</p>
+
+<p>A similar construction
+holds in the other cases
+of perspective figures.</p>
+
+<p>On this depends the
+solution of the following
+general problem.</p>
+
+<p>§ 36. Three pairs of corresponding
+elements in two
+projective rows or pencils
+being given, to determine
+for any element in one
+the corresponding element
+in the other.</p>
+
+<p>We solve this in the two cases of two projective rows and of two
+projective flat pencils in a plane.</p>
+
+<p><i>Problem</i> I.&mdash;Let A, B, C be
+three points in a row s, A&prime;, B&prime;, C&prime;
+the corresponding points in a
+projective row s&prime;, both being in a
+plane; it is required to find for
+any point D in s the corresponding
+point D&prime; in s&prime;.</p>
+
+<p><i>Problem</i> II.&mdash;Let a, b, c be
+three rays in a pencil S, a&prime;, b&prime;, c&prime;
+the corresponding rays in a projective
+pencil S&prime;, both being in
+the same plane; it is required to
+find for any ray d in S the corresponding
+ray d&prime; in S&prime;.</p>
+
+<p>The solution is made to depend on the construction of an auxiliary
+row or pencil which is perspective to both the given ones. This is
+found as follows:&mdash;</p>
+
+<p><i>Solution of Problem</i> I.&mdash;On the line joining two corresponding
+points, say AA&prime; (fig. 11), take any two points, S and S&prime;, as centres
+of auxiliary pencils.
+Join the intersection B<span class="su">1</span>
+of SB and S&prime;B&prime; to the
+intersection C<span class="su">1</span> of SC
+and S&prime;C&prime; by the line s<span class="su">1</span>.
+Then a row on s<span class="su">1</span> will
+be perspective to s with
+S as centre of projection,
+and to s&prime; with S&prime;
+as centre. To find now
+the point D&prime; on s&prime; corresponding
+to a point
+D on s we have only to
+determine the point D<span class="su">1</span>,
+where the line SD cuts
+s<span class="su">1</span>, and to draw S&prime;D<span class="su">1</span>;
+the point where this line
+cuts s&prime; will be the required
+point D&prime;.</p>
+
+<p><i>Proof.</i>&mdash;The rows s
+and s&prime; are both perspective
+to the row s<span class="su">1</span>, hence
+they are projective to
+one another. To A, B,
+C, D on s correspond
+A<span class="su">1</span>, B<span class="su">1</span>, C<span class="su">1</span>, D<span class="su">1</span> on s<span class="su">1</span>, and
+to these correspond A&prime;, B&prime;, C&prime;, D&prime; on s&prime;; so that D and D&prime; are
+corresponding points as required.</p>
+
+<p><span class="pagenum"><a name="page694" id="page694"></a>694</span></p>
+
+<table class="flt" style="float: left; width: 315px;" summary="Illustration">
+<tr><td class="figleft1"><img style="width:251px; height:310px" src="images/img694a.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 12.</span></td></tr>
+<tr><td class="figleft1"><img style="width:265px; height:233px" src="images/img694b.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 13.</span></td></tr></table>
+
+<p><i>Solution of Problem</i> II.&mdash;Through the intersection A of two
+corresponding rays a and a&prime; (fig. 12), take two lines, s and s&prime;, as
+bases of auxiliary rows. Let S<span class="su">1</span>
+be the point where the line b<span class="su">1</span>,
+which joins B and B&prime;, cuts the
+line c<span class="su">1</span>, which joins C and C&prime;.
+Then a pencil S<span class="su">1</span> will be perspective
+to S with s as axis of
+projection. To find the ray d&prime; in
+S&prime; corresponding to a given ray d
+in S, cut d by s at D; project
+this point from S<span class="su">1</span> to D&prime; on s&prime;
+and join D&prime; to S&prime;. This will be
+the required ray.</p>
+
+<p><i>Proof.</i>&mdash;That the pencil S<span class="su">1</span> is
+perspective to S and also to S&prime;
+follows from construction. To
+the lines a<span class="su">1</span>, b<span class="su">1</span>, c<span class="su">1</span>, d<span class="su">1</span> in S<span class="su">1</span> correspond
+the lines a, b, c, d in S and
+the lines a&prime;, b&prime;, c&prime;, d&prime; in S&prime;, so that d
+and d&prime; are corresponding rays.</p>
+
+<p>In the first solution the two
+centres, S, S&prime;, are <i>any</i> two points
+on a line joining any two corresponding
+points, so that the solution
+of the problem allows of a great many different constructions.
+<i>But whatever construction be used, the point</i> D&prime;, <i>corresponding to</i> D,
+<i>must be always the same</i>, according to the theorem in § 29. This
+gives rise to a number of theorems, into which, however, we shall
+not enter. The same remarks hold for the second problem.</p>
+
+<p>§ 37. <i>Homological Triangles.</i>&mdash;As a further application of the
+theorems about perspective rows and pencils we shall prove the
+following important theorem.</p>
+
+<p><i>Theorem.</i>&mdash;If ABC and A&prime;B&prime;C&prime; (fig. 13) be two triangles, such that
+the lines AA&prime;, BB&prime;, CC&prime; meet in a point S, then the intersections of
+BC and B&prime;C&prime;, of CA and C&prime;A&prime;, and of AB and A&prime;B&prime; will lie in a line.
+Such triangles are said to be homological, or in perspective. The
+triangles are &ldquo;co-axial&rdquo; in virtue of the property that the meets of
+corresponding sides are collinear and copolar, since the lines joining
+corresponding vertices are concurrent.</p>
+
+<p><i>Proof.</i>&mdash;Let a, b, c denote the lines AA&prime;, BB&prime;, CC&prime;, which meet at
+S. Then these may be taken as bases of projective rows, so that
+A, A&prime;, S on a correspond to B, B&prime;, S on b, and to C, C&prime;, S on c. As
+the point S is common to all, any two of these rows will be perspective.</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcc">If</td> <td class="tcl">S<span class="su">1</span> be the centre of projection of rows</td> <td class="tcl">b and c,</td></tr>
+
+<tr><td class="tcc">&nbsp;</td> <td class="tcl">S<span class="su">2</span> &emsp;&emsp; &rdquo; &emsp;&emsp;&emsp; &rdquo; &emsp;&emsp;&emsp; &rdquo;</td> <td class="tcl">c and a,</td></tr>
+
+<tr><td class="tcc">&nbsp;</td> <td class="tcl">S<span class="su">3</span> &emsp;&emsp; &rdquo; &emsp;&emsp;&emsp; &rdquo; &emsp;&emsp;&emsp; &rdquo;</td> <td class="tcl">a and b,</td></tr>
+</table>
+
+<p class="noind">and if the line S<span class="su">1</span>S<span class="su">2</span> cuts a in A<span class="su">1</span>, and b in B<span class="su">1</span>, and c in C<span class="su">1</span>, then A<span class="su">1</span>, B<span class="su">1</span>
+will be corresponding points
+in a and b, both corresponding
+to C<span class="su">1</span> in c. But a and b are
+perspective, therefore the line
+A<span class="su">1</span>B<span class="su">1</span>, that is S<span class="su">1</span>S<span class="su">2</span>, joining
+corresponding points must
+pass through the centre of
+projection S<span class="su">3</span> of a and b. In
+other words, S<span class="su">1</span>, S<span class="su">2</span>, S<span class="su">3</span> lie in a
+line. This is Desargues&rsquo; celebrated
+theorem if we state it
+thus:&mdash;</p>
+
+<p><i>Theorem of Desargues.</i>&mdash;If
+each of two triangles has one
+vertex on each of three concurrent
+lines, then the intersections
+of corresponding sides
+lie in a line, those sides
+being called corresponding which are opposite to vertices on the
+same line.</p>
+
+<p>The converse theorem holds also, viz.</p>
+
+<p><i>Theorem.</i>&mdash;If the sides of one triangle meet those of another in
+three points which lie in a line, then the vertices lie on three lines
+which meet in a point.</p>
+
+<p>The proof is almost the same as before.</p>
+
+<p>§ 38. <i>Metrical Relations between Projective Rows.</i>&mdash;Every row
+contains one point which is distinguished from all others, viz.
+the point at infinity. In two projective rows, to the point I at
+infinity in one corresponds a point I&prime; in the other, and to the point
+J&prime; at infinity in the second corresponds a point J in the first. The
+points I&prime; and J are in general finite. If now A and B are any two
+points in the one, A&prime;, B&prime; the corresponding points in the other row,
+then</p>
+
+<p class="center">(AB, JI) = (A&prime;B&prime;, J&prime;I&prime;),</p>
+
+<p class="noind">or</p>
+
+<p class="center">AJ/JB : AI/IB = A&prime;J&prime;/J&prime;B&prime; : A&prime;I&prime;/I&prime;B&prime;.</p>
+
+<p>But, by § 17,</p>
+
+<p class="center">AI/IB = A&prime;J&prime;/J&prime;B&prime; = &minus;1;</p>
+
+<p class="noind">therefore the last equation changes into</p>
+
+<p class="center">AJ · A&prime;I&prime; = BJ · B&prime;I&prime;,</p>
+
+<p class="noind">that is to say&mdash;</p>
+
+<p><i>Theorem.</i>&mdash;The product of the distances of any two corresponding
+points in two projective rows from the points which correspond to
+the points at infinity in the other is constant, viz. AJ · A&prime;I&prime; = k.
+Steiner has called this number k the <i>Power of the correspondence</i>.</p>
+
+<p>[The relation AJ · A&prime;I&prime; = k shows that if J, I&prime; be given then the
+point A&prime; corresponding to a specified point A is readily found; hence
+A, A&prime; generate homographic ranges of which I and J&prime; correspond to
+the points at infinity on the ranges. If we take any two origins O,
+O&prime;, on the ranges and reduce the expression AJ · A&prime;I&prime; = k to its algebraic
+equivalent, we derive an equation of the form &alpha;xx&prime; + &beta;x + &gamma;x&prime;
++ &delta; = 0. Conversely, if a relation of this nature holds, then points
+corresponding to solutions in x, x&prime; form homographic ranges.]</p>
+
+<p>§ 39. <i>Similar Rows.</i>&mdash;If the points at infinity in two projective
+rows correspond so that I&prime; and J are at infinity, this result loses its
+meaning. But if A, B, C be any three points in one, A&prime;, B&prime;, C&prime; the
+corresponding ones on the other row, we have</p>
+
+<p class="center">(AB, CI) = (A&prime;B&prime;, C&prime;I&prime;),</p>
+
+<p class="noind">which reduces to</p>
+
+<p class="center">AC/CB = A&prime;C&prime;/C&prime;B&prime; or AC/A&prime;C&prime; = BC/B&prime;C&prime;,</p>
+
+<p class="noind">that is, corresponding segments are proportional. Conversely, if
+corresponding segments are proportional, then to the point at
+infinity in one corresponds the point at infinity in the other. If we call
+such rows <i>similar</i>, we may state the result thus&mdash;</p>
+
+<p><i>Theorem.</i>&mdash;Two projective rows are similar if to the point at
+infinity in one corresponds the point at infinity in the other, and
+conversely, if two rows are similar then they are projective, and the
+points at infinity are corresponding points.</p>
+
+<p>From this the well-known propositions follow:&mdash;</p>
+
+<p>Two lines are cut proportionally (in similar rows) by a series of
+parallels. The rows are perspective, with centre of projection at
+infinity.</p>
+
+<p>If two similar rows are placed parallel, then the lines joining
+homologous points pass through a common point.</p>
+
+<p>§ 40. If two flat pencils be projective, then there exists in either,
+one single pair of lines at right angles to one another, such that the
+corresponding lines in the other pencil are again at right angles.</p>
+
+<table class="flt" style="float: right; width: 300px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:252px; height:248px" src="images/img694c.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 14.</span></td></tr></table>
+
+<p>To prove this, we place the pencils in perspective position (fig. 14)
+by making one ray coincident
+with its corresponding
+ray. Corresponding rays
+meet then on a line p. And
+now we draw the circle which
+has its centre O on p, and
+which passes through the
+centres S and S&prime; of the two
+pencils. This circle cuts p in
+two points H and K. The
+two pairs of rays, h, k, and
+h&prime;, k&prime;, joining these points to
+S and S&prime; will be pairs of
+corresponding rays at right
+angles. The construction
+gives in general but one
+circle, but if the line p is
+the perpendicular bisector
+of SS&prime;, there exists an infinite
+number, and <i>to every
+right angle in the one pencil corresponds a right angle in the
+other</i>.</p>
+
+<p class="pt2 center sc" style="clear: both;">Principle of Duality</p>
+
+<p>§ 41. It has been stated in § 1 that not only points, but also planes
+and lines, are taken as elements out of which figures are built up.
+We shall now see that the construction of one figure which possesses
+certain properties gives rise in many cases to the construction of
+another figure, by replacing, according to definite rules, elements
+of one kind by those of another. The new figure thus obtained will
+then possess properties which may be stated as soon as those of the
+original figure are known.</p>
+
+<p>We obtain thus a principle, known as the <i>principle of duality</i>
+or of <i>reciprocity</i>, which enables us to construct to any figure not
+containing any measurement in its construction a <i>reciprocal</i> figure,
+as it is called, and to deduce from any theorem a <i>reciprocal</i> theorem,
+for which no further proof is needed.</p>
+
+<p>It is convenient to print reciprocal propositions on opposite sides
+of a page broken into two columns, and this plan will occasionally
+be adopted.</p>
+
+<p>We begin by repeating in this form a few of our former statements:&mdash;</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p>Two points determine a line.</p></td>
+<td class="tcl" style="width: 50%;"><p>Two planes determine a line.</p></td></tr>
+
+<tr><td class="tcl rb3"><p>Three points which are not in a line determine a plane.</p></td>
+<td class="tcl"><p>Three planes which do not pass through a line determine a point.</p></td></tr>
+
+<tr><td class="tcl rb3"><p>A line and a point without it determine a plane.</p></td>
+<td class="tcl"><p>A line and a plane not through it determine a point.</p></td></tr>
+
+<tr><td class="tcl rb3"><p>Two lines in a plane determine a point.</p></td>
+<td class="tcl"><p>Two lines through a point determine a plane.</p></td></tr></table>
+
+<p>These propositions show that it will be possible, when any figure
+is given, to construct a second figure by taking planes instead of
+points, and points instead of planes, but lines where we had lines.</p>
+
+<p><span class="pagenum"><a name="page695" id="page695"></a>695</span></p>
+
+<p>For instance, if in the first figure we take a plane and three points
+in it, we have to take in the second figure a point and three planes
+through it. The three points in the first, together with the three
+lines joining them two and two, form a triangle; the three planes
+in the second and their three lines of intersection form a trihedral
+angle. A triangle and a trihedral angle are therefore reciprocal
+figures.</p>
+
+<p>Similarly, to any figure in a plane consisting of points and lines
+will correspond a figure consisting of planes and lines passing through
+a point S, and hence belonging to the pencil which has S as centre.</p>
+
+<p>The figure reciprocal to four points in space which do not lie
+in a plane will consist of four planes which do not meet in a point.
+In this case each figure forms a tetrahedron.</p>
+
+<p>§ 42. As other examples we have the following:&mdash;</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcl">To a row</td> <td class="tcc">is reciprocal</td> <td class="tcl">an axial pencil,</td></tr>
+
+<tr><td class="tcl">to a flat pencil</td> <td class="tcc">&rdquo;</td> <td class="tcl">a flat pencil,</td></tr>
+
+<tr><td class="tcl">to a field of points and lines</td> <td class="tcc">&rdquo;</td> <td class="tcl">a pencil of planes and lines,</td></tr>
+
+<tr><td class="tcl">to the space of points</td> <td class="tcc">&rdquo;</td> <td class="tcl">the space of planes.</td></tr>
+</table>
+
+<p class="noind">For the row consists of a line and all the points in it, reciprocal to
+it therefore will be a line with all planes through it, that is, an axial
+pencil; and so for the other cases.</p>
+
+<p>This correspondence of reciprocity breaks down, however, if we
+take figures which contain measurement in their construction. For
+instance, there is no figure reciprocal to two planes at <i>right angles</i>,
+because there is no segment in a row which has a magnitude as
+definite as a right angle.</p>
+
+<p>We add a few examples of reciprocal propositions which are easily
+proved.</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p><i>Theorem.</i>&mdash;If A, B, C, D are any four points in space, and if
+ the lines AB and CD meet, then all four points lie in a plane,
+ hence also AC and BD, as well as AD and BC, meet.</p></td>
+
+<td class="tcl" style="width: 50%;"><p><i>Theorem.</i>&mdash;If &alpha;, &beta;, &gamma;, &delta; are four planes in space, and if the
+ lines &alpha;&beta; and &gamma;&delta; meet, then all four planes lie in a point (pencil),
+ hence also &alpha;&gamma; and &beta;&delta;, as well as &alpha;&delta; and &beta;&gamma;, meet.</p></td></tr></table>
+
+<p>Theorem.&mdash;<i>If of any number of lines every one meets every other,
+whilst all do not</i></p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p><i>lie in a point, then all lie in a plane</i>.</p></td>
+
+<td class="tcl" style="width: 50%;"><p><i>lie in a plane, then all lie in a point</i> (<i>pencil</i>).</p></td></tr></table>
+
+<p>§ 43. Reciprocal figures as explained lie both in space of three
+dimensions. If the one is confined to a plane (is formed of elements
+which lie in a plane), then the reciprocal figure is confined to a pencil
+(is formed of elements which pass through a point).</p>
+
+<p>But there is also a more special principle of duality, according to
+which figures are reciprocal which lie both in a plane or both in a
+pencil. In the plane we take points and lines as reciprocal elements,
+for they have this fundamental property in common, that two
+elements of one kind determine one of the other. In the pencil,
+on the other hand, lines and planes have to be taken as reciprocal,
+and here it holds again that two lines or planes determine one plane
+or line.</p>
+
+<p>Thus, to one plane figure we can construct one reciprocal figure
+in the plane, and to each one reciprocal figure in a pencil. We
+mention a few of these. At first we explain a few names:&mdash;</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p>A figure consisting of n points in a plane will be called an n-point.</p></td>
+<td class="tcl" style="width: 50%;"><p>A figure consisting of n lines in a plane will be called an n-side.</p></td></tr>
+
+<tr><td class="tcl rb3"><p>A figure consisting of n planes in a pencil will be called an n-flat.</p></td>
+<td class="tcl"><p>A figure consisting of n lines in a pencil will be called an n-edge.</p></td></tr></table>
+
+<p>It will be understood that an n-side is different from a polygon
+of n sides. The latter has sides of finite length and n vertices, the
+former has sides all of infinite extension, and every point where
+two of the sides meet will be a vertex. A similar difference exists
+between a solid angle and an n-edge or an n-flat. We notice particularly&mdash;</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p>A four-point has six sides, of which two and two are opposite,
+ and three diagonal points, which are intersections of opposite sides.</p></td>
+
+<td class="tcl" style="width: 50%;"><p>A four-side has six vertices, of which two and two are opposite,
+ and three diagonals, which join opposite vertices.</p></td></tr>
+
+<tr><td class="tcl rb3"><p>A four-flat has six edges, of which two and two are opposite,
+ and three diagonal planes, which pass through opposite edges.</p></td>
+
+<td class="tcl"><p>A four-edge has six faces, of which two and two are opposite,
+ and three diagonal edges, which are intersections of opposite faces.</p></td></tr></table>
+
+<p>A four-side is usually called a complete quadrilateral, and a four-point
+a complete quadrangle. The above notation, however, seems
+better adapted for the statement of reciprocal propositions.</p>
+
+<p>§ 44.</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p>If a point moves in a plane it describes a plane curve.</p></td>
+
+<td class="tcl" style="width: 50%;"><p>If a line moves in a plane it envelopes a plane curve (fig. 15).</p></td></tr>
+
+<tr><td class="tcl rb3"><p>If a plane moves in a pencil it envelopes a cone.</p></td>
+
+<td class="tcl"><p>If a line moves in a pencil it describes a cone.</p></td></tr></table>
+
+<p>A curve thus appears as generated either by points, and then we
+call it a &ldquo;locus,&rdquo; or by lines, and then we call it an &ldquo;envelope.&rdquo;
+In the same manner a cone, which means here a surface, appears
+either as the locus of lines passing through a fixed point, the &ldquo;vertex&rdquo;
+of the cone, or as the envelope of planes passing through the same
+point.</p>
+
+<table class="flt" style="float: right; width: 240px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:192px; height:126px" src="images/img695.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 15.</span></td></tr></table>
+
+<p>To a surface as locus of points corresponds, in the same manner,
+a surface as envelope of planes; and to
+a curve in space as locus of points corresponds
+a developable surface as envelope
+of planes.</p>
+
+<p>It will be seen from the above that
+we may, by aid of the principle of
+duality, construct for every figure a
+reciprocal figure, and that to any
+property of the one a reciprocal property
+of the other will exist, as long
+as we consider only properties which
+depend upon nothing but the positions and intersections of the
+different elements and not upon measurement.</p>
+
+<p>For such propositions it will therefore be unnecessary to prove
+more than one of two reciprocal theorems.</p>
+
+<p class="pt2 center sc" style="clear: both;">Generation of Curves and Cones of Second Order
+or Second Class</p>
+
+<p>§ 45. <i>Conics.</i>&mdash;If we have two projective pencils in a plane,
+corresponding rays will meet, and their point of intersection will
+constitute some locus which we have to investigate. Reciprocally,
+if two projective rows in a plane are given, then the lines which join
+corresponding points will envelope some curve. We prove first:&mdash;</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p><i>Theorem.</i>&mdash;If two projective flat pencils lie in a plane, but
+ are neither in perspective nor concentric, then the locus of
+ intersections of corresponding rays is a curve of the second
+ order, that is, no line contains more than two points of the locus.</p></td>
+
+<td class="tcl" style="width: 50%;"><p><i>Theorem.</i>&mdash;If two projective rows lie in a plane, but are
+ neither in perspective nor on a common base, then the envelope
+ of lines joining corresponding points is a curve of the second
+ class, that is, through no point pass more than two of the enveloping lines.</p></td></tr>
+
+<tr><td class="tcl rb3"><p>Proof.&mdash;We draw any line t. This cuts each of the pencils in a
+ row, so that we have on t two rows, and these are projective
+ because the pencils are projective. If corresponding rays
+ of the two pencils meet on the line t, their intersection will be a
+ point in the one row which coincides with its corresponding
+ point in the other. But two projective rows on the same base
+ cannot have more than two points of one coincident with
+ their corresponding points in the other (§ 34).</p></td>
+
+<td class="tcl"><p><i>Proof.</i>&mdash;We take any point T and join it to all points in each
+ row. This gives two concentric pencils, which are projective
+ because the rows are projective. If a line joining corresponding
+ points in the two rows passes through T, it will be a line in the
+ one pencil which coincides with its corresponding line in the
+ other. But two projective concentric flat pencils in the same
+ plane cannot have more than two lines of one coincident with their
+ corresponding line in the other (§ 34).</p></td></tr></table>
+
+<p>It will be seen that the proofs are reciprocal, so that the one may
+be copied from the other by simply interchanging the words point
+and line, locus and envelope, row and pencil, and so on. We shall
+therefore in future prove seldom more than one of two reciprocal
+theorems, and often state one theorem only, the reader being recommended
+to go through the reciprocal proof by himself, and to supply
+the reciprocal theorems when not given.</p>
+
+<p>§ 46. We state the theorems in the pencil reciprocal to the last,
+without proving them:&mdash;</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p><i>Theorem.</i>&mdash;If two projective flat pencils are concentric, but
+ are neither perspective nor coplanar, then the envelope of the
+ planes joining corresponding rays is a cone of the second class;
+ that is, no line through the common centre contains more
+ than two of the enveloping planes.</p></td>
+
+<td class="tcl" style="width: 50%;"><p><i>Theorem.</i>&mdash;If two projective axial pencils lie in the same
+ pencil (their axes meet in a point), but are neither perspective
+ nor co-axial, then the locus of lines joining corresponding
+ planes is a cone of the second order; that is, no plane in the
+ pencil contains more than two of these lines.</p></td></tr></table>
+
+<p>§ 47. Of theorems about cones of second order and cones of second
+class we shall state only very few. We point out, however, the
+following connexion between the curves and cones under consideration:</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p>The lines which join any point in space to the points on a curve
+ of the second order form a cone of the second order.</p></td>
+
+<td class="tcl" style="width: 50%;"><p>Every plane section of a cone of the second order is a curve of
+ the second order.</p></td></tr>
+
+<tr><td class="tcl rb3"><p>The planes which join any point in space to the lines enveloping
+ a curve of the second class envelope themselves a cone of the second class.</p></td>
+
+<td class="tcl"><p>Every plane section of a cone of the second class is a curve of
+ the second class.</p></td></tr></table>
+
+<p>By its aid, or by the principle of duality, it will be easy to obtain
+theorems about them from the theorems about the curves.</p>
+
+<p>We prove the first. A curve of the second order is generated by
+two projective pencils. These pencils, when joined to the point in
+space, give rise to two projective axial pencils, which generate the
+cone in question as the locus of the lines where corresponding planes
+meet.</p>
+
+<p><span class="pagenum"><a name="page696" id="page696"></a>696</span></p>
+
+<p>§48.</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p><i>Theorem.</i>&mdash;The curve of second order which is generated by two
+ projective flat pencils passes through the centres of the two pencils.</p></td>
+
+<td class="tcl" style="width: 50%;"><p><i>Theorem.</i>&mdash;The envelope of second class which is generated
+ by two projective rows contains the bases of these rows as enveloping
+ lines or tangents.</p></td></tr>
+
+<tr><td class="tcl rb3"><p><i>Proof.</i>&mdash;If S and S&prime; are the two pencils, then to the ray SS&prime; or p&prime;
+ in the pencil S&prime; corresponds in the pencil S a ray p, which is
+ different from p&prime;, for the pencils are not perspective. But p and
+ p&prime; meet at S, so that S is a point on the curve, and similarly S&prime;.</p></td>
+
+<td class="tcl"><p><i>Proof.</i>&mdash;If s and s&prime; are the two rows, then to the point ss&prime; or P&prime;
+ as a point in s&prime; corresponds in s a point P, which is not coincident
+ with P&prime;, for the rows are not perspective. But P and P&prime; are
+ joined by s, so that s is one of the enveloping lines, and similarly s&prime;.</p></td></tr></table>
+
+<p>It follows that every line in one of the two pencils cuts the curve
+in two points, viz. once at the centre S of the pencil, and once
+where it cuts its corresponding ray in the other pencil. These two
+points, however, coincide, if the line is cut by its corresponding
+line at S itself. The line p in S, which corresponds to the line
+SS&prime; in S&prime;, is therefore the only line through S which has but one
+point in common with the curve, or which cuts the curve in two
+coincident points. Such a line is called a <i>tangent</i> to the curve,
+touching the latter at the point S, which is called the &ldquo;point of
+contact.&rdquo;</p>
+
+<p>In the same manner we get in the reciprocal investigation the
+result that through every point in one of the rows, say in s, two
+tangents may be drawn to the curve, the one being s, the other the
+line joining the point to its corresponding point in s&prime;. There is,
+however, one point P in s for which these two lines coincide. Such
+a point in one of the tangents is called the &ldquo;point of contact&rdquo; of the
+tangent. We thus get&mdash;</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p><i>Theorem.</i>&mdash;To the line joining the centres of the projective
+ pencils as a line in one pencil corresponds in the other the
+ tangent at its centre.</p></td>
+
+<td class="tcl" style="width: 50%;"><p><i>Theorem.</i>&mdash;To the point of intersection of the bases of two
+ projective rows as a point in one row corresponds in the other the
+ <i>point of contact</i> of its base.</p></td></tr></table>
+
+<p>§ 49. Two projective pencils are determined if three pairs of
+corresponding lines are given. Hence if a<span class="su">1</span>, b<span class="su">1</span>, c<span class="su">1</span> are three lines in a
+pencil S<span class="su">1</span>, and a<span class="su">2</span>, b<span class="su">2</span>, c<span class="su">2</span> the corresponding lines in a projective pencil
+S<span class="su">2</span>, the correspondence and therefore the curve of the second order
+generated by the points of intersection of corresponding rays is
+determined. Of this curve we know the two centres S<span class="su">1</span> and S<span class="su">2</span>,
+and the three points a<span class="su">1</span>a<span class="su">2</span>, b<span class="su">1</span>b<span class="su">2</span>, c<span class="su">1</span>c<span class="su">2</span>, hence five points in all. This
+and the reciprocal considerations enable us to solve the following
+two problems:</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p><i>Problem.</i>&mdash;To construct a curve of the second order, of which five
+ points S<span class="su">1</span>, S<span class="su">2</span>, A, B, C are given.</p></td>
+
+<td class="tcl" style="width: 50%;"><p><i>Problem.</i>&mdash;To construct a curve of the second class, of which five
+ tangents u<span class="su">1</span>, u<span class="su">2</span>, a, b, c are given.</p></td></tr></table>
+
+<p>In order to solve the left-hand problem, we take two of the given
+points, say S<span class="su">1</span> and S<span class="su">2</span>, as centres of pencils. These we make projective
+by taking the rays a<span class="su">1</span>, b<span class="su">1</span>, c<span class="su">1</span>, which join S<span class="su">1</span> to A, B, C respectively,
+as corresponding to the rays a<span class="su">2</span>, b<span class="su">2</span>, c<span class="su">2</span>, which join S<span class="su">2</span> to A, B, C
+respectively, so that three rays meet their corresponding rays at
+the given points A, B, C. This determines the correspondence of
+the pencils which will generate a curve of the second order passing
+through A, B, C and through the centres S<span class="su">1</span> and S<span class="su">2</span>, hence through
+the five given points. To find more points on the curve we have to
+construct for any ray in S<span class="su">1</span> the corresponding ray in S<span class="su">2</span>. This has
+been done in § 36. But we repeat the construction in order to deduce
+further properties from it. We also solve the right-hand problem.
+Here we select two, viz. u<span class="su">1</span>, u<span class="su">2</span> of the five given lines, u<span class="su">1</span>, u<span class="su">2</span>, a, b, c,
+as bases of two rows, and the points A<span class="su">1</span>, B<span class="su">1</span>, C<span class="su">1</span> where a, b, c cut u<span class="su">1</span>
+as corresponding to the points A<span class="su">2</span>, B<span class="su">2</span>, C<span class="su">2</span> where a, b, c cut u<span class="su">2</span>.</p>
+
+<p>We get then the following solutions of the two problems:</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p><i>Solution.</i>&mdash;Through the point
+ A draw any two lines, u<span class="su">1</span> and u<span class="su">2</span>
+ (fig. 16), the first u<span class="su">1</span> to cut the
+ pencil S<span class="su">1</span> in a row AB<span class="su">1</span>C<span class="su">1</span>, the
+ other u<span class="su">2</span> to cut the pencil S<span class="su">2</span> in a
+ row AB<span class="su">2</span>C<span class="su">2</span>. These two rows will
+ be perspective, as the point A
+ corresponds to itself, and the
+ centre of projection will be the
+ point S, where the lines B<span class="su">1</span>B<span class="su">2</span>
+ and C<span class="su">1</span>C<span class="su">2</span> meet. To find now for
+ any ray d<span class="su">1</span> in S<span class="su">1</span> its corresponding
+ ray d<span class="su">2</span> in S<span class="su">2</span>, we determine the
+ point D<span class="su">1</span> where d<span class="su">1</span> cuts u<span class="su">1</span>, project
+ this point from S to D<span class="su">2</span> on u<span class="su">2</span> and
+ join S<span class="su">2</span> to D<span class="su">2</span>. This will be the
+ required ray d<span class="su">2</span> which cuts d<span class="su">1</span> at
+ some point D on the curve.</p></td>
+
+<td class="tcl" style="width: 50%;"><p><i>Solution.</i>&mdash;In the line a take
+ any two points S<span class="su">1</span> and S<span class="su">2</span> as
+ centres of pencils (fig. 17), the
+ first S<span class="su">1</span> (A<span class="su">1</span>B<span class="su">1</span>C<span class="su">1</span>) to project the
+ row u<span class="su">1</span>, the other S<span class="su">2</span> (A<span class="su">2</span>B<span class="su">2</span>C<span class="su">2</span>) to
+ project the row u<span class="su">2</span>. These two
+ pencils will be perspective, the
+ line S<span class="su">1</span>A<span class="su">1</span> being the same as the
+ corresponding line S<span class="su">2</span>A<span class="su">2</span>, and the
+ axis of projection will be the line
+ u, which joins the intersection B
+ of S<span class="su">1</span>B<span class="su">1</span> and S<span class="su">2</span>B<span class="su">2</span> to the intersection
+ C of S<span class="su">1</span>C<span class="su">1</span> and S<span class="su">2</span>C<span class="su">2</span>. To find
+ now for any point D<span class="su">1</span> in u<span class="su">1</span> the
+ corresponding point D<span class="su">2</span> in u<span class="su">2</span>, we
+ draw S<span class="su">1</span>D<span class="su">1</span> and project the point
+ D where this line cuts u from S<span class="su">2</span>
+ to u<span class="su">2</span>. This will give the required
+ point D<span class="su">2</span>, and the line d joining D<span class="su">1</span>
+ to D<span class="su">2</span> will be a new tangent to the
+ curve.</p></td></tr></table>
+
+<p>§ 50. These constructions prove, when rightly interpreted, very
+important properties of the curves in question.</p>
+
+<table class="nobctr" style="clear: both;" summary="Illustration">
+<tr><td class="figcenter"><img style="width:348px; height:319px" src="images/img696a.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 16.</span></td></tr></table>
+
+<p>If in fig. 16 we draw in the pencil S<span class="su">1</span> the ray k<span class="su">1</span> which passes
+through the auxiliary centre S, it will be found that the corresponding
+ray k<span class="su">2</span> cuts it on u<span class="su">2</span>. Hence&mdash;</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p><i>Theorem.</i>&mdash;In the above construction the bases of the auxiliary
+ rows u<span class="su">1</span> and u<span class="su">2</span> cut the curve
+ where they cut the rays S<span class="su">2</span>S and
+ S<span class="su">1</span>S respectively.</p></td>
+
+<td class="tcl" style="width: 50%;"><p><i>Theorem.</i>&mdash;In the above construction (fig. 17) the tangents to
+ the curve from the centres of the auxiliary pencils S<span class="su">1</span> and S<span class="su">2</span> are the
+ lines which pass through u<span class="su">2</span>u and
+ u<span class="su">1</span>u respectively.</p></td></tr></table>
+
+<p>As A is any given point on the curve, and u<span class="su">1</span> any line through
+it, we have solved the problems:</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p><i>Problem.</i>&mdash;To find the second point in which any line through a
+known point on the curve cuts the curve.</p></td>
+
+<td class="tcl" style="width: 50%;"><p><i>Problem.</i>&mdash;To find the second tangent which can be drawn
+from any point in a given tangent to the curve.</p></td></tr></table>
+
+<p>If we determine in S<span class="su">1</span> (fig. 16) the ray corresponding to the ray
+S<span class="su">2</span>S<span class="su">1</span> in S<span class="su">2</span>, we get the tangent at S<span class="su">1</span>. Similarly, we can determine
+the point of contact of the tangents u<span class="su">1</span> or u<span class="su">2</span> in fig. 17.</p>
+
+<table class="nobctr" style="clear: both;" summary="Illustration">
+<tr><td class="figcenter"><img style="width:386px; height:266px" src="images/img696b.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 17.</span></td></tr></table>
+
+<table class="flt" style="float: right; width: 300px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:248px; height:183px" src="images/img696c.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 18.</span></td></tr></table>
+
+<p>§ 51. If five points are given, of which not three are in a line,
+then we can, as has just been shown, always draw a curve of the
+second order through them; we select two of the points as centres of
+projective pencils, and then one such curve is determined. It will
+be presently shown that we get always the same curve if two other
+points are taken as centres of pencils, that therefore five points
+<i>determine</i> one curve of the second order, and reciprocally, that five
+tangents determine one curve of the second class. Six points taken
+at random will therefore not lie on a curve of the second order. In
+order that this may be the case a certain condition has to be satisfied,
+and this condition is easily obtained
+from the construction in
+§ 49, fig. 16. If we consider the
+conic determined by the five
+points A, S<span class="su">1</span>, S<span class="su">2</span>, K, L, then the
+point D will be on the curve if,
+and only if, the points on D<span class="su">1</span>, S,
+D<span class="su">2</span> be in a line.</p>
+
+<p>This may be stated differently
+if we take AKS<span class="su">1</span>DS<span class="su">2</span>L (figs. 16
+and 18) as a hexagon inscribed
+in the conic, then AK and DS<span class="su">2</span>
+will be opposite sides, so will be
+KS<span class="su">1</span> and S<span class="su">2</span>L, as well as S<span class="su">1</span>D and
+LA. The first two meet in D<span class="su">2</span>,
+the others in S and D<span class="su">1</span> respectively. We may therefore state the
+required condition, together with the reciprocal one, as follows:&mdash;</p>
+
+<p><span class="pagenum"><a name="page697" id="page697"></a>697</span></p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p><i>Pascal&rsquo;s Theorem.</i>&mdash;If a hexagon be inscribed in a curve of the
+ second order, then the intersections of opposite sides are three points in a line.</p></td>
+
+<td class="tcl" style="width: 50%;"><p><i>Brianchon&rsquo;s Theorem.</i>&mdash;If a hexagon be circumscribed about
+ a curve of the second class, then the lines joining opposite vertices
+ are three lines meeting in a point.</p></td></tr></table>
+
+<p>These celebrated theorems, which are known by the names of
+their discoverers, are perhaps the most fruitful in the whole theory
+of conics. Before we go over to their applications we have to show
+that we obtain the same curve if we take, instead of S<span class="su">1</span>, S<span class="su">2</span>, any two
+other points on the curve as centres of projective pencils.</p>
+
+<p>§ 52. We know that the curve depends only upon the correspondence
+between the pencils S<span class="su">1</span> and S<span class="su">2</span>, and not upon the special construction
+used for finding new points on the curve. The point A
+(fig. 16 or 18), through which the two auxiliary rows u<span class="su">1</span>, u<span class="su">2</span> were
+drawn, may therefore be changed to any other point on the curve.
+Let us now suppose the curve drawn, and keep the points S<span class="su">1</span>, S<span class="su">2</span>,
+K, L and D, and hence also the point S fixed, whilst we move A
+along the curve. Then the line AL will describe a pencil about
+L as centre, and the point D<span class="su">1</span> a row on S<span class="su">1</span>D perspective to the
+pencil L. At the same time AK describes a pencil about K and D<span class="su">2</span>
+a row perspective to it on S<span class="su">2</span>D. But by Pascal&rsquo;s theorem D<span class="su">1</span> and
+D<span class="su">2</span> will always lie in a line with S, so that the rows described by D<span class="su">1</span>
+and D<span class="su">2</span> are perspective. It follows that the pencils K and L will
+themselves be projective, corresponding rays meeting on the curve.
+This proves that we get the same curve whatever pair of the five
+given points we take as centres of projective pencils. Hence&mdash;</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p>Only one curve of the second order can be drawn which passes through five given points.</p></td>
+
+<td class="tcl" style="width: 50%;"><p>Only one curve of the second class can be drawn which touches five given lines.</p></td></tr></table>
+
+<p>We have seen that if on a curve of the second order two points
+coincide at A, the line joining them becomes the tangent at A.
+If, therefore, a point on the curve and its tangent are given, this
+will be equivalent to having given two points on the curve. Similarly,
+if on the curve of second class a tangent and its point of
+contact are given, this will be equivalent to two given tangents.</p>
+
+<p>We may therefore extend the last theorem:</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p>Only one curve of the second order can be drawn, of which
+ four points and the tangent at one of them, or three points and the
+ tangents at two of them, are given.</p></td>
+
+<td class="tcl" style="width: 50%;"><p>Only one curve of the second class can be drawn, of which four
+ tangents and the point of contact at one of them, or three tangents
+ and the points of contact at two of them, are given.</p></td></tr></table>
+
+<p>§ 53. At the same time it has been proved:</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p>If all points on a curve of the second order be joined to any
+ two of them, then the two pencils thus formed are projective, those
+ rays being corresponding which meet on the curve. Hence&mdash;</p></td>
+
+<td class="tcl" style="width: 50%;"><p>All tangents to a curve of second class are cut by any two of
+ them in projective rows, those being corresponding points which
+ lie on the same tangent. Hence&mdash;</p></td></tr>
+
+<tr><td class="tcl rb3"><p>The cross-ratio of four rays joining a point S on a curve of
+ second order to four fixed points A, B, C, D in the curve is independent
+ of the position of S, and is called the cross-ratio of the
+ four points A, B, C, D.</p></td>
+
+<td class="tcl"><p>The cross-ratio of the four points in which any tangent u is
+ cut by four fixed tangents a, b, c, d is independent of the position of
+ u, and is called the cross-ratio of the four tangents a, b, c, d.</p></td></tr>
+
+<tr><td class="tcl rb3"><p>If this cross-ratio equals &minus;1 the four points are said to be
+ four harmonic points.</p></td>
+
+<td class="tcl"><p>If this cross-ratio equals &minus;1 the four tangents are said to be
+ four harmonic tangents.</p></td></tr></table>
+
+<p>We have seen that a curve of second order, as generated by
+projective pencils, has at the centre of each pencil one tangent;
+and further, that any point on the curve may be taken as centre of
+such pencil. Hence&mdash;</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p>A curve of second order has at every point one tangent.</p></td>
+
+<td class="tcl" style="width: 50%;"><p>A curve of second class has on every tangent a point of contact.</p></td></tr></table>
+
+<p>§ 54. We return to Pascal&rsquo;s and Brianchon&rsquo;s theorems and their
+applications, and shall, as before, state the results both for curves
+of the second order and curves of the second class, but prove them
+only for the former.</p>
+
+<p>Pascal&rsquo;s theorem may be used when five points are given to find
+more points on the curve, viz. it enables us to find the point where
+any line through one of the given points cuts the curve again. It
+is convenient, in making use of Pascal&rsquo;s theorem, to number the
+points, to indicate the order in which they are to be taken in forming
+a hexagon, which, by the way, may be done in 60 different ways.
+It will be seen that 1 2 (leaving out 3) 4 5 are opposite sides,
+so are 2 3 and (leaving out 4) 5 6, and also 3 4 and (leaving
+out 5) 6 1.</p>
+
+<p>If the points 1 2 3 4 5 are given, and we want a 6th point on a
+line drawn through 1, we know all the sides of the hexagon with
+the exception of 5 6, and this is found by Pascal&rsquo;s theorem.</p>
+
+<p>If this line should happen to pass through 1, then 6 and 1 coincide,
+or the line 6 1 is the tangent at 1. And always if two consecutive
+vertices of the hexagon approach nearer and nearer, then the side
+joining them will ultimately become a tangent.</p>
+
+<p>We may therefore consider a pentagon inscribed in a curve of
+second order and the tangent at one of its vertices as a hexagon,
+and thus get the theorem:</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p>Every pentagon inscribed in a curve of second order has the
+ property that the intersections of two pairs of non-consecutive
+ sides lie in a line with the point where the fifth side cuts the tangent
+ at the opposite vertex.</p></td>
+
+<td class="tcl" style="width: 50%;"><p>Every pentagon circumscribed about a curve of the second class
+ has the property that the lines which join two pairs of non-consecutive
+ vertices meet on that line which joins the fifth vertex
+ to the point of contact of the opposite side.</p></td></tr></table>
+
+<p>This enables us also to solve the following problems.</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p>Given five points on a curve of second order to construct the
+ tangent at any one of them.</p></td>
+
+<td class="tcl" style="width: 50%;"><p>Given five tangents to a curve of second class to construct the
+ point of contact of any one of them.</p></td></tr></table>
+
+<table class="nobctr" style="clear: both;" summary="Illustration">
+<tr><td class="figcenter"><img style="width:390px; height:354px" src="images/img697a.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 19.</span></td></tr></table>
+
+<p>If two pairs of adjacent vertices coincide, the hexagon becomes a
+quadrilateral, with tangents at two vertices. These we take to be
+opposite, and get the following theorems:</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p>If a quadrilateral be inscribed in a curve of second order, the
+ intersections of opposite sides, and also the intersections of the
+ tangents at opposite vertices, lie in a line (fig. 19).</p></td>
+
+<td class="tcl" style="width: 50%;"><p>If a quadrilateral be circumscribed about a curve of second
+ class, the lines joining opposite vertices, and also the lines joining
+ points of contact of opposite sides, meet in a point.</p></td></tr></table>
+
+<table class="nobctr" style="clear: both;" summary="Illustration">
+<tr><td class="figcenter"><img style="width:365px; height:294px" src="images/img697b.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 20.</span></td></tr></table>
+
+<p>If we consider the hexagon made up of a triangle and the tangents
+at its vertices, we get&mdash;</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p>If a triangle is inscribed in a curve of the second order, the
+ points in which the sides are cut by the tangents at the opposite
+ vertices meet in a point.</p></td>
+
+<td class="tcl" style="width: 50%;"><p>If a triangle be circumscribed about a curve of second class,
+ the lines which join the vertices to the points of contact of the
+ opposite sides meet in a point (fig. 20).</p></td></tr></table>
+
+<p>§ 55. Of these theorems, those about the quadrilateral give rise to
+a number of others. Four points A, B, C, D may in three different
+ways be formed into a quadrilateral, for we may take them in the
+order ABCD, or ACBD, or ACDB, so that either of the points
+B, C, D may be taken as the vertex opposite to A. Accordingly we
+may apply the theorem in three different ways.</p>
+
+<p>Let A, B, C, D be four points on a curve of second order (fig. 21),
+and let us take them as forming a quadrilateral by taking the points
+in the order ABCD, so that A, C and also B, D are pairs of opposite
+vertices. Then P, Q will be the points where opposite sides meet,
+<span class="pagenum"><a name="page698" id="page698"></a>698</span>
+and E, F the intersections of tangents at opposite vertices. The
+four points P, Q, E, F lie therefore in a line. The quadrilateral
+ACBD gives us in the same way the four points Q, R, G, H in a line,
+and the quadrilateral ABDC a line containing the four points R, P,
+I, K. These three lines form a triangle PQR.</p>
+
+<p>The relation between the points and lines in this figure may be
+expressed more clearly if we consider ABCD as a four-point inscribed
+in a conic, and the tangents at these points as a four-side circumscribed
+about it,&mdash;viz. it will be seen that P, Q, R are the diagonal points
+of the four-point ABCD, whilst the sides of the triangle PQR are
+the diagonals of the circumscribing four-side. Hence the theorem&mdash;</p>
+
+<p><i>Any four-point on a curve of the second order and the four-side
+formed by the tangents at these points stand in this relation that the
+diagonal points of the four-point lie in the diagonals of the four-side.</i>
+And conversely,</p>
+
+<p><i>If a four-point and a circumscribed four-side stand in the above
+relation, then a curve of the second order may be described which passes
+through the four points and touches there the four sides of these figures.</i></p>
+
+<table class="nobctr" style="clear: both;" summary="Illustration">
+<tr><td class="figcenter"><img style="width:396px; height:707px" src="images/img698a.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 21.</span></td></tr></table>
+
+<p>That the last part of the theorem is true follows from the fact
+that the four points A, B, C, D and the line a, as tangent at A, determine
+a curve of the second order, and the tangents to this curve at
+the other points B, C, D are given by the construction which leads
+to fig. 21.</p>
+
+<p>The theorem reciprocal to the last is&mdash;</p>
+
+<p><i>Any four-side circumscribed about a curve of second class and the
+four-point formed by the points of contact stand in this relation that the
+diagonals of the four-side pass through the diagonal points of the
+four-point.</i> And conversely,</p>
+
+<p><i>If a four-side and an inscribed four-point stand in the above relation,
+then a curve of the second class may be described which touches the sides
+of the four-side at the points of the four-point.</i></p>
+
+<p>§ 56. The four-point and the four-side in the two reciprocal
+theorems are alike. Hence if we have a four-point ABCD and a
+four-side abcd related in the manner described, then not only may
+a curve of the second order be drawn, but also a curve of the second
+class, which both touch the lines a, b, c, d at the points A, B, C, D.</p>
+
+<p>The curve of second order is already more than determined by the
+points A, B, C and the tangents a, b, c at A, B and C. The point D
+may therefore be <i>any</i> point on this curve, and d any tangent to the
+curve. On the other hand the curve of the second class is more
+than determined by the three tangents a, b, c and their points of
+contact A, B, C, so that d is any tangent to this curve. It follows
+that every tangent to the curve of second order is a tangent of a
+curve of the second class having the same point of contact. In
+other words, the curve of second order is a curve of second class,
+and <i>vice versa</i>. Hence the important theorems&mdash;</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p><i>Every curve of second order is a curve of second class.</i></p></td>
+
+<td class="tcl" style="width: 50%;"><p><i>Every curve of second class is a curve of second order.</i></p></td></tr></table>
+
+<p>The curves of second order and of second class, having thus been
+proved to be identical, shall henceforth be called by the common
+name of <i>Conics</i>.</p>
+
+<p>For these curves hold, therefore, all properties which have been
+proved for curves of second order or of second class. We may
+therefore now state Pascal&rsquo;s and Brianchon&rsquo;s theorem thus&mdash;</p>
+
+<p><i>Pascal&rsquo;s Theorem.</i>&mdash;If a hexagon be inscribed in a conic, then
+the intersections of opposite sides lie in a line.</p>
+
+<p><i>Brianchon&rsquo;s Theorem.</i>&mdash;If a hexagon be circumscribed about a
+conic, then the diagonals forming opposite centres meet in a point.</p>
+
+<p>§ 57. If we suppose in fig. 21 that the point D together with the
+tangent d moves along the curve, whilst A, B, C and their tangents
+a, b, c remain fixed, then the ray DA will describe a pencil about
+A, the point Q a projective row on the fixed line BC, the point F
+the row b, and the ray EF a pencil about E. But EF passes always
+through Q. Hence the pencil described by AD is projective to the
+pencil described by EF, and therefore to the row described by F on
+b. At the same time the line BD describes a pencil about B projective
+to that described by AD (§ 53). Therefore the pencil BD
+and the row F on b are projective. Hence&mdash;</p>
+
+<p><i>If on a conic a point</i> A <i>be taken and the tangent a at this point, then
+the cross-ratio of the four rays which join</i> A <i>to any four points on the
+curve is equal to the cross-ratio of the points in which the tangents at
+these points cut the tangent at</i> A.</p>
+
+<p>§ 58. There are theorems about cones of second order and second
+class in a pencil which are reciprocal to the above, according to § 43.
+We mention only a few of the more important ones.</p>
+
+<p>The locus of intersections of corresponding planes in two projective
+axial pencils whose axes meet is a cone of the second order.</p>
+
+<p>The envelope of planes which join corresponding lines in two
+projective flat pencils, not in the same plane, is a cone of the second
+class.</p>
+
+<p>Cones of second order and cones of second class are identical.</p>
+
+<p>Every plane cuts a cone of the second order in a conic.</p>
+
+<p><i>A cone of second order is uniquely determined by five of its edges
+or by five of its tangent planes, or by four edges and the tangent plane
+at one of them, &amp;c. &amp;c.</i></p>
+
+<p><i>Pascal&rsquo;s Theorem.</i>&mdash;If a solid angle of six faces be inscribed in a
+cone of the second order, then the intersections of opposite faces
+are three lines in a plane.</p>
+
+<p><i>Brianchon&rsquo;s Theorem.</i>&mdash;If a solid angle of six edges be circumscribed
+about a cone of the second order, then the planes through
+opposite edges meet in a line.</p>
+
+<p>Each of the other theorems about conics may be stated for cones
+of the second order.</p>
+
+<table class="flt" style="float: right; width: 360px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:310px; height:314px" src="images/img698b.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 22.</span></td></tr></table>
+
+<p>§ 59. <i>Projective Definitions of the Conics.</i>&mdash;We now consider the
+shape of the conics. We know that any line in the plane of the conic,
+and hence that the line at infinity, either has no point in common
+with the curve, or one (counting for two coincident points) or two
+distinct points. If the line at infinity has no point on the curve the
+latter is altogether finite, and is called an <i>Ellipse</i> (fig. 21). If the line
+at infinity has only one point in common with the conic, the latter
+extends to infinity, and has the line at infinity a tangent. It is
+called a <i>Parabola</i> (fig. 22). If, lastly, the line at infinity cuts the
+curve in two points, it
+consists of two separate
+parts which each extend
+in two branches to the
+points at infinity where
+they meet. The curve is
+in this case called an
+<i>Hyperbola</i> (see fig. 20).
+The tangents at the
+two points at infinity
+are finite because the
+line at infinity is not
+a tangent. They are
+called <i>Asymptotes</i>. The
+branches of the hyperbola
+approach these lines
+indefinitely as a point on
+the curves moves to infinity.</p>
+
+<p>§ 60. That the circle
+belongs to the curves of
+the second order is seen
+at once if we state in
+a slightly different form the theorem that in a circle all angles at
+the circumference standing upon the same arc are equal. If two
+points S<span class="su">1</span>, S<span class="su">2</span> on a circle be joined to any other two points A and B
+on the circle, then the angle included by the rays S<span class="su">1</span>A and S<span class="su">1</span>B is
+equal to that between the rays S<span class="su">2</span>A and S<span class="su">2</span>B, so that as A moves
+along the circumference the rays S<span class="su">1</span>A and S<span class="su">2</span>A describe equal and
+therefore projective pencils. The circle can thus be generated by
+two projective pencils, and is a curve of the second order.</p>
+
+<p><span class="pagenum"><a name="page699" id="page699"></a>699</span></p>
+
+<p>If we join a point in space to all points on a circle, we get a (circular)
+cone of the second order (§ 43). Every plane section of this cone is a
+conic. This conic will be an ellipse, a parabola, or an hyperbola,
+according as the line at infinity in the plane has no, one or two points
+in common with the conic in which the plane at infinity cuts the
+cone. It follows that our curves of second order may be obtained
+as sections of a circular cone, and that they are identical with the
+&ldquo;Conic Sections&rdquo; of the Greek mathematicians.</p>
+
+<p>§ 61. Any two tangents to a parabola are cut by all others in
+projective rows; but the line at infinity being one of the tangents,
+the points at infinity on the rows are corresponding points, and the
+rows therefore similar. Hence the theorem&mdash;</p>
+
+<p><i>The tangents to a parabola cut each other proportionally.</i></p>
+
+<p class="pt2 center sc">Pole and Polar</p>
+
+<p>§ 62. We return once again to fig. 21, which we obtained in § 55.</p>
+
+<p>If a four-side be circumscribed about and a four-point inscribed
+in a conic, so that the vertices of the second are the points of contact
+of the sides of the first, then the triangle formed by the diagonals
+of the first is the same as that formed by the diagonal points of the
+other.</p>
+
+<p>Such a triangle will be called a <i>polar-triangle</i> of the conic, so that
+PQR in fig. 21 is a polar-triangle. It has the property that on the
+side p opposite P meet the tangents at A and B, and also those at C
+and D. From the harmonic properties of four-points and four-sides
+it follows further that the points L, M, where it cuts the lines AB
+and CD, are harmonic conjugates with regard to AB and CD
+respectively.</p>
+
+<p>If the point P is given, and we draw a line through it, cutting
+the conic in A and B, then the point Q harmonic conjugate to P
+with regard to AB, and the point H where the tangents at A and B
+meet, are determined. But they lie both on p, and therefore this
+line is determined. If we now draw a second line through P, cutting
+the conic in C and D, then the point M harmonic conjugate to P
+with regard to CD, and the point G where the tangents at C and D
+meet, must also lie on p. As the first line through P already determines
+p, the second may be any line through P. Now every two
+lines through P determine a four-point ABCD on the conic, and
+therefore a polar-triangle which has one vertex at P and its opposite
+side at p. This result, together with its reciprocal, gives the
+theorems&mdash;</p>
+
+<p><i>All polar-triangles which have one vertex in common have also the
+opposite side in common.</i></p>
+
+<p><i>All polar-triangles which have one side in common have also the
+opposite vertex in common.</i></p>
+
+<p>§ 63. To any point P in the plane of, but not on, a conic corresponds
+thus one line p as the side opposite to P in all polar-triangles which
+have one vertex at P, and reciprocally to every line p corresponds
+one point P as the vertex opposite to p in all triangles which have p
+as one side.</p>
+
+<p>We call the line p the <i>polar</i> of P, and the point P the <i>pole</i> of the
+line p with regard to the conic.</p>
+
+<p>If a point lies on the conic, we call the tangent at that point its
+polar; and reciprocally we call the point of contact the pole of
+tangent.</p>
+
+<p>§ 64. From these definitions and former results follow&mdash;</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;"><p>The polar of any point P not on the conic is a line p, which has
+ the following properties:&mdash;</p></td>
+
+<td class="tcl" style="width: 50%;"><p>The pole of any line p not a tangent to the conic is a point
+ P, which has the following properties:&mdash;</p></td></tr>
+
+<tr><td class="tcl rb3">1. On every line through P which cuts the conic, the polar
+ of P contains the harmonic conjugate of P with regard to those
+ points on the conic.</td>
+
+<td class="tcl">1. Of all lines through a point on p from which two tangents
+ may be drawn to the conic, the pole P contains the line which is
+ harmonic conjugate to p, with regard to the two tangents.</td></tr>
+
+<tr><td class="tcl rb3">2. If tangents can be drawn from P, their points of contact lie
+ on p.</td>
+
+<td class="tcl">2. If p cuts the conic, the tangents at the intersections
+ meet at P.</td></tr>
+
+<tr><td class="tcl rb3">3. Tangents drawn at the points where any line through P
+ cuts the conic meet on p; and conversely,</td>
+
+<td class="tcl">3. The point of contact of tangents drawn from any point
+ on p to the conic lie in a line with P; and conversely,</td></tr>
+
+<tr><td class="tcl rb3">4. If from any point on p, tangents be drawn, their points
+ of contact will lie in a line with P.</td>
+
+<td class="tcl">4. Tangents drawn at points where any line through P cuts the
+ conic meet on p.</td></tr>
+
+<tr><td class="tcl rb3">5. Any four-point on the conic which has one diagonal point at
+ P has the other two lying on p.</td>
+
+<td class="tcl">5. Any four-side circumscribed about a conic which has one
+ diagonal on p has the other two meeting at P.</td></tr></table>
+
+<p>The truth of 2 follows from 1. If T be a point where p cuts the
+conic, then one of the points where PT cuts the conic, and which
+are harmonic conjugates with regard to PT, coincides with T; hence
+the other does&mdash;that is, PT touches the curve at T.</p>
+
+<p>That 4 is true follows thus: If we draw from a point H on the
+polar one tangent a to the conic, join its point of contact A to the
+pole P, determine the second point of intersection B of this line with
+the conic, and draw the tangent at B, it will pass through H, and
+will therefore be the second tangent which may be drawn from H to
+the curve.</p>
+
+<p>§ 65. The second property of the polar or pole gives rise to the
+theorem&mdash;</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;">From a point in the plane of a conic, two, one or no tangents
+ may be drawn to the conic, as its polar has two,
+ one, or no points in common with the curve.</td>
+
+<td class="tcl" style="width: 50%;">A line in the plane of a conic has two, one or no points in
+ common with the conic, according as two, one or no tangents
+ can be drawn from its pole to the conic.</td></tr></table>
+
+<p>Of any point in the plane of a conic we say that it was <i>without</i>,
+on or <i>within</i> the curve according as two, one or no tangents to the
+curve pass through it. The points on the conic separate those within
+the conic from those without. That this is true for a circle is known
+from elementary geometry. That it also holds for other conics
+follows from the fact that every conic may be considered as the
+projection of a circle, which will be proved later on.</p>
+
+<p>The fifth property of pole and polar stated in § 64 shows how
+to find the polar of any point and the pole of any line by aid of the
+straight-edge only. Practically it is often convenient to draw three
+secants through the pole, and to determine only one of the diagonal
+points for two of the four-points formed by pairs of these lines and
+the conic (fig. 22).</p>
+
+<p>These constructions also solve the problem&mdash;</p>
+
+<p>From a point without a conic, to draw the two tangents to the
+conic by aid of the straight-edge only.</p>
+
+<p>For we need only draw the polar of the point in order to find the
+points of contact.</p>
+
+<p>§ 66. The property of a polar-triangle may now be stated thus&mdash;</p>
+
+<p>In a polar-triangle each side is the polar of the opposite vertex,
+and each vertex is the pole of the opposite side.</p>
+
+<table class="flt" style="float: right; width: 340px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:286px; height:303px" src="images/img699.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 23.</span></td></tr></table>
+
+<p>If P is one vertex of a polar-triangle, then the other vertices, Q
+and R, lie on the polar p of P. One of these vertices we may choose
+arbitrarily. For if from
+any point Q on the polar
+a secant be drawn cutting
+the conic in A and D (fig.
+23), and if the lines joining
+these points to P cut the
+conic again at B and C,
+then the line BC will pass
+through Q. Hence P and
+Q are two of the vertices
+on the polar-triangle which
+is determined by the four-point
+ABCD. The third
+vertex R lies also on the
+line p. It follows, therefore,
+also&mdash;</p>
+
+<p><i>If</i> Q <i>is a point on the polar
+of</i> P, <i>then</i> P <i>is a point on the
+polar of</i> Q; and reciprocally,</p>
+
+<p><i>If</i> q <i>is a line through the
+pole of</i> p, <i>then</i> p <i>is a line
+through the pole of</i> q.</p>
+
+<p>This is a very important theorem. It may also be stated
+thus&mdash;</p>
+
+<p><i>If a point moves along a line describing a row, its polar turns about
+the pole of the line describing a pencil.</i></p>
+
+<p><i>This pencil is projective to the row, so that the cross-ratio of four
+poles in a row equals the cross-ratio of its four polars, which pass
+through the pole of the row.</i></p>
+
+<p>To prove the last part, let us suppose that P, A and B in fig. 23
+remain fixed, whilst Q moves along the polar p of P. This will
+make CD turn about P and move R along p, whilst QD and RD
+describe projective pencils about A and B. Hence Q and R describe
+projective rows, and hence PR, which is the polar of Q, describes a
+pencil projective to either.</p>
+
+<p>§ 67. Two points, of which one, and therefore each, lies on the
+polar of the other, are said to be <i>conjugate with regard to the conic</i>;
+and two lines, of which one, and therefore each, passes through the
+pole of the other, are said to be <i>conjugate with regard to the conic</i>.
+Hence all points conjugate to a point P lie on the polar of P; all lines
+conjugate to a line p pass through the pole of p.</p>
+
+<p>If the line joining two conjugate poles cuts the conic, then the
+poles are harmonic conjugates with regard to the points of intersection;
+hence one lies within the other without the conic, and all
+points conjugate to a point within a conic lie without it.</p>
+
+<p>Of a polar-triangle any two vertices are conjugate poles, any two
+sides conjugate lines. If, therefore, one side cuts a conic, then
+one of the two vertices which lie on this side is within and the other
+without the conic. The vertex opposite this side lies also without,
+for it is the pole of a line which cuts the curve. In this case therefore
+one vertex lies within, the other two without. If, on the
+other hand, we begin with a side which does not cut the conic,
+then its pole lies within and the other vertices without. Hence&mdash;</p>
+
+<p>Every polar-triangle has one and only one vertex within the conic.</p>
+
+<p>We add, without a proof, the theorem&mdash;</p>
+
+<p>The four points in which a conic is cut by two conjugate polars
+are four harmonic points in the conic.</p>
+
+<p>§ 68. If two conics intersect in four points (they cannot have
+more points in common, § 52), there exists one and only one
+<span class="pagenum"><a name="page700" id="page700"></a>700</span>
+four-point which is inscribed in both, and therefore one polar-triangle
+common to both.</p>
+
+<p><i>Theorem.</i>&mdash;Two conics which intersect in four points have always
+one and only one common polar-triangle; and reciprocally,</p>
+
+<p>Two conics which have four common tangents have always one
+and only one common polar-triangle.</p>
+
+<p class="pt2 center sc">Diameters and Axes of Conics</p>
+
+<p>§ 69. <i>Diameters.</i>&mdash;The theorems about the harmonic properties
+of poles and polars contain, as special cases, a number of important
+metrical properties of conics. These are obtained if either the pole
+or the polar is moved to infinity,&mdash;it being remembered that the
+harmonic conjugate to a point at infinity, with regard to two points
+A, B, is the middle point of the segment AB. The most important
+properties are stated in the following theorems:&mdash;</p>
+
+<p><i>The middle points of parallel chords of a conic lie in a line&mdash;viz. on
+the polar to the point at infinity on the parallel chords.</i></p>
+
+<p>This line is called a <i>diameter</i>.</p>
+
+<p><i>The polar of every point at infinity is a diameter.</i></p>
+
+<p><i>The tangents at the end points of a diameter are parallel, and are
+parallel to the chords bisected by the diameter.</i></p>
+
+<p><i>All diameters pass through a common point, the pole of the line at
+infinity.</i></p>
+
+<p><i>All diameters of a parabola are parallel</i>, the pole to the line at
+infinity being the point where the curve touches the line at infinity.</p>
+
+<p>In case of the ellipse and hyperbola, the pole to the line at infinity
+is a finite point called the <i>centre</i> of the curve.</p>
+
+<p><i>A centre of a conic bisects every chord through it.</i></p>
+
+<p><i>The centre of an ellipse is within the curve</i>, for the line at infinity
+does not cut the ellipse.</p>
+
+<p><i>The centre of an hyperbola is without the curve</i>, because the line at
+infinity cuts the curve. Hence also&mdash;</p>
+
+<p><i>From the centre of an hyperbola two tangents can be drawn to the
+curve which have their point of contact at infinity.</i> These are called
+<i>Asymptotes</i> (§ 59).</p>
+
+<p><i>To construct a diameter</i> of a conic, draw two parallel chords and
+join their middle points.</p>
+
+<p><i>To find the centre</i> of a conic, draw two diameters; their intersection
+will be the centre.</p>
+
+<p>§ 70. <i>Conjugate Diameters.</i>&mdash;A polar-triangle with one vertex at
+the centre will have the opposite side at infinity. The other two
+sides pass through the centre, and are called <i>conjugate diameters</i>,
+each being the polar of the point at infinity on the other.</p>
+
+<p><i>Of two conjugate diameters each bisects the chords parallel to the
+other, and if one cuts the curve, the tangents at its ends are parallel to
+the other diameter.</i></p>
+
+<p>Further&mdash;</p>
+
+<p><i>Every parallelogram inscribed in a conic has its sides parallel to
+two conjugate diameters</i>; and</p>
+
+<p><i>Every parallelogram circumscribed about a conic has as diagonals two
+conjugate diameters.</i></p>
+
+<p>This will be seen by considering the parallelogram in the first
+case as an inscribed four-point, in the other as a circumscribed
+four-side, and determining in each case the corresponding polar-triangle.
+The first may also be enunciated thus&mdash;</p>
+
+<p><i>The lines which join any point on an ellipse or an hyperbola to the
+ends of a diameter are parallel to two conjugate diameters.</i></p>
+
+<p>§ 71. <i>If every diameter is perpendicular to its conjugate the conic is
+a circle.</i></p>
+
+<p>For the lines which join the ends of a diameter to any point on
+the curve include a right angle.</p>
+
+<p><i>A conic which has more than one pair of conjugate diameters at right
+angles to each other is a circle.</i></p>
+
+<table class="flt" style="float: right; width: 260px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:209px; height:221px" src="images/img700a.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 24.</span></td></tr></table>
+
+<p>Let AA&prime; and BB&prime; (fig. 24) be one pair of conjugate diameters at
+right angles to each other, CC and DD&prime; a second pair. If we draw
+through the end point A of one
+diameter a chord AP parallel to
+DD&prime;, and join P to A&prime;, then PA and
+PA&prime; are, according to § 70, parallel to
+two conjugate diameters. But PA is
+parallel to DD&prime;, hence PA&prime; is parallel
+to CC, and therefore PA and PA&prime;
+are perpendicular. If we further
+draw the tangents to the conic at A
+and A&prime;, these will be perpendicular
+to AA&prime;, they being parallel to the
+conjugate diameter BB&prime;. We know
+thus five points on the conic, viz. the
+points A and A&prime; with their tangents,
+and the point P. Through these a
+circle may be drawn having AA&prime; as
+diameter; and as through five points
+one conic only can be drawn, this circle must coincide with the
+given conic.</p>
+
+<p>§ 72. <i>Axes.</i>&mdash;Conjugate diameters perpendicular to each other
+are called <i>axes</i>, and the points where they cut the curve <i>vertices</i>
+of the conic.</p>
+
+<p>In a circle every diameter is an axis, every point on it is a vertex;
+and any two lines at right angles to each other may be taken as a
+pair of axes of any circle which has its centre at their intersection.</p>
+
+<table class="flt" style="float: left; width: 340px;" summary="Illustration">
+<tr><td class="figleft1"><img style="width:289px; height:261px" src="images/img700b.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 25.</span></td></tr></table>
+
+<p>If we describe on a diameter AB of an ellipse or hyperbola a circle
+concentric to the conic, it will cut the latter in A and B (fig. 25).
+Each of the semicircles in which it is divided by AB will be partly
+within, partly without the curve, and must cut the latter therefore
+again in a point. The circle and the conic have thus four points
+A, B, C, D, and therefore
+one polar-triangle, in common
+(§ 68). Of this the
+centre is one vertex, for
+the line at infinity is the
+polar to this point, both
+with regard to the circle
+and the other conic. The
+other two sides are conjugate
+diameters of both,
+hence perpendicular to
+each other. This gives&mdash;</p>
+
+<p>An ellipse as well as an
+hyperbola has one pair of
+axes.</p>
+
+<p>This reasoning shows at
+the same time <i>how to construct
+the axis of an ellipse
+or of an hyperbola</i>.</p>
+
+<p><i>A parabola has one axis</i>,
+if we define an axis as a diameter perpendicular to the chords
+which it bisects. It is easily constructed. The line which bisects
+any two parallel chords is a diameter. Chords perpendicular to it
+will be bisected by a parallel diameter, and this is the axis.</p>
+
+<p>§ 73. The first part of the right-hand theorem in § 64 may be
+stated thus: any two conjugate lines through a point P without a
+conic are harmonic conjugates with regard to the two tangents
+that may be drawn from P to the conic.</p>
+
+<p>If we take instead of P the centre C of an hyperbola, then the
+conjugate lines become conjugate diameters, and the tangents
+asymptotes. Hence&mdash;</p>
+
+<p><i>Any two conjugate diameters of an hyperbola are harmonic conjugates
+with regard to the asymptotes.</i></p>
+
+<p>As the axes are conjugate diameters at right angles to one another,
+it follows (§ 23)&mdash;</p>
+
+<p><i>The axes of an hyperbola bisect the angles between the asymptotes.</i></p>
+
+<table class="nobctr" style="clear: both;" summary="Illustration">
+<tr><td class="figcenter"><img style="width:352px; height:337px" src="images/img700c.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 26.</span></td></tr></table>
+
+<p>Let O be the centre of the hyperbola (fig. 26), t any secant which
+cuts the hyperbola in C, D and the asymptotes in E, F, then the
+line OM which bisects the chord CD is a diameter conjugate to the
+diameter OK which is parallel to the secant t, so that OK and OM
+are harmonic with regard to the asymptotes. The point M therefore
+bisects EF. But by construction M bisects CD. It follows
+that DF = EC, and ED = CF; or</p>
+
+<p><i>On any secant of an hyperbola the segments between the curve and the
+asymptotes are equal.</i></p>
+
+<p>If the chord is changed into a tangent, this gives&mdash;</p>
+
+<p><i>The segment between the asymptotes on any tangent to an hyperbola
+is bisected by the point of contact.</i></p>
+
+<p>The first part allows a simple solution of the problem to find any
+number of points on an hyperbola, of which the asymptotes and one
+point are given. This is equivalent to three points and the tangents
+at two of them. This construction requires measurement.</p>
+
+<p>§ 74. For the parabola, too, follow some metrical properties. A
+diameter PM (fig. 27) bisects every chord conjugate to it, and the
+pole P of such a chord BC lies on the diameter. But a diameter cuts
+the parabola once at infinity. Hence&mdash;</p>
+
+<p><i>The segment</i> PM <i>which joins the middle point</i> M <i>of a chord of a parabola
+to the pole</i> P <i>of the chord is bisected by the parabola at</i> A.</p>
+
+<table class="flt" style="float: right; width: 340px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:293px; height:282px" src="images/img701a.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 27.</span></td></tr></table>
+
+<p>§ 75. Two asymptotes and any two tangents to an hyperbola
+may be considered as a quadrilateral circumscribed about the
+<span class="pagenum"><a name="page701" id="page701"></a>701</span>
+hyperbola. But in such a quadrilateral the intersections of the
+diagonals and the points of contact of opposite sides lie in a line
+(§ 54). If therefore DEFG
+(fig. 28) is such a quadrilateral,
+then the diagonals
+DF and GE will meet on
+the line which joins the
+points of contact of the
+asymptotes, that is, on the
+line at infinity; hence they
+are parallel. From this
+the following theorem is
+a simple deduction:</p>
+
+<p><i>All triangles formed by a
+tangent and the asymptotes
+of an hyperbola are equal in
+area.</i></p>
+
+<p>If we draw at a point P
+(fig. 28) on an hyperbola
+a tangent, the part HK
+between the asymptotes
+is bisected at P. The
+parallelogram PQOQ&prime;
+formed by the asymptotes and lines parallel to them through
+P will be half the triangle OHK, and will therefore be constant.
+If we now take the asymptotes OX and OY as oblique
+axes of co-ordinates, the lines OQ and QP will be the co-ordinates of
+P, and will satisfy the equation xy = const. = a².</p>
+
+<table class="nobctr" style="clear: both;" summary="Illustration">
+<tr><td class="figcenter"><img style="width:323px; height:333px" src="images/img701b.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 28.</span></td></tr></table>
+
+<p><i>For the asymptotes as axes of co-ordinates the equation of the hyperbola
+is</i> xy = const.</p>
+
+<p class="pt2 center sc">Involution</p>
+
+<table class="flt" style="float: right; width: 270px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:221px; height:49px" src="images/img701c.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 29.</span></td></tr></table>
+
+<p>§ 76. If we have two projective rows, ABC on u and A&prime;B&prime;C&prime; on
+u&prime;, and place their bases on the same line, then each point in this
+line counts twice, once as a point in the row u and once as a point
+in the row u&prime;. In fig. 29 we denote the points as points in the one
+row by letters above the line A, B, C ..., and as points in the second
+row by A&prime;, B&prime;, C&prime; ... below the
+line. Let now A and B&prime; be the
+same point, then to A will correspond
+a point A&prime; in the second,
+and to B&prime; a point B in the first
+row. In general these points A&prime;
+and B will be different. It may, however, happen that they coincide.
+Then the correspondence is a peculiar one, as the following theorem
+shows:</p>
+
+<p><i>If two projective rows lie on the same base, and if it happens that to one
+point in the base the same point corresponds, whether we consider the
+point as belonging to the first or to the second row, then the same will
+happen for every point in the base&mdash;that is to say, to every point in the
+line corresponds the same point in the first as in the second row.</i></p>
+
+<table class="flt" style="float: right; width: 280px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:226px; height:45px" src="images/img701d.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 30.</span></td></tr></table>
+
+<p>In order to determine the correspondence, we may assume three
+pairs of corresponding points in two projective rows. Let then
+A&prime;, B&prime;, C&prime;, in fig. 30, correspond to
+A, B, C, so that A and B&prime;, and also
+B and A&prime;, denote the same point.
+Let us further denote the point
+C&prime; when considered as a point in
+the first row by D; then it is to
+be proved that the point D&prime;, which corresponds to D, is the same
+point as C. We know that the cross-ratio of four points is equal
+to that of the corresponding row. Hence</p>
+
+<p class="center">(AB, CD) = (A&prime;B&prime;, C&prime;D&prime;)</p>
+
+<p>but replacing the dashed letters by those undashed ones which
+denote the same points, the second cross-ratio equals (BA, DD&prime;),
+which, according to § 15, equals (AB, D&prime;D); so that the equation
+becomes</p>
+
+<p class="center">(AB, CD) = (AB, D&prime;D).</p>
+
+<p>This requires that C and D&prime; coincide.</p>
+
+<p>§ 77. Two projective rows on the same base, which have the above
+property, that to every point, whether it be considered as a point in
+the one or in the other row, corresponds the same point, are said
+to be in <i>involution</i>, or to form an <i>involution</i> of points on the line.</p>
+
+<p>We mention, but without proving it, that any two projective
+rows may be placed so as to form an involution.</p>
+
+<p>An involution may be said to consist of a row of pairs of points,
+to every point A corresponding a point A&prime;, and to A&prime; again the
+point A. These points are said to be conjugate, or, better, one point
+is termed the &ldquo;mate&rdquo; of the other.</p>
+
+<p>From the definition, according to which an involution may be
+considered as made up of two projective rows, follow at once the
+following important properties:</p>
+
+<p>1. The cross-ratio of four points equals that of the four conjugate
+points.</p>
+
+<p>2. If we call a point which coincides with its mate a &ldquo;focus&rdquo;
+or &ldquo;double point&rdquo; of the involution, we may say: An involution
+has either two foci, or one, or none, and is called respectively a
+hyperbolic, parabolic or elliptic involution (§ 34).</p>
+
+<p>3. In <span class="correction" title="amended from a">an</span> hyperbolic involution any two conjugate points are
+harmonic conjugates with regard to the two foci.</p>
+
+<p>For if A, A&prime; be two conjugate points, F<span class="su">1</span>, F<span class="su">2</span> the two foci, then to the
+points F<span class="su">1</span>, F<span class="su">2</span>, A, A&prime; in the one row correspond the points F<span class="su">1</span>, F<span class="su">2</span>, A&prime;, A
+in the other, each focus corresponding to itself. Hence (F<span class="su">1</span>F<span class="su">2</span>, AA&prime;) =
+(F<span class="su">1</span>F<span class="su">2</span>, A&prime;A)&mdash;that is, we may interchange the two points AA&prime; without
+altering the value of the cross-ratio, which is the characteristic
+property of harmonic conjugates (§ 18).</p>
+
+<p>4. The point conjugate to the point at infinity is called the
+&ldquo;centre&rdquo; of the involution. Every involution has a centre, unless
+the point at infinity be a focus, in which case we may say that
+the centre is at infinity.</p>
+
+<p>In an hyperbolic involution the centre is the middle point between
+the foci.</p>
+
+<p>5. The product of the distances of two conjugate points A, A&prime;
+from the centre O is constant: OA · OA&prime; = c.</p>
+
+<p>For let A, A&prime; and B, B&prime; be two pairs of conjugate points,  the
+centre, I the point at infinity, then</p>
+
+<p class="center">(AB, OI) = (A&prime;B&prime;, IO),</p>
+
+<p class="noind">or</p>
+
+<p class="center">OA · OA&prime; = OB · OB&prime;.</p>
+
+<p>In order to determine the distances of the foci from the centre,
+we write F for A and A&prime; and get</p>
+
+<p class="center">OF² = c; OF = ±&radic;c.</p>
+
+<p class="noind">Hence if c is positive OF is real, and has two values, equal and
+opposite. The involution is hyperbolic.</p>
+
+<p>If c = 0, OF = 0, and the two foci both coincide with the centre.
+If c is negative, &radic;c becomes imaginary, and there are no foci.
+Hence we may write&mdash;</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcl">In an hyperbolic involution,</td> <td class="tcl">OA · OA&prime; = k²,</td></tr>
+
+<tr><td class="tcl">In a parabolic involution,</td> <td class="tcl">OA · OA&prime; = 0,</td></tr>
+
+<tr><td class="tcl">In an elliptic involution,</td> <td class="tcl">OA · OA&prime; = &minus;k².</td></tr>
+</table>
+
+<p>From these expressions it follows that conjugate points A, A&prime; in an
+hyperbolic involution lie on the same side of the centre, and in an
+elliptic involution on opposite sides of the centre, and that in a
+parabolic involution one coincides with the centre.</p>
+
+<p>In the first case, for instance, OA · OA&prime; is positive; hence OA
+and OA&prime; have the same sign.</p>
+
+<p>It also follows that two segments, AA&prime; and BB&prime;, between pairs of
+conjugate points have the following positions: in an hyperbolic
+involution they lie either one altogether within or altogether without
+each other; in a parabolic involution they have one point in common;
+and in an elliptic involution they overlap, each being partly within
+and partly without the other.</p>
+
+<p><i>Proof.</i>&mdash;We have OA . OA&prime; = OB · OB&prime; = k² in case of an hyperbolic
+involution. Let A and B be the points in each pair which are
+nearer to the centre O. If now A, A&prime; and B, B&prime; lie on the same side of
+O, and if B is nearer to O than A, so that OB &lt; OA, then OB&prime; &gt; OA&prime;;
+hence B&prime; lies farther away from O than A&prime;, or the segment AA&prime; lies
+within BB&prime;. And so on for the other cases.</p>
+
+<p>6. An involution is determined&mdash;</p>
+
+<table class="reg" summary="poem"><tr><td> <div class="poemr">
+<p>(&alpha;) By two pairs of conjugate points. Hence also</p>
+<p>(&beta;) By one pair of conjugate points and the centre;</p>
+<p>(&gamma;) By the two foci;</p>
+<p>(&delta;) By one focus and one pair of conjugate points;</p>
+<p>(&epsilon;) By one focus and the centre.</p>
+</div> </td></tr></table>
+
+<p>7. The condition that A, B, C and A&prime;, B&prime;, C&prime; may form an involution
+may be written in one of the forms&mdash;</p>
+
+<p class="center">(AB, CC&prime;) = (A&prime;B&prime;, C&prime;C),</p>
+
+<p class="noind">or</p>
+
+<p class="center">(AB, CA&prime;) = (A&prime;B&prime;, C&prime;A),</p>
+
+<p class="noind">or</p>
+
+<p class="center">(AB, C&prime;A&prime;) = (A&prime;B&prime;, CA),</p>
+
+<p class="noind">for each expresses that in the two projective rows in which A, B, C
+<span class="pagenum"><a name="page702" id="page702"></a>702</span>
+and A&prime;, B&prime;, C&prime; are conjugate points two conjugate elements may be
+interchanged.</p>
+
+<p>8. Any three pairs. A, A&prime;, B, B&prime;, C, C&prime;, of conjugate points are
+connected by the relations:</p>
+
+<table class="math0" summary="math">
+<tr><td>AB&prime; · BC&prime; · CA&prime;</td>
+<td rowspan="2">=</td> <td>AB&prime; · BC · C&prime;A&prime;</td>
+<td rowspan="2">=</td> <td>AB · B&prime;C&prime; · CA&prime;</td>
+<td rowspan="2">=</td> <td>AB · B&prime;C · C&prime;A&prime;</td>
+<td rowspan="2">= &minus;1.</td></tr>
+<tr><td class="denom">A&prime;B · B&prime;C · C&prime;A</td> <td class="denom">A&prime;B · B&prime;C&prime; · CA</td>
+<td class="denom">A&prime;B&prime; · BC · C&prime;A</td> <td class="denom">A&prime;B&prime; · BC&prime; · CA</td></tr></table>
+
+<p>These relations readily follow by working out the relations in (7)
+(above).</p>
+
+<p>§ 78. <i>Involution of a quadrangle.&mdash;The sides of any four-point are
+cut by any line in six points in involution, opposite sides being cut in
+conjugate points.</i></p>
+
+<p>Let A<span class="su">1</span>B<span class="su">1</span>C<span class="su">1</span>D<span class="su">1</span> (fig. 31) be the four-point. If its sides be cut by
+the line p in the points A, A&prime;, B, B&prime;, C, C&prime;, if further, C<span class="su">1</span>D<span class="su">1</span> cuts the
+line A<span class="su">1</span>B<span class="su">1</span> in C<span class="su">2</span>, and if we project the row A<span class="su">1</span>B<span class="su">1</span>C<span class="su">2</span>C to p once from
+D<span class="su">1</span> and once from C<span class="su">1</span>, we get (A&prime;B&prime;, C&prime;C) = (BA, C&prime;C).</p>
+
+<p>Interchanging in the last cross-ratio the letters in each pair we get
+(A&prime;B&prime;, C&prime;C) = (AB, CC&prime;). Hence by § 77 (7) the points are in involution.</p>
+
+<p>The theorem may also be stated thus:</p>
+
+<p><i>The three points in which any line cuts the sides of a triangle and the
+projections, from any point in the plane, of the vertices of the triangle
+on to the same line are six points in involution.</i></p>
+
+<table class="nobctr" style="clear: both;" summary="Illustration">
+<tr><td class="figcenter"><img style="width:421px; height:325px" src="images/img702a.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 31.</span></td></tr></table>
+
+<p>Or again&mdash;</p>
+
+<p>The projections from any point on to any line of the six vertices
+of a four-side are six points in involution, the projections of opposite
+vertices being conjugate points.</p>
+
+<p>This property gives a simple means to construct, by aid of the
+straight edge only, in an involution of which two pairs of conjugate
+points are given, to any point its conjugate.</p>
+
+<p>§ 79. <i>Pencils in Involution.</i>&mdash;The theory of involution may at once
+be extended from the row to the flat and the axial pencil&mdash;viz. we say
+that there is an involution in a flat or in an axial pencil if any line
+cuts the pencil in an involution of points. An involution in a pencil
+consists of pairs of conjugate rays or planes; it has two, one or no
+<i>focal rays</i> (double lines) or <i>planes</i>, but nothing corresponding to a
+centre.</p>
+
+<p>An involution in a flat pencil contains always one, and in general
+only one, pair of conjugate rays which are perpendicular to one
+another. For in two projective flat pencils exist always two corresponding
+right angles (§ 40).</p>
+
+<p>Each involution in an axial pencil contains in the same manner
+one pair of conjugate planes at right angles to one another.</p>
+
+<p>As a rule, there exists but one pair of conjugate lines or planes
+at right angles to each other. But it is possible that there are
+more, and then there is an infinite number of such pairs. An involution
+in a flat pencil, in which every ray is perpendicular to its
+conjugate ray, is said to be <i>circular</i>. That such involution is
+possible is easily seen thus: if in two concentric flat pencils each
+ray on one is made to correspond to that ray on the other which
+is perpendicular to it, then the two pencils are projective, for if
+we turn the one pencil through a right angle each ray in one coincides
+with its corresponding ray in the other. But these two projective
+pencils are in involution.</p>
+
+<p>A circular involution has no focal rays, because no ray in a pencil
+coincides with the ray perpendicular to it.</p>
+
+<p>§ 80. <i>Every elliptical involution in a row may be considered as a
+section of a circular involution.</i></p>
+
+<p>In an elliptical involution any two segments AA&prime; and BB&prime; lie
+partly within and partly without each other (fig. 32). Hence two
+circles described on AA&prime; and BB&prime; as diameters will intersect in two
+points E and E&prime;. The line EE&prime; cuts the base of the involution at a
+point O, which has the property that OA . OA&prime; = OB · OB&prime;, for
+each is equal to OE . OE&prime;. The point O is therefore the centre of
+the involution. If we wish to construct to any point C the conjugate
+point C&prime;, we may draw the circle through CEE&prime;. This will cut the
+base in the required point C&prime; for OC · OC&prime; = OA · OA&prime;. But EC and
+EC&prime; are at right angles. Hence the involution which is obtained
+by joining E or E&prime; to the points
+in the given involution is circular.
+This may also be expressed
+thus:</p>
+
+<table class="flt" style="float: right; width: 310px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:257px; height:158px" src="images/img702b.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 32.</span></td></tr></table>
+
+<p><i>Every elliptical involution has
+the property that there are two
+definite points in the plane from
+which any two conjugate points
+are seen under a right angle.</i></p>
+
+<p>At the same time the following
+problem has been solved:</p>
+
+<p>To determine the centre and
+also the point corresponding
+to any given point in an elliptical involution of which two pairs of
+conjugate points are given.</p>
+
+<p>§ 81. <i>Involution Range on a Conic.</i>&mdash;By the aid of § 53, the points
+on a conic may be made to correspond to those on a line, so that the
+row of points on the conic is projective to a row of points on a line.
+We may also have two projective rows on the same conic, and these
+will be in involution as soon as one point on the conic has the same
+point corresponding to it all the same to whatever row it belongs.
+An involution of points on a conic will have the property (as follows
+from its definition, and from § 53) that the lines which join conjugate
+points of the involution to any point on the conic are conjugate lines
+of an involution in a pencil, and that a fixed tangent is cut by the
+tangents at conjugate points on the conic in points which are again
+conjugate points of an involution on the fixed tangent. For such
+involution on a conic the following theorem holds:</p>
+
+<p><i>The lines which join corresponding points in an involution on a conic
+all pass through a fixed point; and reciprocally, the points of intersection
+of conjugate lines in an involution among tangents to a conic
+lie on a line.</i></p>
+
+<table class="flt" style="float: left; width: 400px;" summary="Illustration">
+<tr><td class="figleft1"><img style="width:350px; height:288px" src="images/img702c.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 33</span></td></tr></table>
+
+<p>We prove the first part only. The involution is determined by
+two pairs of conjugate points, say by A, A&prime; and B, B&prime; (fig. 33). Let
+AA&prime; and BB&prime;
+meet in P. If we
+join the points in
+involution to any
+point on the conic,
+and the conjugate
+points to another
+point on the conic,
+we obtain two
+projective pencils.
+We take A and
+A&prime; as centres of
+these pencils, so
+that the pencils
+A(A&prime;BB&prime;) and
+A&prime;(AB&prime;B) are projective,
+and in
+perspective position,
+because AA&prime;
+corresponds to
+A&prime;A. Hence corresponding
+rays
+meet in a line, of which two points are found by joining AB&prime; to
+A&prime;B and AB to A&prime;B&prime;. It follows that the <i>axis</i> of perspective is the
+polar of the point P, where AA&prime; and BB&prime; meet. If we now wish
+to construct to any other point C on the conic the corresponding
+point C&prime;, we join C to A&prime; and the point where this line cuts p to A.
+The latter line cuts the conic again in C&prime;. But we know from the
+theory of pole and polar that the line CC&prime; passes through P. The
+point of concurrence is called the &ldquo;pole of the involution,&rdquo; and
+the line of collinearity of the meets is called the &ldquo;axis of the
+involution.&rdquo;</p>
+
+<p class="pt2 center sc" style="clear: both;">Involution Determined by a Conic on a Line.&mdash;Foci</p>
+
+<p>§ 82. The polars, with regard to a conic, of points in a row p form
+a pencil P projective to the row (§ 66). This pencil cuts the base of
+the row p in a projective row.</p>
+
+<p>If A is a point in the given row, A&prime; the point where the polar of
+A cuts p, then A and A&prime; will be corresponding points. If we take
+A&prime; a point in the first row, then the polar of A&prime; will pass through
+A, so that A corresponds to A&prime;&mdash;in other words, the rows are in
+involution. The conjugate points in this involution are conjugate
+points with regard to the conic. Conjugate points coincide only if
+the polar of a point A passes through A&mdash;that is, if A lies on the
+conic. Hence&mdash;</p>
+
+<p><i>A conic determines on every line in its plane an involution, in which
+those points are conjugate which are also conjugate with regard to the
+conic.</i></p>
+
+<p><i>If the line cuts the conic the involution is hyperbolic, the points of
+intersection being the foci.</i></p>
+
+<p><i>If the line touches the conic the involution is parabolic, the two foci
+coinciding at the point of contact.</i></p>
+
+<p><i>If the line does not cut the conic the involution is elliptic, having no
+foci.</i></p>
+
+<p><span class="pagenum"><a name="page703" id="page703"></a>703</span></p>
+
+<p>If, on the other hand, we take a point P in the plane of a conic,
+we get to each line a through P one conjugate line which joins P
+to the pole of a. These pairs of conjugate lines through P form an
+involution in the pencil at P. The focal rays of this involution are
+the tangents drawn from P to the conic. This gives the theorem
+reciprocal to the last, viz:&mdash;</p>
+
+<p><i>A conic determines in every pencil in its plane an involution, corresponding
+lines being conjugate lines with regard to the conic.</i></p>
+
+<p><i>If the point is without the conic the involution is hyperbolic, the
+tangents from the points being the focal rays.</i></p>
+
+<p><i>If the point lies on the conic the involution is parabolic, the tangent
+at the point counting for coincident focal rays.</i></p>
+
+<p><i>If the point is within the conic the involution is elliptic, having no
+focal rays.</i></p>
+
+<p>It will further be seen that the involution determined by a conic
+on any line p is a section of the involution, which is determined by
+the conic at the pole P of p.</p>
+
+<p>§ 83. <i>Foci.</i>&mdash;The centre of a pencil in which the conic determines
+a circular involution is called a &ldquo;focus&rdquo; of the conic.</p>
+
+<p>In other words, a focus is such a point that every line through it is
+perpendicular to its conjugate line. The polar to a focus is called a
+<i>directrix</i> of the conic.</p>
+
+<p>From the definition it follows that <i>every focus lies on an axis</i>, for
+the line joining a focus to the centre of the conic is a diameter to
+which the conjugate lines are perpendicular; and <i>every line joining
+two foci is an axis</i>, for the perpendiculars to this line through the foci
+are conjugate to it. These conjugate lines pass through the pole of
+the line, the pole lies therefore at infinity, and the line is a diameter,
+hence by the last property an axis.</p>
+
+<p>It follows that all <i>foci lie on one axis</i>, for no line joining a point
+in one axis to a point in the other can be an axis.</p>
+
+<p>As the conic determines in the pencil which has its centre at a focus
+a circular involution, no tangents can be drawn from the focus to
+the conic. Hence <i>each focus lies within a conic</i>; and <i>a directrix does
+not cut the conic</i>.</p>
+
+<p>Further properties are found by the following considerations:</p>
+
+<p>§ 84. Through a point P one line p can be drawn, which is with
+regard to a given conic conjugate to a given line q, viz. that line
+which joins the point P to the pole of the line q. If the line q is made
+to describe a pencil about a point Q, then the line p will describe a
+pencil about P. These two pencils will be projective, for the line
+p passes through the pole of q, and whilst q describes the pencil Q,
+its pole describes a projective row, and this row is perspective to
+the pencil P.</p>
+
+<p>We now take the point P on an axis of the conic, draw any line
+p through it, and from the pole of p draw a perpendicular q to p.
+Let q cut the axis in Q. Then, in the pencils of conjugate lines,
+which have their centres at P and Q, the lines p and q are conjugate
+lines at right angles to one another. Besides, to the axis as a ray
+in either pencil will correspond in the other the perpendicular to the
+axis (§ 72). The conic generated by the intersection of corresponding
+lines in the two pencils is therefore the circle on PQ as diameter,
+<i>so that every line in P is perpendicular to its corresponding line
+in Q</i>.</p>
+
+<p>To every point P on an axis of a conic corresponds thus a point
+Q, such that conjugate lines through P and Q are perpendicular.</p>
+
+<p>We shall show that these <i>point-pairs</i> P, Q <i>form an involution</i>.
+To do this let us move P along the axis, and with it the line p,
+keeping the latter parallel to itself. Then P describes a row, p a
+perspective pencil (of parallels), and the pole of p a projective row.
+At the same time the line q describes a pencil of parallels perpendicular
+to p, and perspective to the row formed by the pole of p. The point
+Q, therefore, where q cuts the axis, describes a row projective to the
+row of points P. The two points P and Q describe thus two projective
+rows on the axis; and not only does P as a point in the first
+row correspond to Q, but also Q as a point in the first corresponds
+to P. The two rows therefore form an involution. <i>The centre of
+this involution, it is easily seen, is the centre of the conic.</i></p>
+
+<p><i>A focus of this involution has the property that any two conjugate
+lines through it are perpendicular; hence, it is a focus to the conic.</i></p>
+
+<p>Such involution exists on each axis. But only one of these can
+have foci, because all foci lie on the same axis. The involution on
+one of the axes is elliptic, and appears (§ 80) therefore as the section
+of two circular involutions in two pencils whose centres lie in the
+other axis. These centres are foci, hence the one axis contains two
+foci, the other axis none; <i>or every central conic has two foci which lie
+on one axis equidistant from the centre</i>.</p>
+
+<p>The axis which contains the foci is called the <i>principal axis</i>; in
+case of an hyperbola it is the axis which cuts the curve, because the
+foci lie within the conic.</p>
+
+<p>In case of the parabola there is but one axis. The involution
+on this axis has its centre at infinity. One focus is therefore at
+infinity, the one focus only is finite. <i>A parabola has only one
+focus.</i></p>
+
+<table class="nobctr" style="clear: both;" summary="Illustration">
+<tr><td class="figcenter"><img style="width:356px; height:210px" src="images/img703a.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 34.</span></td></tr></table>
+
+<p>§ 85. If through any point P (fig. 34) on a conic the tangent PT
+and the normal PN (<i>i.e.</i> the perpendicular to the tangent through
+the point of contact) be drawn, these will be conjugate lines with
+regard to the conic, and at right angles to each other. They will
+therefore cut the principal axis in two points, which are conjugate
+in the involution considered in § 84; hence they are harmonic
+conjugates with regard to the foci. If therefore the two foci F<span class="su">1</span> and
+F<span class="su">2</span> be joined to P, these lines will be harmonic with regard to the
+tangent and normal. As the latter are perpendicular, they will
+bisect the angles between the other pair. Hence&mdash;</p>
+
+<p><i>The lines joining any point on a conic to the two foci are equally
+inclined to the tangent and normal at that point.</i></p>
+
+<p>In case of the parabola this becomes&mdash;</p>
+
+<p><i>The line joining any point on a parabola to the focus and the diameter
+through the point, are equally inclined to the tangent and normal at
+that point.</i></p>
+
+<p>From the definition of a focus it follows that&mdash;</p>
+
+<p><i>The segment of a tangent between the directrix and the point of
+contact is seen from the focus belonging to the directrix under a right
+angle</i>, because the lines joining the focus to the ends of this
+segment are conjugate with regard to the conic, and therefore
+perpendicular.</p>
+
+<p>With equal ease the following theorem is proved:</p>
+
+<p><i>The two lines which join the points of contact of two tangents each
+to one focus, but not both to the same, are seen from the intersection of
+the tangents under equal angles.</i></p>
+
+<p>§ 86. Other focal properties of a conic are obtained by the following
+considerations:</p>
+
+<table class="flt" style="float: right; width: 370px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:317px; height:550px" src="images/img703b.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 35.</span></td></tr></table>
+
+<p>Let F (fig. 35) be a focus to a conic, f the corresponding directrix,
+A and B the points of contact of two tangents meeting at T, and P
+the point where the
+line AB cuts the directrix.
+Then TF will be
+the polar of P (because
+polars of F and T meet
+at P). Hence TF and
+PF are conjugate lines
+through a focus, and
+therefore perpendicular.
+They are further harmonic
+conjugates with
+regard to FA and FB
+(§§ 64 and 13), so that
+they bisect the angles
+formed by these lines.
+This by the way
+proves&mdash;</p>
+
+<p><i>The segments between
+the point of intersection
+of two tangents to a conic
+and their points of contact
+are seen from a focus
+under equal angles.</i></p>
+
+<p>If we next draw
+through A and B lines
+parallel to TF, then the
+points A<span class="su">1</span>, B<span class="su">1</span> where
+these cut the directrix
+will be harmonic conjugates
+with regard to P
+and the point where FT
+cuts the directrix. The
+lines FT and FP bisect
+therefore also the angles
+between FA<span class="su">1</span> and FB<span class="su">1</span>.
+From this it follows
+easily that the triangles
+FAA<span class="su">1</span> and FBB<span class="su">1</span> are
+equiangular, and therefore similar, so that FA : AA<span class="su">1</span> = FB : BB<span class="su">1</span>.</p>
+
+<p>The triangles AA<span class="su">1</span>A<span class="su">2</span> and BB<span class="su">1</span>B<span class="su">2</span> formed by drawing perpendiculars
+from A and B to the directrix are also similar, so that AA<span class="su">1</span> : AA<span class="su">2</span> =
+= BB<span class="su">1</span> : BB<span class="su">2</span>. This, combined with the above proportion, gives
+FA : AA<span class="su">2</span> = FB : BB<span class="su">2</span>. Hence the theorem:</p>
+
+<p><i>The ratio of the distances of any point on a conic from a focus and
+the corresponding directrix is constant.</i></p>
+
+<p>To determine this ratio we consider its value for a vertex on the
+principal axis. In an ellipse the focus lies between the two vertices
+on this axis, hence the focus is nearer to a vertex than to the corresponding
+directrix. Similarly, in an hyperbola a vertex is nearer
+<span class="pagenum"><a name="page704" id="page704"></a>704</span>
+to the directrix than to the focus. In a parabola the vertex lies
+halfway between directrix and focus.</p>
+
+<p>It follows in an ellipse the ratio between the distance of a point
+from the focus to that from the directrix is less than unity, in the
+parabola it equals unity, and in the hyperbola it is greater than
+unity.</p>
+
+<p>It is here the same which focus we take, because the two foci
+lie symmetrical to the axis of the conic. If now P is any point on
+the conic having the distances r<span class="su">1</span> and r<span class="su">2</span> from the foci and the distances
+d<span class="su">1</span> and d<span class="su">2</span> from the corresponding directrices, then r<span class="su">1</span>/d<span class="su">1</span> = r<span class="su">2</span>/d<span class="su">2</span> = e,
+where e is constant. Hence also (r<span class="su">1</span> ± r<span class="su">2</span>) / (d<span class="su">1</span> ± d<span class="su">2</span>) = e.</p>
+
+<p>In the ellipse, which lies between the directrices, d<span class="su">1</span> + d<span class="su">2</span> is constant,
+therefore also r<span class="su">1</span> + r<span class="su">2</span>. In the hyperbola on the other hand d<span class="su">1</span> &minus; d<span class="su">2</span> is
+constant, equal to the distance between the directrices, therefore
+in this case r<span class="su">1</span> &minus; r<span class="su">2</span> is constant.</p>
+
+<p>If we call the distances of a point on a conic from the focus its
+focal distances we have the theorem:</p>
+
+<p><i>In an ellipse the sum of the focal distances is constant; and in an
+hyperbola the difference of the focal distances is constant.</i></p>
+
+<p><i>This constant sum or difference equals in both cases the length of
+the principal axis.</i></p>
+
+<p class="pt2 center sc">Pencil of Conics</p>
+
+<p>§ 87. Through four points A, B, C, D in a plane, of which no three
+lie in a line, an infinite number of conics may be drawn, viz. through
+these four points and any fifth one single conic. This system of
+conics is called a pencil of conics. Similarly, all conics touching four
+fixed lines form a system such that any fifth tangent determines one
+and only one conic. We have here the theorems:</p>
+
+<table class="nobctr" summary="Contents">
+<tr><td class="tcl rb3" style="width: 50%;">The pairs of points in which any line is cut by a system of
+ conics through four fixed points are in involution.</td>
+
+<td class="tcl" style="width: 50%;">The pairs of tangents which can be drawn from a point to
+ a system of conics touching four fixed lines are in involution.</td></tr></table>
+
+<table class="nobctr" style="clear: both;" summary="Illustration">
+<tr><td class="figcenter"><img style="width:371px; height:298px" src="images/img704a.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 36.</span></td></tr></table>
+
+<p>We prove the first theorem only. Let ABCD (fig. 36) be the
+four-point, then any line t will cut two opposite sides AC, BD in
+the points E, E&prime;, the pair AD, BC in points F, F&prime;, and any conic
+of the system in M, N, and we have A(CD, MN) = B(CD, MN).</p>
+
+<p>If we cut these pencils by t we get</p>
+
+<p class="center">(EF, MN) = (F&prime;E&prime;, MN)</p>
+
+<p class="noind">or</p>
+
+<p class="center">(EF, MN) = (E&prime;F&prime;, NM).</p>
+
+<p>But this is, according to § 77 (7), the condition that M, N are
+corresponding points in the involution determined by the point pairs
+E, E&prime;, F, F&prime; in which the line t cuts pairs of opposite sides of the
+four-point ABCD. This involution is independent of the particular
+conic chosen.</p>
+
+<p>§ 88. There follow several important theorems:</p>
+
+<p><i>Through four points two, one, or no conics may be drawn which touch
+any given line, according as the involution determined by the given
+four-point on the line has real, coincident or imaginary foci.</i></p>
+
+<p><i>Two, one, or no conics may be drawn which touch four given lines
+and pass through a given point, according as the involution determined
+by the given four-side at the point has real, coincident or imaginary
+focal rays.</i></p>
+
+<p>For the conic through four points which touches a given line has
+its point of contact at a focus of the involution determined by the
+four-point on the line.</p>
+
+<p>As a special case we get, by taking the line at infinity:</p>
+
+<p><i>Through four points of which none is at infinity either two or no
+parabolas may be drawn.</i></p>
+
+<p>The problem of drawing a conic through four points and touching
+a given line is solved by determining the points of contact on the
+line, that is, by determining the foci of the involution in which the
+line cuts the sides of the four-point. The corresponding remark
+holds for the problem of drawing the conics which touch four lines
+and pass through a given point.</p>
+
+<p class="pt2 center sc">Ruled Quadric Surfaces</p>
+
+<p>§ 89. We have considered hitherto projective rows which lie in
+the same plane, in which case lines joining corresponding points
+envelop a conic. We shall now consider projective rows whose
+bases do not meet. In this case, corresponding points will be joined
+by lines which do not lie in a plane, but on some surface, which
+like every surface generated by lines is called a <i>ruled</i> surface. This
+surface clearly contains the bases of the two rows.</p>
+
+<p>If the points in either row be joined to the base of the other, we
+obtain two axial pencils which are also projective, those planes
+being corresponding which pass through corresponding points in the
+given rows. If A&prime;, A be two corresponding points, &alpha;, &alpha;&prime; the planes in
+the axial pencils passing through them, then AA&prime; will be the line
+of intersection of the corresponding planes &alpha;, &alpha;&prime; and also the line
+joining corresponding points in the rows.</p>
+
+<p>If we cut the whole figure by a plane this will cut the axial pencils
+in two projective flat pencils, and the curve of the second order
+generated by these will be the curve in which the plane cuts the
+surface. Hence</p>
+
+<p><i>The locus of lines joining corresponding points in two projective
+rows which do not lie in the same plane is a surface which contains the
+bases of the rows, and which can also be generated by the lines of intersection
+of corresponding planes in two projective axial pencils. This
+surface is cut by every plane in a curve of the second order, hence either
+in a conic or in a line-pair. No line which does not lie altogether on
+the surface can have more than two points in common with the surface,
+which is therefore said to be of the second order or is called a ruled
+quadric surface.</i></p>
+
+<p>That no line which does not lie on the surface can cut the surface
+in more than two points is seen at once if a plane be drawn through
+the line, for this will cut the surface in a conic. It follows also that
+a line which contains more than two points of the surface lies altogether
+on the surface.</p>
+
+<p>§ 90. Through any point in space one line can always be drawn
+cutting two given lines which do not themselves meet.</p>
+
+<p>If therefore three lines in space be given of which no two meet,
+then through every point in either one line may be drawn cutting
+the other two.</p>
+
+<p><i>If a line moves so that it always cuts three given lines of which no
+two meet, then it generates a ruled quadric surface.</i></p>
+
+<p>Let a, b, c be the given lines, and p, q, r ... lines cutting them in the
+points A, A&prime;, A&Prime; ...; B, B&prime;, B&Prime; ...; C, C&prime;, C&Prime; ... respectively; then
+the planes through a containing p, q, r, and the planes through b containing
+the same lines, may be taken as corresponding planes in two
+axial pencils which are projective, because both pencils cut the line
+c in the same row, C, C&prime;, C&Prime; ...; the surface can therefore be generated
+by projective axial pencils.</p>
+
+<p>Of the lines p, q, r ... no two can meet, for otherwise the lines
+a, b, c which cut them would also lie in their plane. There is a single
+infinite number of them, for one passes through each point of a.
+These lines are said to form a set of lines on the surface.</p>
+
+<p>If now three of the lines p, q, r be taken, then every line d cutting
+them will have three points in common with the surface, and will
+therefore lie altogether on it. This gives rise to a second set of lines
+on the surface. From what has been said the theorem follows:</p>
+
+<p><i>A ruled quadric surface contains two sets of straight lines. Every
+line of one set cuts every line of the other, but no two lines of the same
+set meet.</i></p>
+
+<p><i>Any two lines of the same set may be taken as bases of two projective
+rows, or of two projective pencils which generate the surface. They are
+cut by the lines of the other set in two projective rows.</i></p>
+
+<p>The plane at infinity like every other plane cuts the surface either
+in a conic proper or in a line-pair. In the first case the surface is
+called an <i>Hyperboloid of one sheet</i>, in the second an <i>Hyperbolic
+Paraboloid</i>.</p>
+
+<p>The latter may be generated by a line cutting three lines of which
+one lies at infinity, that is, cutting two lines and remaining parallel
+to a given plane.</p>
+
+<p class="pt2 center sc">Quadric Surfaces</p>
+
+<p>§ 91. The conics, the cones of the second order, and the ruled
+quadric surfaces complete the figures which can be generated by
+projective rows or flat and axial pencils, that is, by those aggregates
+of elements which are of one dimension (§§ 5, 6). We shall
+now consider the simpler figures which are generated by aggregates of
+two dimensions. The space at our disposal will not, however, allow
+us to do more than indicate a few of the results.</p>
+
+<p>§ 92. We establish a correspondence between the lines and planes
+in pencils in space, or reciprocally between the points and lines in
+two or more planes, but consider principally pencils.</p>
+
+<p>In two pencils we may either make planes correspond to planes
+and lines to lines, or else planes to lines and lines to planes. If
+hereby the condition be satisfied that to a flat, or axial, pencil
+corresponds in the first case a projective flat, or axial, pencil, and in
+the second a projective axial, or flat, pencil, the pencils are said to be
+<i>projective</i> in the first case and <i>reciprocal</i> in the second.</p>
+
+<p>For instance, two pencils which join two points S<span class="su">1</span> and S<span class="su">2</span> to the
+different points and lines in a given plane &pi; are projective (and
+in perspective position), if those lines and planes be taken as
+<span class="pagenum"><a name="page705" id="page705"></a>705</span>
+corresponding which meet the plane &pi; in the same point or in the
+same line. In this case every plane through both centres S<span class="su">1</span> and S<span class="su">2</span>
+of the two pencils will correspond to itself. If these pencils are
+brought into any other position they will be projective (but not
+perspective).</p>
+
+<p><i>The correspondence between two projective pencils is uniquely
+determined, if to four rays (or planes) in the one the corresponding
+rays (or planes) in the other are given, provided that no three rays of
+either set lie in a plane.</i></p>
+
+<p>Let a, b, c, d be four rays in the one, a&prime;, b&prime;, c&prime;, d&prime; the corresponding
+rays in the other pencil. We shall show that we can find for every
+ray e in the first a single corresponding ray e&prime; in the second. To
+the axial pencil a (b, c, d ...) formed by the planes which join a to
+b, c, d ..., respectively corresponds the axial pencil a&prime; (b&prime;, c&prime;, d&prime; ... ),
+and this correspondence is determined. Hence, the plane a&prime;e&prime; which
+corresponds to the plane ae is determined. Similarly the plane
+b&prime;e&prime; may be found and both together determine the ray e&prime;.</p>
+
+<p>Similarly the correspondence between two reciprocal pencils is
+determined if for four rays in the one the corresponding planes in
+the other are given.</p>
+
+<p>§ 93. We may now combine&mdash;</p>
+
+<div class="list">
+<p>1. Two reciprocal pencils.</p>
+</div>
+
+<div class="list1">
+<p>Each ray cuts its corresponding plane in a point, the locus
+of these points is a quadric surface.</p>
+</div>
+
+<div class="list">
+<p>2. Two projective pencils.</p>
+</div>
+
+<div class="list1">
+<p>Each plane cuts its corresponding plane in a line, but a
+ray as a rule does not cut its corresponding ray. The
+locus of points where a ray cuts its corresponding ray
+is a twisted cubic. The lines where a plane cuts its
+corresponding plane are secants.</p>
+</div>
+
+<div class="list">
+<p>3. Three projective pencils.</p>
+</div>
+
+<div class="list1">
+<p>The locus of intersection of corresponding planes is a
+cubic surface.</p>
+</div>
+
+<p>Of these we consider only the first two cases.</p>
+
+<p>§ 94. If two pencils are reciprocal, then to a plane in either corresponds
+a line in the other, to a flat pencil an axial pencil, and so on.
+Every line cuts its corresponding plane in a point. If S<span class="su">1</span> and S<span class="su">2</span> be
+the centres of the two pencils, and P be a point where a line a<span class="su">1</span> in the
+first cuts its corresponding plane &alpha;<span class="su">2</span>, <i>then the line</i> b<span class="su">2</span> <i>in the pencil</i> S<span class="su">2</span>
+<i>which passes through</i> P <i>will meet its corresponding plane &beta;<span class="su">1</span> in</i> P. For
+b<span class="su">2</span> is a line in the plane &alpha;<span class="su">2</span>. The corresponding plane &beta;<span class="su">1</span> must therefore
+pass through the line a<span class="su">1</span>, hence through P.</p>
+
+<p>The points in which the lines in S<span class="su">1</span> cut the planes corresponding
+to them in S<span class="su">2</span> are therefore the same as the points in which the lines
+in S<span class="su">2</span> cut the planes corresponding to them in S<span class="su">1</span>.</p>
+
+<p><i>The locus of these points is a surface which is cut by a plane in a
+conic or in a line-pair and by a line in not more than two points unless
+it lies altogether on the surface. The surface itself is therefore called a
+quadric surface, or a surface of the second order.</i></p>
+
+<p>To prove this we consider any line p in space.</p>
+
+<p>The flat pencil in S<span class="su">1</span> which lies in the plane drawn through p
+and the corresponding axial pencil in S<span class="su">2</span> determine on p two projective
+rows, and those points in these which coincide with their
+corresponding points lie on the surface. But there exist only two,
+or one, or no such points, unless every point coincides with its
+corresponding point. In the latter case the line lies altogether on
+the surface.</p>
+
+<p>This proves also that a plane cuts the surface in a curve of the
+second order, as no line can have more than two points in common
+with it. To show that this is a curve of the same kind as those
+considered before, we have to show that it can be generated by
+projective flat pencils. We prove first that this is true for any
+plane through the centre of one of the pencils, and afterwards that
+every point on the surface may be taken as the centre of such pencil.
+Let then &alpha;<span class="su">1</span> be a plane through S<span class="su">1</span>. To the flat pencil in S<span class="su">1</span> which
+it contains corresponds in S<span class="su">2</span> a projective axial pencil with axis
+a<span class="su">2</span> and this cuts &alpha;<span class="su">1</span> in a second flat pencil. These two flat pencils
+in &alpha;<span class="su">1</span> are projective, and, in general, neither concentric nor perspective.
+They generate therefore a conic. But if the line a<span class="su">2</span> passes
+through S<span class="su">1</span> the pencils will have S<span class="su">1</span> as common centre, and may
+therefore have two, or one, or no lines united with their corresponding
+lines. The section of the surface by the plane &alpha;<span class="su">1</span> will be accordingly
+a line-pair or a single line, or else the plane &alpha;<span class="su">1</span> will have only the
+point S<span class="su">1</span> in common with the surface.</p>
+
+<p>Every line l<span class="su">1</span> through S<span class="su">1</span> cuts the surface in two points, viz. first
+in S<span class="su">1</span> and then at the point where it cuts its corresponding plane.
+If now the corresponding plane passes through S<span class="su">1</span>, as in the case
+just considered, then the two points where l<span class="su">1</span> cuts the surface coincide
+at S<span class="su">1</span>, and the line is called a tangent to the surface with S<span class="su">1</span> as point
+of contact. Hence if l<span class="su">1</span> be a tangent, it lies in that plane &tau;<span class="su">1</span> which
+corresponds to the line S<span class="su">2</span>S<span class="su">1</span> as a line in the pencil S<span class="su">2</span>. The section
+of this plane has just been considered. It follows that&mdash;</p>
+
+<p><i>All tangents to quadric surface at the centre of one of the reciprocal
+pencils lie in a plane which is called the tangent plane to the surface
+at that point as point of contact.</i></p>
+
+<p><i>To the line joining the centres of the two pencils as a line in one
+corresponds in the other the tangent plane at its centre.</i></p>
+
+<p><i>The tangent plane to a quadric surface either cuts the surface in
+two lines, or it has only a single line, or else only a single point in
+common with the surface.</i></p>
+
+<p><i>In the first case the point of contact is said to be hyperbolic, in the
+second parabolic, in the third elliptic.</i></p>
+
+<p>§ 95. It remains to be proved that every point S on the surface
+may be taken as centre of one of the pencils which generate the
+surface. Let S be any point on the surface &Phi;&prime; generated by the
+reciprocal pencils S<span class="su">1</span> and S<span class="su">2</span>. We have to establish a reciprocal
+correspondence between the pencils S and S<span class="su">1</span>, so that the surface
+generated by them is identical with &Phi;. To do this we draw two
+planes &alpha;<span class="su">1</span> and &beta;<span class="su">1</span> through S<span class="su">1</span>, cutting the surface &Phi; in two conics
+which we also denote by &alpha;<span class="su">1</span> and &beta;<span class="su">1</span>. These conics meet at S<span class="su">1</span>, and
+at some other point T where the line of intersection of &alpha;<span class="su">1</span> and &beta;<span class="su">1</span>
+cuts the surface.</p>
+
+<p>In the pencil S we draw some plane &sigma; which passes through T,
+but not through S<span class="su">1</span> or S<span class="su">2</span>. It will cut the two conics first at T, and
+therefore each at some other point which we call A and B respectively.
+These we join to S by lines a and b, and now establish the
+required correspondence between the pencils S<span class="su">1</span> and S as follows:&mdash;To
+S<span class="su">1</span>T shall correspond the plane &sigma;, to the plane &alpha;<span class="su">1</span> the line a, and
+to &beta;<span class="su">1</span> the line b, hence to the flat pencil in &alpha;<span class="su">1</span> the axial pencil a.
+These pencils are made projective by aid of the conic in &alpha;<span class="su">1</span>.</p>
+
+<p>In the same manner the flat pencil in &beta;<span class="su">1</span> is made projective to the
+axial pencil b by aid of the conic in &beta;<span class="su">1</span>, corresponding elements being
+those which meet on the conic. This determines the correspondence,
+for we know for more than four rays in S<span class="su">1</span> the corresponding planes
+in S. The two pencils S and S<span class="su">1</span> thus made reciprocal generate a
+quadric surface &Phi;&prime;, which passes through the point S and through
+the two conics &alpha;<span class="su">1</span> and &beta;<span class="su">1</span>.</p>
+
+<p>The two surfaces &Phi; and &Phi;&prime; have therefore the points S and S<span class="su">1</span> and
+the conics &alpha;<span class="su">1</span> and &beta;<span class="su">1</span> in common. To show that they are identical,
+we draw a plane through S and S<span class="su">2</span>, cutting each of the conics &alpha;<span class="su">1</span> and
+&beta;<span class="su">1</span> in two points, which will always be possible. This plane cuts
+&Phi; and &Phi;&prime; in two conics which have the point S and the points where
+it cuts &alpha;<span class="su">1</span> and &beta;<span class="su">1</span> in common, that is five points in all. The conics
+therefore coincide.</p>
+
+<p>This proves that all those points P on &Phi;&prime; lie on &Phi; which have the
+property that the plane SS<span class="su">2</span>P cuts the conics &alpha;<span class="su">1</span>, &beta;<span class="su">1</span> in two points
+each. If the plane SS<span class="su">2</span>P has not this property, then we draw a plane
+SS<span class="su">1</span>P. This cuts each surface in a conic, and these conics have in
+common the points S, S<span class="su">1</span>, one point on each of the conics &alpha;<span class="su">1</span>, &beta;<span class="su">1</span>, and
+one point on one of the conics through S and S<span class="su">2</span> which lie on both
+surfaces, hence five points. They are therefore coincident, and our
+theorem is proved.</p>
+
+<p>§ 96. The following propositions follow:&mdash;</p>
+
+<p><i>A quadric surface has at every point a tangent plane.</i></p>
+
+<p><i>Every plane section of a quadric surface is a conic or a line-pair.</i></p>
+
+<p><i>Every line which has three points in common with a quadric surface
+lies on the surface.</i></p>
+
+<p><i>Every conic which has five points in common with a quadric surface
+lies on the surface.</i></p>
+
+<p><i>Through two conics which lie in different planes, but have two points
+in common, and through one external point always one quadric surface
+may be drawn.</i></p>
+
+<p>§ 97. <i>Every plane which cuts a quadric surface in a line-pair is a
+tangent plane.</i> For every line in this plane through the centre of
+the line-pair (the point of intersection of the two lines) cuts the
+surface in two coincident points and is therefore a tangent to the
+surface, <i>the centre of the line-pair being the point of contact</i>.</p>
+
+<p><i>If a quadric surface contains a line, then every plane through this
+line cuts the surface in a line-pair (or in two coincident lines).</i> For
+this plane cannot cut the surface in a conic. Hence:&mdash;</p>
+
+<p><i>If a quadric surface contains one line p then it contains an infinite
+number of lines, and through every point</i> Q <i>on the surface, one line</i>
+q <i>can be drawn which cuts</i> p. For the plane through the point Q
+and the line p cuts the surface in a line-pair which must pass through
+Q and of which p is one line.</p>
+
+<p><i>No two such lines</i> q <i>on the surface can meet</i>. For as both meet p
+their plane would contain p and therefore cut the surface in a
+triangle.</p>
+
+<p><i>Every line which cuts three lines</i> q <i>will be on the surface</i>; for it
+has three points in common with it.</p>
+
+<p><i>Hence the quadric surfaces which contain lines are the same as the
+ruled quadric surfaces considered in</i> §§ 89-93, but with one important
+exception. In the last investigation we have left out of consideration
+the possibility of a plane having only one line (two coincident
+lines) in common with a quadric surface.</p>
+
+<p>§ 98. To investigate this case we suppose first that there is one
+point A on the surface through which two different lines a, b can be
+drawn, which lie altogether on the surface.</p>
+
+<p>If P is any other point on the surface which lies neither on a nor
+b, then the plane through P and a will cut the surface in a second
+line a&prime; which passes through P and which cuts a. Similarly there
+is a line b&prime; through P which cuts b. These two lines a&prime; and b&prime; <i>may</i>
+coincide, but then they must coincide with PA.</p>
+
+<p>If this happens for one point P, it happens for every other point
+Q. For if two different lines could be drawn through Q, then by the
+same reasoning the line PQ would be altogether on the surface,
+hence two lines would be drawn through P against the assumption.
+From this follows:&mdash;</p>
+
+<p><i>If there is one point on a quadric surface through which one, but only
+one, line can be drawn on the surface, then through every point one line</i>
+<span class="pagenum"><a name="page706" id="page706"></a>706</span>
+<i>can be drawn, and all these lines meet in a point. The surface is a cone
+of the second order</i>.</p>
+
+<p><i>If through one point on a quadric surface, two, and only two, lines
+can be drawn on the surface, then through every point two lines may
+be drawn, and the surface is  ruled quadric surface.</i></p>
+
+<p><i>If through one point on a quadric surface no line on the surface can
+be drawn, then the surface contains no lines.</i></p>
+
+<p>Using the definitions at the end of § 95, we may also say:&mdash;</p>
+
+<p><i>On a quadric surface the points are all hyperbolic, or all parabolic,
+or all elliptic.</i></p>
+
+<p>As an example of a quadric surface with elliptical points, we
+mention the sphere which may be generated by two reciprocal
+pencils, where to each line in one corresponds the plane perpendicular
+to it in the other.</p>
+
+<p>§ 99. <i>Poles and Polar Planes.</i>&mdash;The theory of poles and polars
+with regard to a conic is easily extended to quadric surfaces.</p>
+
+<p>Let P be a point in space not on the surface, which we suppose
+not to be a cone. On every line through P which cuts the surface
+in two points we determine the harmonic conjugate Q of P with
+regard to the points of intersection. Through one of these lines we
+draw two planes &alpha; and &beta;. The locus of the points Q in &alpha; is a line a,
+the polar of P with regard to the conic in which &alpha; cuts the surface.
+Similarly the locus of points Q in &beta; is a line b. This cuts a, because
+the line of intersection of &alpha; and &beta; contains but one point Q. The
+locus of all points Q therefore is a plane. <i>This plane is called the
+polar plane of the point</i> P, <i>with regard to the quadric surface. If</i> P
+<i>lies on the surface we take the tangent plane of P as its polar.</i></p>
+
+<p>The following propositions hold:&mdash;</p>
+
+<p>1. <i>Every point has a polar plane</i>, which is constructed by drawing
+the polars of the point with regard to the conics in which two planes
+through the point cut the surface.</p>
+
+<p>2. <i>If</i> Q <i>is a point in the polar of</i> P, <i>then</i> P <i>is a point in the polar
+of</i> Q, because this is true with regard to the conic in which a plane
+through PQ cuts the surface.</p>
+
+<p>3. <i>Every plane is the polar plane of one point, which is called the
+Pole of the plane.</i></p>
+
+<p>The pole to a plane is found by constructing the polar planes of
+three points in the plane. Their intersection will be the pole.</p>
+
+<p>4. <i>The points in which the polar plane of P cuts the surface are
+points of contact of tangents drawn from P to the surface</i>, as is easily
+seen. Hence:&mdash;</p>
+
+<p>5. <i>The tangents drawn from a point P to a quadric surface form a
+cone of the second order</i>, for the polar plane of P cuts it in a conic.</p>
+
+<p>6. <i>If the pole describes a line a, its polar plane will turn about
+another line</i> a&prime;, as follows from 2. <i>These lines a and a&prime; are said to be
+conjugate with regard to the surface.</i></p>
+
+<p>§ 100. The pole of the line at infinity is called the <i>centre</i> of the
+surface. If it lies at the infinity, the plane at infinity is a tangent
+plane, and the surface is called a <i>paraboloid</i>.</p>
+
+<p><i>The polar plane to any point at infinity passes through the centre,
+and is called a diametrical plane.</i></p>
+
+<p><i>A line through the centre is called a diameter. It is bisected at the
+centre. The line conjugate to it lies at infinity.</i></p>
+
+<p><i>If a point moves along a diameter its polar plane turns about the
+conjugate line at infinity</i>; that is, <i>it moves parallel to itself, its centre
+moving on the first line.</i></p>
+
+<p><i>The middle points of parallel chords lie in a plane</i>, viz. in the polar
+plane of the point at infinity through which the chords are drawn.</p>
+
+<p><i>The centres of parallel sections lie in a diameter which is a line
+conjugate to the line at infinity in which the planes meet.</i></p>
+
+<p class="pt2 center sc">Twisted Cubics</p>
+
+<p>§ 101. If two pencils with centres S<span class="su">1</span> and S<span class="su">2</span> are made projective,
+then to a ray in one corresponds a ray in the other, to a plane a
+plane, to a flat or axial pencil a projective flat or axial pencil, and
+so on.</p>
+
+<p>There is a double infinite number of lines in a pencil. We shall
+see that a single infinite number of lines in one pencil meets its
+corresponding ray, and that the points of intersection form a curve
+in space.</p>
+
+<p>Of the double infinite number of planes in the pencils each will
+meet its corresponding plane. This gives a system of a double
+infinite number of lines in space. We know (§ 5) that there is a
+quadruple infinite number of lines in space. From among these we
+may select those which satisfy one or more given conditions. The
+systems of lines thus obtained were first systematically investigated
+and classified by Plücker, in his <i>Geometrie des Raumes</i>. He uses the
+following names:&mdash;</p>
+
+<p>A <i>treble infinite</i> number of lines, that is, all lines which satisfy one
+condition, are said to form a <i>complex of lines</i>; <i>e.g.</i> all lines cutting
+a given line, or all lines touching a surface.</p>
+
+<p>A <i>double infinite</i> number of lines, that is, all lines which satisfy
+two conditions, or which are common to two complexes, are said to
+form a <i>congruence of lines</i>; <i>e.g.</i> all lines in a plane, or all lines
+cutting two curves, or all lines cutting a given curve twice.</p>
+
+<p>A <i>single infinite</i> number of lines, that is, all lines which satisfy
+three conditions, or which belong to three complexes, form a <i>ruled
+surface</i>; <i>e.g.</i> one set of lines on a ruled quadric surface, or developable
+surfaces which are formed by the tangents to a curve.</p>
+
+<p>It follows that all lines in which corresponding planes in two
+projective pencils meet form a congruence. We shall see this congruence
+consists of all lines which cut a twisted cubic twice, or of
+all <i>secants</i> to a twisted cubic.</p>
+
+<p>§ 102. Let l<span class="su">1</span> be the line S<span class="su">1</span>S<span class="su">2</span> as a line in the pencil S<span class="su">1</span>. To it
+corresponds a line l<span class="su">2</span> in S<span class="su">2</span>. <i>At each of the centres two corresponding
+lines meet.</i> The two axial pencils with l<span class="su">1</span> and l<span class="su">2</span> as axes are projective,
+and, as, their axes meet at S<span class="su">2</span>, the intersections of corresponding
+planes form a cone of the second order (§ 58), with S<span class="su">2</span> as
+centre. If &pi;<span class="su">1</span> and &pi;<span class="su">2</span> be corresponding planes, then their intersection
+will be a line p<span class="su">2</span> which passes through S<span class="su">2</span>. Corresponding to it in
+S<span class="su">1</span> will be a line p<span class="su">1</span> which lies in the plane &pi;<span class="su">1</span>, and which therefore
+meets p<span class="su">2</span> at some point P. Conversely, if p<span class="su">2</span> be any line in S<span class="su">2</span> which
+meets its corresponding line p<span class="su">1</span> at a point P, then to the plane l<span class="su">2</span>p<span class="su">2</span>
+will correspond the plane l<span class="su">1</span>p<span class="su">1</span>, that is, the plane S<span class="su">1</span>S<span class="su">2</span>P. These
+planes intersect in p<span class="su">2</span>, so that p<span class="su">2</span> is a line on the quadric cone generated
+by the axial pencils l<span class="su">1</span> and l<span class="su">2</span>. Hence:&mdash;</p>
+
+<p><i>All lines in one pencil which meet their corresponding lines in the
+other form a cone of the second order which has its centre at the centre
+of the first pencil, and passes through the centre of the second.</i></p>
+
+<p>From this follows that the points in which corresponding rays
+meet lie on two cones of the second order which have the ray joining
+their centres in common, and form therefore, together with the line
+S<span class="su">1</span>S<span class="su">2</span> or l<span class="su">1</span>, the intersection of these cones. Any plane cuts each of the
+cones in a conic. These two conics have necessarily that point in
+common in which it cuts the line l<span class="su">1</span>, and therefore besides either
+one or three other points. It follows that the curve is of the third
+order as a plane may cut it in three, but not in more than three,
+points. Hence:&mdash;</p>
+
+<p><i>The locus of points in which corresponding lines on two projective
+pencils meet is a curve of the third order or a &ldquo;twisted cubic&rdquo; k, which
+passes through the centres of the pencils, and which appears as the
+intersection of two cones of the second order, which have one line in
+common.</i></p>
+
+<p><i>A line belonging to the congruence determined by the pencils is a
+secant of the cubic; it has two, or one, or no points in common with
+this cubic, and is called accordingly a secant proper, a tangent, or a
+secant improper of the cubic.</i> A secant improper may be considered,
+to use the language of coordinate geometry, as a secant with
+imaginary points of intersection.</p>
+
+<p>§ 103. If a<span class="su">1</span> and a<span class="su">2</span> be any two corresponding lines in the two
+pencils, then corresponding planes in the axial pencils having a<span class="su">1</span> and
+a<span class="su">2</span> as axes generate a ruled quadric surface. If P be any point on
+the cubic k, and if p<span class="su">1</span>, p<span class="su">2</span> be the corresponding rays in S<span class="su">1</span> and S<span class="su">2</span> which
+meet at P, then to the plane a<span class="su">1</span>p<span class="su">1</span> in S<span class="su">1</span> corresponds a<span class="su">2</span>p<span class="su">2</span> in S<span class="su">2</span>. These
+therefore meet in a line through P.</p>
+
+<p>This may be stated thus:&mdash;</p>
+
+<p><i>Those secants of the cubic which cut a ray</i> a<span class="su">1</span>, <i>drawn through the
+centre</i> S<span class="su">1</span> <i>of one pencil, form a ruled quadric surface which passes through
+both centres, and which contains the twisted cubic</i> k. <i>Of such surfaces
+an infinite number exists. Every ray through</i> S<span class="su">1</span> <i>or</i> S<span class="su">2</span> <i>which is not a
+secant determines one of them.</i></p>
+
+<p>If, however, the rays a<span class="su">1</span> and a<span class="su">2</span> are secants meeting at A, then the
+ruled quadric surface becomes a cone of the second order, having
+A as centre. Or <i>all lines of the congruence which pass through a point
+on the twisted cubic k form a cone of the second order</i>. In other words,
+the projection of a twisted cubic from any point in the curve on to
+any plane is a conic.</p>
+
+<p>If a<span class="su">1</span> is not a secant, but made to pass through any point Q in
+space, the ruled quadric surface determined by a<span class="su">1</span> will pass through
+Q. <i>There will therefore be one line of the congruence passing through</i>
+Q, <i>and only one.</i> For if two such lines pass through Q, then the lines
+S<span class="su">1</span>Q and S<span class="su">2</span>Q will be corresponding lines; hence Q will be a point on
+the cubic k, and an infinite number of secants will pass through it.
+Hence:&mdash;</p>
+
+<p><i>Through every point in space not on the twisted cubic one and only
+one secant to the cubic can be drawn.</i></p>
+
+<p>§ 104. The fact that all the secants through a point on the cubic
+form a quadric cone shows that the centres of the projective pencils
+generating the cubic are not distinguished from any other points on
+the cubic. If we take any two points S, S&prime; on the cubic, and draw
+the secants through each of them, we obtain two quadric cones,
+which have the line SS&prime; in common, and which intersect besides
+along the cubic. If we make these two pencils having S and S&prime; as
+centres projective by taking four rays on the one cone as corresponding
+to the four rays on the other which meet the first on the
+cubic, the correspondence is determined. These two pencils will
+generate a cubic, and the two cones of secants having S and S&prime; as
+centres will be identical with the above cones, for each has five
+rays in common with one of the first, viz. the line SS&prime; and the four
+lines determined for the correspondence; therefore these two cones
+intersect in the original cubic. This gives the theorem:&mdash;</p>
+
+<p><i>On a twisted cubic any two points may be taken as centres of projective
+pencils which generate the cubic, corresponding planes being
+those which meet on the same secant.</i></p>
+
+<p>Of the two projective pencils at S and S&prime; we may keep the first
+fixed, and move the centre of the other along the curve. The pencils
+will hereby remain projective, and a plane &alpha; in S will be cut by its
+corresponding plane &alpha;&prime; always in the same secant a. Whilst S&prime;
+moves along the curve the plane &alpha;&prime; will turn about a, describing an
+axial pencil.</p>
+
+<p><span class="pagenum"><a name="page707" id="page707"></a>707</span></p>
+
+<p><span class="sc">Authorities.</span>&mdash;In this article we have given a purely geometrical
+theory of conics, cones of the second order, quadric surfaces, &amp;c. In
+doing so we have followed, to a great extent, Reye&rsquo;s <i>Geometrie der
+Lage</i>, and to this excellent work those readers are referred who wish
+for a more exhaustive treatment of the subject. Other works
+especially valuable as showing the development of the subject are:
+Monge, <i>Géométrie descriptive</i>: Carnot, <i>Géométrie de position</i>
+(1803), containing a theory of transversals; Poncelet&rsquo;s great work
+<i>Traité des propriétés projectives des figures</i> (1822); Möbins, <i>Barycentrischer
+Calcul</i> (1826); Steiner, <i>Abhängigkeit geometrischer
+Gestalten</i> (1832), containing the first full discussion of the projective
+relations between rows, pencils, &amp;c.; Von Staudt, <i>Geometrie der
+Lage</i> (1847) and <i>Beiträge zur Geometrie der Lage</i> (1856-1860), in
+which a system of geometry is built up from the beginning without
+any reference to number, so that ultimately a number itself gets
+a geometrical definition, and in which imaginary elements are
+systematically introduced into pure geometry; Chasles, <i>Aperēu
+historique</i> (1837), in which the author gives a brilliant account of
+the progress of modern geometrical methods, pointing out the
+advantages of the different purely geometrical methods as compared
+with the analytical ones, but without taking as much account of
+the German as of the French authors; Id., <i>Rapport sur les progrčs
+de la géométrie</i> (1870), a continuation of the <i>Aperēu</i>; Id., <i>Traité de
+géométrie supérieure</i> (1852); Cremona, <i>Introduzione ad una teoria
+geometrica delle curve piane</i> (1862) and its continuation <i>Preliminari
+di una teoria geometrica delle superficie</i> (German translations by
+Curtze). As more elementary books, we mention: Cremona,
+<i>Elements of Projective Geometry</i>, translated from the Italian by
+C. Leudesdorf (2nd ed., 1894); J.W. Russell, <i>Pure Geometry</i> (2nd ed.,
+1905).</p>
+</div>
+<div class="author">(O. H.)</div>
+
+<p class="pt2 center sc">III. Descriptive Geometry</p>
+
+<p>This branch of geometry is concerned with the methods for
+representing solids and other figures in three dimensions by
+drawings in one plane. The most important method is that
+which was invented by Monge towards the end of the 18th
+century. It is based on parallel projections to a plane by rays
+perpendicular to the plane. Such a projection is called orthographic
+(see <span class="sc"><a href="#artlinks">Projection</a></span>, § 18). If the plane is horizontal the
+projection is called the plan of the figure, and if the plane is
+vertical the elevation. In Monge&rsquo;s method a figure is represented
+by its plan and elevation. It is therefore often called drawing
+in plan and elevation, and sometimes simply orthographic
+projection.</p>
+
+<div class="condensed">
+<p>§ 1. We suppose then that we have two planes, one horizontal,
+the other vertical, and these we call the planes of plan and of elevation
+respectively, or the horizontal and the vertical plane, and
+denote them by the letters &pi;<span class="su">1</span> and &pi;<span class="su">2</span>. Their line of intersection is
+called the axis, and will be denoted by xy.</p>
+
+<p>If the surface of the drawing paper is taken as the plane of the
+plan, then the vertical plane will be the plane perpendicular to it
+through the axis xy. To bring this also into the plane of the drawing
+paper we turn it about the axis till it coincides with the horizontal
+plane. This process of turning one plane down till it coincides with
+another is called <i>rabatting</i> one to the other. Of course there is no
+necessity to have one of the two planes horizontal, but even when
+this is not the case it is convenient to retain the above names.</p>
+
+<table class="nobctr" style="clear: both;" summary="Illustration">
+<tr><td class="figcenter" colspan="2"><img style="width:466px; height:205px" src="images/img707a.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 37.</span></td>
+<td class="caption"><span class="sc">Fig. 38.</span></td></tr></table>
+
+<p>The whole arrangement will be better understood by referring to
+fig. 37. A point A in space is there projected by the perpendicular
+AA<span class="su">1</span> and AA<span class="su">2</span> to the planes &pi;<span class="su">1</span> and &pi;<span class="su">2</span> so that A<span class="su">1</span> and A<span class="su">2</span> are the
+horizontal and vertical projections of A.</p>
+
+<p>If we remember that a line is perpendicular to a plane that is
+perpendicular to every line in the plane if only it is perpendicular
+to any two intersecting lines in the plane, we see that the axis which
+is perpendicular both to AA<span class="su">1</span> and to AA<span class="su">2</span> is also perpendicular to
+A<span class="su">1</span>A<span class="su">0</span> and to A<span class="su">2</span>A<span class="su">0</span> because these four lines are all in the same plane.
+Hence, if the plane &pi;<span class="su">2</span> be turned about the axis till it coincides with
+the plane &pi;<span class="su">1</span>, then A<span class="su">2</span>A<span class="su">0</span> will be the continuation of A<span class="su">1</span>A<span class="su">0</span>. This
+position of the planes is represented in fig. 38, in which the line A<span class="su">1</span>A<span class="su">2</span>
+is perpendicular to the axis x.</p>
+
+<p>Conversely any two points A<span class="su">1</span>, A<span class="su">2</span> in a line perpendicular to the
+axis will be the projections of some point in space when the plane
+&pi;<span class="su">2</span> is turned about the axis till it is perpendicular to the plane &pi;<span class="su">1</span>,
+because in this position the two perpendiculars to the planes &pi;<span class="su">1</span>
+and &pi;<span class="su">2</span> through the points A<span class="su">1</span> and A<span class="su">2</span> will be in a plane and therefore
+meet at some point A.</p>
+
+<p><i>Representation of Points.</i>&mdash;We have thus the following method
+of representing in a single plane the position of points in space:&mdash;<i>we
+take in the plane a line xy as the axis, and then any pair of points
+A<span class="su">1</span>, A<span class="su">2</span> in the plane on a line perpendicular to the axis represent a
+point A in space</i>. If the line A<span class="su">1</span>A<span class="su">2</span> cuts the axis at A<span class="su">0</span>, and if at A<span class="su">1</span>
+a perpendicular be erected to the plane, then the point A will be in
+it at a height A<span class="su">1</span>A = A<span class="su">0</span>A<span class="su">2</span> above the plane. This gives the position
+of the point A relative to the plane &pi;<span class="su">1</span>. In the same way, if in a
+perpendicular to &pi;<span class="su">2</span> through A<span class="su">2</span> a point A be taken such that A<span class="su">2</span>A =
+A<span class="su">0</span>A<span class="su">1</span>, then this will give the point A relative to the plane &pi;<span class="su">2</span>.</p>
+
+<table class="flt" style="float: right; width: 230px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:175px; height:182px" src="images/img707b.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 39.</span></td></tr></table>
+
+<p>§ 2. The two planes &pi;<span class="su">1</span>, &pi;<span class="su">2</span> in their original position divide space
+into four parts. These are called the four quadrants. We suppose
+that the plane &pi;<span class="su">2</span> is turned as indicated in
+fig. 37, so that the point P comes to Q and
+R to S, then the quadrant in which the
+point A lies is called the first, and we say
+that in the first quadrant a point lies above
+the horizontal and in front of the vertical
+plane. Now we go round the axis in the
+sense in which the plane &pi;<span class="su">2</span> is turned and
+come in succession to the second, third
+and fourth quadrant. In the second a
+point lies above the plane of the plan and
+behind the plane of elevation, and so on.
+In fig. 39, which represents a side view of
+the planes in fig. 37 the quadrants are
+marked, and in each a point with its projection
+is taken. Fig. 38 shows how these are represented when
+the plane &pi;<span class="su">2</span> is turned down. We see that</p>
+
+<p><i>A point lies in the first quadrant if the plan lies below, the elevation
+above the axis; in the second if plan and elevation both lie above; in
+the third if the plan lies above, the elevation below; in the fourth if plan
+and elevation both lie below the axis.</i></p>
+
+<p><i>If a point lies in the horizontal plane</i>, its elevation lies in the axis
+and the plan coincides with the point itself. <i>If a point lies in the
+vertical plane</i>, its plan lies in the axis and the elevation coincides
+with the point itself. <i>If a point lies in the axis</i>, both its plan and
+elevation lie in the axis and coincide with it.</p>
+
+<p>Of each of these propositions, which will easily be seen to be true,
+the converse holds also.</p>
+
+<p>§ 3. <i>Representation of a Plane.</i>&mdash;As we are thus enabled to represent
+points in a plane, we can represent any finite figure by representing
+its separate points. It is, however, not possible to represent a plane
+in this way, for the projections of its points completely cover the
+planes &pi;<span class="su">1</span> and &pi;<span class="su">2</span>, and no plane would appear different from any other.
+But any plane &alpha; cuts each of the planes &pi;<span class="su">1</span>, &pi;<span class="su">2</span> in a line. These are
+called the traces of the plane. They cut each other in the axis at the
+point where the latter cuts the plane &alpha;.</p>
+
+<p><i>A plane is determined by its two traces, which are two lines that meet
+on the axis</i>, and, conversely, <i>any two lines which meet on the axis
+determine a plane</i>.</p>
+
+<p><i>If the plane is parallel to the axis its traces are parallel to the axis.</i>
+Of these one may be at infinity; then the plane will cut one of the
+planes of projection at infinity and will be parallel to it. Thus a
+plane parallel to the horizontal plane of the plan has only one finite
+trace, viz. that with the plane of elevation.</p>
+
+<table class="flt" style="float: right; width: 300px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:246px; height:207px" src="images/img707c.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 40.</span></td></tr></table>
+
+<p><i>If the plane passes through the axis both its traces coincide with the
+axis.</i> This is the only case in which the representation of the plane
+by its two traces fails. A third plane of projection is therefore
+introduced, which is best taken perpendicular to the other two.
+We call it simply the third plane and denote it by &pi;<span class="su">3</span>. As it is
+perpendicular to &pi;<span class="su">1</span>, it may be
+taken as the plane of elevation,
+its line of intersection &gamma; with &pi;<span class="su">1</span>
+being the axis, and be turned
+down to coincide with &pi;<span class="su">1</span>. This
+is represented in fig. 40. OC is
+the axis xy whilst OA and OB
+are the traces of the third plane.
+They lie in one line &gamma;. The plane
+is rabatted about &gamma; to the horizontal
+plane. A plane &alpha; through
+the axis xy will then show in it
+a trace &alpha;<span class="su">3</span>. In fig. 40 the lines OC
+and OP will thus be the traces
+of a plane through the axis xy,
+which makes an angle POQ with
+the horizontal plane.</p>
+
+<p>We can also find the trace
+which any other plane makes
+with &pi;<span class="su">3</span>. In rabatting the plane
+&pi;<span class="su">3</span> its trace OB with the plane &pi;<span class="su">2</span> will come to the position OD.
+Hence a plane &beta; having the traces CA and CB will have with the
+third plane the trace &beta;<span class="su">3</span>, or AD if OD = OB.</p>
+
+<p><span class="pagenum"><a name="page708" id="page708"></a>708</span></p>
+
+<p>It also follows immediately that&mdash;</p>
+
+<p><i>If a plane &alpha; is perpendicular to the horizontal plane, then every point
+in it has its horizontal projection in the horizontal trace of the plane</i>,
+as all the rays projecting these points lie in the plane itself.</p>
+
+<p><i>Any plane which is perpendicular to the horizontal plane has its
+vertical trace perpendicular to the axis.</i></p>
+
+<p><i>Any plane which is perpendicular to the vertical plane has its horizontal
+trace perpendicular to the axis and the vertical projections of all
+points in the plane lie in this trace.</i></p>
+
+<p>§ 4. <i>Representation of a Line.</i>&mdash;A line is determined either by two
+points in it or by two planes through it. We get accordingly two
+representations of it either by projections or by traces.</p>
+
+<p>First.&mdash;<i>A line a is represented by its projections</i> a<span class="su">1</span> <i>and</i> a<span class="su">2</span> <i>on the
+two planes</i> &pi;<span class="su">1</span> <i>and</i> &pi;<span class="su">2</span>. These may be any two lines, for, bringing
+the planes &pi;<span class="su">1</span>, &pi;<span class="su">2</span> into their original position, the planes through these
+lines perpendicular to &pi;<span class="su">1</span> and &pi;<span class="su">2</span> respectively will intersect in some line
+a which has a<span class="su">1</span>, a<span class="su">2</span> as its projections.</p>
+
+<p>Secondly.&mdash;<i>A line a is represented by its traces&mdash;that is, by the points
+in which it cuts the two planes</i> &pi;<span class="su">1</span>, &pi;<span class="su">2</span>. Any two points may be taken
+as the traces of a line in space, for it is determined when the planes
+are in their original position as the line joining the two traces. This
+representation becomes undetermined if the two traces coincide in
+the axis. In this case we again use a third plane, or else the projections
+of the line.</p>
+
+<p>The fact that there are different methods of representing points
+and planes, and hence two methods of representing lines, suggests
+the principle of duality (section ii., <i>Projective Geometry</i>, § 41). It
+is worth while to keep this in mind. It is also worth remembering
+that traces of planes or lines always lie in the planes or lines which
+they represent. Projections do not as a rule do this excepting when
+the point or line projected lies in one of the planes of projection.</p>
+
+<p>Having now shown how to represent points, planes and lines,
+we have to state the conditions which must hold in order that these
+elements may lie one in the other, or else that the figure formed by
+them may possess certain metrical properties. It will be found that
+the former are very much simpler than the latter.</p>
+
+<p>Before we do this, however, we shall explain the notation used;
+for it is of great importance to have a systematic notation. We
+shall denote points in space by capitals A, B, C; planes in space
+by Greek letters &alpha;, &beta;, &gamma;; lines in space by small letters a, b, c;
+horizontal projections by suffixes 1, like A<span class="su">1</span>, a<span class="su">1</span>; vertical projections
+by suffixes 2, like A<span class="su">2</span>, a<span class="su">2</span>; traces by single and double dashes &alpha;&prime; &alpha;&Prime;,
+a&prime;, a&Prime;. Hence P<span class="su">1</span> will be the horizontal projection of a point P in
+space; a line a will have the projections a<span class="su">1</span>, a<span class="su">2</span> and the traces a&prime; and
+a&Prime;; a plane &alpha; has the traces &alpha;&prime; and &alpha;&Prime;.</p>
+
+<p>§ 5. <i>If a point lies in a line, the projections of the point lie in the
+projections of the line.</i></p>
+
+<p><i>If a line lies in a plane, the traces of the line lie in the traces of the
+plane.</i></p>
+
+<p>These propositions follow at once from the definitions of the
+projections and of the traces.</p>
+
+<p>If a point lies in two lines its projections must lie in the projections
+of both. Hence</p>
+
+<p><i>If two lines, given by their projections, intersect, the intersection of
+their <span class="correction" title="amended from plans">planes</span> and the intersection of their elevations must lie in a line
+perpendicular to the axis</i>, because they must be the projections of
+the point common to the two lines.</p>
+
+<p>Similarly&mdash;<i>If two lines given by their traces lie in the same plane
+or intersect, then the lines joining their horizontal and vertical traces
+respectively must meet on the axis</i>, because they must be the traces
+of the plane through them.</p>
+
+<p>§ 6. <i>To find the projections of a line which joins two points A, B
+given by their projections</i> A<span class="su">1</span>, A<span class="su">2</span> <i>and</i> B<span class="su">1</span>, B<span class="su">2</span>, we join A<span class="su">1</span>, B<span class="su">1</span> and A<span class="su">2</span>,
+B<span class="su">2</span>; these will be the projections required. For example, the
+traces of a line are two points in the line whose projections are
+known or at all events easily found. They are the traces themselves
+and the feet of the perpendiculars from them to the axis.</p>
+
+<table class="flt" style="float: right; width: 300px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:246px; height:207px" src="images/img708a.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 41.</span></td></tr></table>
+
+<p>Hence <i>if</i> a&prime; a&Prime; (fig. 41) <i>are the traces of a line a, and if the perpendiculars
+from them cut the axis in</i> P <i>and</i> Q <i>respectively, then the
+line</i> a&prime;Q <i>will be the horizontal and</i>
+a&Prime;P <i>the vertical projection of the
+line</i>.</p>
+
+<p>Conversely, if the projections
+a<span class="su">1</span>, a<span class="su">2</span> of a line are given, and if
+these cut the axis in Q and P
+respectively, then <i>the perpendiculars</i>
+Pa&prime; <i>and</i> Qa&Prime; <i>to the axis
+drawn through these points cut the
+projections</i> a<span class="su">1</span> <i>and</i> a<span class="su">2</span> <i>in the traces</i>
+a&prime; <i>and</i> a&Prime;.</p>
+
+<p><i>To find the line of intersection of
+two planes</i>, we observe that this
+line lies in both planes; its traces
+must therefore lie in the traces
+of both. Hence the points where the horizontal traces of the given
+planes meet will be the horizontal, and the point where the vertical
+traces meet the vertical trace of the line required.</p>
+
+<p>§ 7. <i>To decide whether a point</i> A, <i>given by its projections, lies in
+a plane &alpha;, given by its traces</i>, we draw a line p by joining A to some
+point in the plane &alpha; and determine its traces. If these lie in the
+traces of the plane, then the line, and therefore the point A, lies
+in the plane; otherwise not. This is conveniently done by joining
+A<span class="su">1</span> to some point p&prime; in the trace &alpha;&prime;; this gives p<span class="su">1</span>; and the point
+where the perpendicular from p&prime; to the axis cuts the latter we join
+to A<span class="su">2</span>; this gives p<span class="su">2</span>. If the vertical trace of this line lies in the
+vertical trace of the plane, then, and then only, does the line p, and
+with it the point A, lie in the plane &alpha;.</p>
+
+<p>§ 8. <i>Parallel planes have parallel traces</i>, because parallel planes are
+cut by any plane, hence also by &pi;<span class="su">1</span> and by &pi;<span class="su">2</span>, in parallel lines.</p>
+
+<p><i>Parallel lines have parallel projections</i>, because points at infinity
+are projected to infinity.</p>
+
+<p><i>If a line is parallel to a plane, then lines through the traces of the
+line and parallel to the traces of the plane must meet on the axis</i>, because
+these lines are the traces of a plane parallel to the given plane.</p>
+
+<p>§ 9. <i>To draw a plane through two intersecting lines or through two
+parallel lines</i>, we determine the traces of the lines; the lines joining
+their horizontal and vertical traces respectively will be the horizontal
+and vertical traces of the plane. They will meet, at a finite point
+or at infinity, on the axis if the lines do intersect.</p>
+
+<p><i>To draw a plane through a line and a point without the line</i>, we
+join the given point to any point in the line and determine the plane
+through this and the given line.</p>
+
+<p><i>To draw a plane through three points which are not in a line</i>, we
+draw two of the lines which each join two of the given points and
+draw the plane through them. If the traces of all three lines AB,
+BC, CA be found, these must lie in two lines which meet on the
+axis.</p>
+
+<p>§ 10. We have in the last example got more points, or can easily
+get more points, than are necessary for the determination of the
+figure required&mdash;in this case the traces of the plane. This will
+happen in a great many constructions and is of considerable importance.
+It may happen that some of the points or lines obtained
+are not convenient in the actual construction. The horizontal
+traces of the lines AB and AC may, for instance, fall very near
+together, in which case the line joining them is not well defined.
+Or, one or both of them may fall beyond the drawing paper, so that
+they are practically non-existent for the construction. In this case
+the traces of the line BC may be used. Or, if the vertical traces of
+AB and AC are both in convenient position, so that the vertical
+trace of the required plane is found and one of the horizontal traces
+is got, then we may join the latter to the point where the vertical
+trace cuts the axis.</p>
+
+<p>The draughtsman must remember that the lines which he draws
+are not mathematical lines without thickness, and therefore every
+drawing is affected by some errors. It is therefore very desirable
+to be able constantly to check the latter. Such checks always
+present themselves when the same result can be obtained by different
+constructions, or when, as in the above case, some lines must meet
+on the axis, or if three points must lie in a line. A careful draughtsman
+will always avail himself of these checks.</p>
+
+<p>§ 11. <i>To draw a plane through a given point parallel to a given
+plane &alpha;</i>, we draw through the point two lines which are parallel to
+the plane &alpha;, and determine the plane through them; or, as we
+know that the traces of the required plane are parallel to those of
+the given one (§ 8), we need only draw one line l through the point
+parallel to the plane and find one of its traces, say the vertical trace
+l&Prime;; a line through this parallel to the vertical trace of &alpha; will be the
+vertical trace &beta;&Prime; of the required plane &beta;, and a line parallel to the
+horizontal trace of &alpha; meeting &beta;&Prime; on the axis will be the horizontal
+trace &beta;&prime;.</p>
+
+<table class="flt" style="float: right; width: 340px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:286px; height:183px" src="images/img708b.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 42.</span></td></tr></table>
+
+<p>Let A<span class="su">1</span> A<span class="su">2</span> (fig. 42) be the given point, &alpha;&prime; &alpha;&Prime; the given plane, a
+line l<span class="su">1</span> through A<span class="su">1</span>, parallel to &alpha;&prime; and a horizontal line l<span class="su">2</span> through
+A<span class="su">2</span> will be the projections of
+a line l through A parallel
+to the plane, because the
+horizontal plane through
+this line will cut the plane
+&alpha; in a line c which has its
+horizontal projection c<span class="su">1</span>
+parallel to &alpha;&prime;.</p>
+
+<p>§ 12. We now come to
+the metrical properties of
+figures.</p>
+
+<p><i>A line is perpendicular
+to a plane if the projections
+of the line are perpendicular
+to the traces of the plane.</i> We prove it for the horizontal
+projection. If a line p is perpendicular to a plane &alpha;, every plane
+through p is perpendicular to &alpha;; hence also the vertical plane which
+projects the line p to p<span class="su">1</span>. As this plane is perpendicular both to the
+horizontal plane and to the plane &alpha;, it is also perpendicular to their
+intersection&mdash;that is, to the horizontal trace of &alpha;. It follows that
+every line in this projecting plane, therefore also p<span class="su">1</span>, the plan of p, is
+perpendicular to the horizontal trace of &alpha;.</p>
+
+<p><i>To draw a plane through a given point A perpendicular to a given
+line p</i>, we first draw through some point O in the axis lines &gamma;&prime;, &gamma;&Prime;
+perpendicular respectively to the projections p<span class="su">1</span> and p<span class="su">2</span> of the given
+line. These will be the traces of a plane &gamma; which is perpendicular
+to the given line. We next draw through the given point A a plane
+parallel to the plane &gamma;; this will be the plane required.</p>
+
+<p><span class="pagenum"><a name="page709" id="page709"></a>709</span></p>
+
+<p>Other metrical properties depend on the determination of the real
+size or shape of a figure.</p>
+
+<p>In general the projection of a figure differs both in size and shape
+from the figure itself. But figures in a plane parallel to a plane
+of projection will be identical with their projections, and will thus
+be given in their true dimensions. In other cases there is the
+problem, constantly recurring, either to find the true shape and
+size of a plane figure when plan and elevation are given, or, conversely,
+to find the latter from the known true shape of the figure
+itself. To do this, the plane is turned about one of its traces till it
+is laid down into that plane of projection to which the trace belongs.
+This is technically called rabatting the plane respectively into the
+plane of the plan or the elevation. As there is no difference in the
+treatment of the two cases, we shall consider only the case of rabatting
+a plane &alpha; into the plane of the plan. The plan of the figure is
+a parallel (orthographic) projection of the figure itself. The results
+of parallel projection (see <span class="sc"><a href="#artlinks">Projection</a></span>, §§ 17 and 18) may therefore
+now be used. The trace &alpha;&prime; will hereby take the place of what
+formerly was called the axis of projection. Hence we see that corresponding
+points in the plan and in the rabatted plane are joined by
+lines which are perpendicular to the trace &alpha;&prime; and that corresponding
+lines meet on this trace. We also see that the correspondence is
+completely determined if we know for one point or one line in the
+plan the corresponding point or line in the rabatted plane.</p>
+
+<p>Before, however, we treat of this we consider some special cases.</p>
+
+<p>§ 13. <i>To determine the distance between two points A, B given by their
+projections</i> A<span class="su">1</span>, B<span class="su">1</span> <i>and</i> A<span class="su">2</span>, B<span class="su">2</span>, <i>or, in other words, to determine the true
+length of a line the plan and elevation of which are given.</i></p>
+
+<table class="flt" style="float: right; width: 310px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:242px; height:225px" src="images/img709a.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 43.</span></td></tr>
+<tr><td class="figright1"><img style="width:257px; height:216px" src="images/img709b.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 44.</span></td></tr></table>
+
+<p><i>Solution.</i>&mdash;The two points A, B in space lie vertically above their
+plans A<span class="su">1</span>, B<span class="su">1</span> (fig. 43) and A<span class="su">1</span>A = A<span class="su">0</span>A<span class="su">2</span>, B<span class="su">1</span>B =
+B<span class="su">0</span>B<span class="su">2</span>. The four points
+A, B, A<span class="su">1</span>, B<span class="su">1</span> therefore form a plane
+quadrilateral on the base A<span class="su">1</span>B<span class="su">1</span> and
+having right angles at the base.
+This plane we rabatt about A<span class="su">1</span>B<span class="su">1</span>
+by drawing A<span class="su">1</span>A and B<span class="su">1</span>B perpendicular
+to A<span class="su">1</span>B<span class="su">1</span> and making
+A<span class="su">1</span>A = A<span class="su">0</span>A<span class="su">2</span>, B<span class="su">1</span>B = B<span class="su">0</span>B<span class="su">2</span>. Then
+AB will give the length required.</p>
+
+<p>The construction might have
+been performed in the elevation
+by making A<span class="su">2</span>A = A<span class="su">0</span>A<span class="su">1</span> and
+B<span class="su">2</span>B = B<span class="su">0</span>B<span class="su">1</span> on lines perpendicular
+to A<span class="su">2</span>B<span class="su">2</span>. Of course AB must have
+the same length in both cases.</p>
+
+<p>This figure may be turned into
+a model. Cut the paper along
+A<span class="su">1</span>A, AB and BB<span class="su">1</span>, and fold the
+piece A<span class="su">1</span>ABB<span class="su">1</span> over along A<span class="su">1</span>B<span class="su">1</span> till
+it stands upright at right angles to the horizontal plane. The points
+A, B will then be in their true position in space relative to
+&pi;<span class="su">1</span>. Similarly
+if B<span class="su">2</span>BAA<span class="su">2</span> be cut out and turned along A<span class="su">2</span>B<span class="su">2</span> through a right
+angle we shall get AB in its true position relative to the plane
+&pi;<span class="su">2</span>. Lastly we fold the whole plane of the paper along the axis x
+till the plane &pi;<span class="su">2</span> is at right angles to &pi;<span class="su">1</span>. In this position the two
+sets of points AB will coincide if the drawing has been accurate.</p>
+
+<p>Models of this kind can be made in many cases and their construction
+cannot be too highly recommended in order to realize
+orthographic projection.</p>
+
+<p>§ 14. <i>To find the angle between two given lines</i> a, b <i>of which the
+projections</i> a<span class="su">1</span>, b<span class="su">1</span> <i>and</i> a<span class="su">2</span>, b<span class="su">2</span> <i>are given.</i></p>
+
+<p><i>Solution.</i>&mdash;Let a<span class="su">1</span>, b<span class="su">1</span> (fig. 44) meet in P<span class="su">1</span>, a<span class="su">2</span>, b<span class="su">2</span> in T, then if the line
+P<span class="su">1</span>T is not perpendicular to the axis the two lines will not meet. In
+this case we draw a line parallel
+to b to meet the line a. This is
+easiest done by drawing first the
+line P<span class="su">1</span>P<span class="su">2</span> perpendicular to the
+axis to meet a<span class="su">2</span> in P<span class="su">2</span>, and then
+drawing through P<span class="su">2</span> a line c<span class="su">2</span>
+parallel to b<span class="su">2</span>; then b<span class="su">1</span>, c<span class="su">2</span> will be
+the projections of a line c which
+is parallel to b and meets a in P.
+The plane &alpha; which these two
+lines determine we rabatt to the
+plan. We determine the traces
+a&prime; and c&prime; of the lines a and c;
+then a&prime;c&prime; is the trace &alpha;&prime; of their
+plane. On rabatting the point
+P comes to a point S on the line
+P<span class="su">1</span>Q perpendicular to a&prime;c&prime;, so
+that QS = QP. But QP is the hypotenuse of a triangle PP<span class="su">1</span>Q with
+a right angle P<span class="su">1</span>. This we construct by making QR = P<span class="su">0</span>P<span class="su">2</span>; then
+P<span class="su">1</span>R = PQ. The lines a&prime;S and c&prime;S will therefore include angles equal
+to those made by the given lines. It is to be remembered that two
+lines include two angles which are supplementary. Which of these
+is to be taken in any special case depends upon the circumstances.</p>
+
+<p><i>To determine the angle between a line and a plane</i>, we draw through
+any point in the line a perpendicular to the plane (§ 12) and determine
+the angle between it and the given line. The complement of this
+angle is the required one.</p>
+
+<p><i>To determine the angle between two planes</i>, we draw through any
+point two lines perpendicular to the two planes and determine the
+angle between the latter as above.</p>
+
+<p>In special cases it is simpler to determine at once the angle between
+the two planes by taking a plane section perpendicular to the intersection
+of the two planes and rabatt this. This is especially the
+case if one of the planes is the horizontal or vertical plane of projection.</p>
+
+<p>Thus in fig. 45 the angle P<span class="su">1</span>QR is the angle which the plane &alpha;
+makes with the horizontal plane.</p>
+
+<p>§ 15. We return to the general case of rabatting a plane &alpha; of
+which the traces &alpha;&prime; &alpha;&Prime; are given.</p>
+
+<table class="flt" style="float: left; width: 350px;" summary="Illustration">
+<tr><td class="figleft1"><img style="width:300px; height:238px" src="images/img709c.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 45.</span></td></tr></table>
+
+<p>Here it will be convenient to determine first the position which
+the trace &alpha;&Prime;&mdash;which is a line in &alpha;&mdash;assumes when rabatted. Points
+in this line coincide with their elevations. Hence it is given in
+its true dimension, and we can measure off along it the true distance
+between two points in it. If therefore (fig. 45) P is any point in &alpha;&Prime;
+originally coincident with
+its elevation P<span class="su">2</span>, and if O
+is the point where &alpha;&Prime; cuts
+the axis xy, so that O is
+also in &alpha;&prime;, then the point P
+will after rabatting the
+plane assume such a position
+that OP = OP<span class="su">2</span>. At
+the same time the plan is
+an orthographic projection
+of the plane &alpha;. Hence the
+line joining P to the plan
+P<span class="su">1</span> will after rabatting be
+perpendicular to &alpha;&prime;. But
+P<span class="su">1</span> is known; it is the foot
+of the perpendicular from
+P<span class="su">2</span> to the axis xy. We
+draw therefore, to find P,
+from P<span class="su">1</span> a perpendicular P<span class="su">1</span>Q to &alpha;&prime; and find on it a point P such that
+OP = OP<span class="su">2</span>. Then the line OP will be the position of &alpha;&Prime; when
+rabatted. This line corresponds therefore to the plan of
+&alpha;&Prime;&mdash;that
+is, to the axis xy, corresponding points on these lines being those
+which lie on a perpendicular to &alpha;&prime;.</p>
+
+<p>We have thus one pair of corresponding lines and can now find
+for any point B<span class="su">1</span> in the plan the corresponding point B in the rabatted
+plane. We draw a line through B<span class="su">1</span>, say B<span class="su">1</span>P<span class="su">1</span>, cutting &alpha;&prime; in C. To it
+corresponds the line CP, and the point where this is cut by the projecting
+ray through B<span class="su">1</span>, perpendicular to &alpha;&prime;, is the required point B.</p>
+
+<p>Similarly any figure in the rabatted plane can be found when the
+plan is known; but this is usually found in a different manner
+without any reference to the general theory of parallel projection.
+As this method and the reasoning employed for it have their peculiar
+advantages, we give it also.</p>
+
+<p>Supposing the planes &pi;<span class="su">1</span> and &pi;<span class="su">2</span> to be in their positions in space
+perpendicular to each other, we take a section of the whole figure
+by a plane perpendicular to the trace &alpha;&prime; about which we are going
+to rabatt the plane &alpha;. Let this section pass through the point Q in
+&alpha;&prime;. Its traces will then be the lines QP<span class="su">1</span> and P<span class="su">1</span>P<span class="su">2</span> (fig. 9). These
+will be at right angles, and will therefore, together with the section
+QP<span class="su">2</span> of the plane &alpha;, form a right-angled triangle QP<span class="su">1</span>P<span class="su">2</span> with the
+right angle at P<span class="su">1</span>, and having the sides P<span class="su">1</span>Q and P<span class="su">1</span>P<span class="su">2</span> which both
+are given in their true lengths. This triangle we rabatt about its
+base P<span class="su">1</span>Q, making P<span class="su">1</span>R = P<span class="su">1</span>P<span class="su">2</span>. The line QR will then give the true
+length of the line QP in space. If now the plane &alpha; be turned about
+&alpha;&prime; the point P will describe a circle about Q as centre with radius
+QP = QR, in a plane perpendicular to the trace &alpha;&prime;. Hence when the
+plane &alpha; has been rabatted into the horizontal plane the point P will
+lie in the perpendicular P<span class="su">1</span>Q to &alpha;&prime;, so that QP = QR.</p>
+
+<p>If A<span class="su">1</span> is the plan of a point A in the plane &alpha;, and if A<span class="su">1</span> lies in QP<span class="su">1</span>,
+then the point A will lie vertically above A<span class="su">1</span> in the line QP. On
+turning down the triangle QP<span class="su">1</span>P<span class="su">2</span>, the point A will come to A<span class="su">0</span>, the
+line A<span class="su">1</span>A<span class="su">0</span> being perpendicular to QP<span class="su">1</span>. Hence A will be a point in
+QP such that QA = QA<span class="su">0</span>.</p>
+
+<p>If B<span class="su">1</span> is the plan of another point, but such that A<span class="su">1</span>B<span class="su">1</span> is parallel
+to &alpha;&prime;, then the corresponding line AB will also be parallel to &alpha;&prime;.
+Hence, if through A a line AB be drawn parallel to &alpha;&prime;, and B<span class="su">1</span>B
+perpendicular to &alpha;&prime;, then their intersection gives the point B. Thus
+of any point given in plan the real position in the plane &alpha;, when
+rabatted, can be found by this second method. This is the one
+most generally given in books on geometrical drawing. The first
+method explained is, however, in most cases preferable as it gives
+the draughtsman a greater variety of constructions. It requires a
+somewhat greater amount of theoretical knowledge.</p>
+
+<p>If instead of our knowing the plan of a figure the latter is itself
+given, then the process of finding the plan is the reverse of the
+above and needs little explanation. We give an example.</p>
+
+<p>§ 16. <i>It is required to draw the plan and elevation of a polygon of
+which the real shape and position in a given plane &alpha; are known.</i></p>
+
+<table class="flt" style="float: right; width: 410px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:358px; height:511px" src="images/img710a.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 46.</span></td></tr></table>
+
+<p>We first rabatt the plane &alpha; (fig. 46) as before so that P<span class="su">1</span> comes to
+P, hence OP<span class="su">1</span> to OP. Let the given polygon in &alpha; be the figure
+ABCDE. We project, not the vertices, but the sides. To project
+the line AB, we produce it to cut &alpha;&prime; in F and OP in G, and draw GG<span class="su">1</span>
+perpendicular to &alpha;&prime;; then G<span class="su">1</span> corresponds to G, therefore FG<span class="su">1</span> to FG.
+In the same manner we might project all the other sides, at least
+<span class="pagenum"><a name="page710" id="page710"></a>710</span>
+those which cut OF and OP in convenient points. It will be best,
+however, first to produce all the sides to cut OP and &alpha;&prime; and then to
+draw all the projecting rays through A, B, C ... perpendicular to
+&alpha;&prime;, and in the same
+direction the lines
+G, G<span class="su">1</span>, &amp;c. By
+drawing FG we
+get the points A<span class="su">1</span>,
+B<span class="su">1</span> on the projecting
+ray through A
+and B. We then
+join B to the point
+M where BC produced
+meets the
+trace &alpha;&prime;. This
+gives C<span class="su">1</span>. So we
+go on till we have
+found E<span class="su">1</span>. The
+line A<span class="su">1</span> E<span class="su">1</span> must
+then meet AE in
+&alpha;&prime;, and this gives
+a check. If one
+of the sides cuts
+&alpha;&prime; or OP beyond
+the drawing paper
+this method fails,
+but then we may
+easily find the projection
+of some
+other line, say of
+a diagonal, or
+directly the projection
+of a point,
+by the former
+methods. The
+diagonals may
+also serve to check
+the drawing, for two corresponding diagonals must meet in the
+trace &alpha;&prime;.</p>
+
+<p>Having got the plan we easily find the elevation. The elevation
+of G is above G<span class="su">1</span> in &alpha;&Prime;, and that of F is at F<span class="su">2</span> in the axis. This
+gives the elevation F<span class="su">2</span>G<span class="su">2</span> of FG and in it we get A<span class="su">2</span>B<span class="su">2</span> in the verticals
+through A<span class="su">1</span> and B<span class="su">1</span>. As a check we have OG = OG<span class="su">2</span>. Similarly the
+elevation of the other sides and vertices are found.</p>
+
+<p>§ 17. We proceed to give some applications of the above principles
+to the representation of solids and of the solution of problems
+connected with them.</p>
+
+<p><i>Of a pyramid are given its base, the length of the perpendicular from
+the vertex to the base, and the point where this perpendicular cuts the
+base; it is required first to develop the whole surface of the pyramid
+into one plane, and second to determine its section by a plane which
+cuts the plane of the base in a given line and makes a given angle
+with it.</i></p>
+
+<p>1. As the planes of projection are not given we can take them as we
+like, and we select them in such a manner that the solution becomes
+as simple as possible. We take the plane of the base as the horizontal
+plane and the vertical plane perpendicular to the plane of the section.
+Let then (fig. 47) ABCD be the base of the pyramid, V<span class="su">1</span> the plan of
+the vertex, then the elevations of A, B, C, D will be in the axis at
+A<span class="su">2</span>, B<span class="su">2</span>, C<span class="su">2</span>, D<span class="su">2</span>, and the vertex at some point V<span class="su">2</span> above V<span class="su">1</span> at a known
+distance from the axis. The lines V<span class="su">1</span>A, V<span class="su">1</span>B, &amp;c., will be the plans
+and the lines V<span class="su">2</span>A<span class="su">2</span>, V<span class="su">2</span>B<span class="su">2</span>, &amp;c., the elevations of the edges of the
+pyramid, of which thus plan and elevation are known.</p>
+
+<p>We develop the surface into the plane of the base by turning
+each lateral face about its lower edge into the horizontal plane by
+the method used in § 14. If one face has been turned down, say
+ABV to ABP, then the point Q to which the vertex of the next
+face BCV comes can be got more simply by finding on the line
+V<span class="su">1</span>Q perpendicular to BC the point Q such that BQ = BP, for these
+lines represent the same edge BV of the pyramid. Next R is
+found by making CR = CQ, and so on till we have got the last vertex&mdash;in
+this case S. The fact that AS must equal AP gives a convenient
+check.</p>
+
+<p>2. The plane &alpha; whose section we have to determine has its horizontal
+trace given perpendicular to the axis, and its vertical trace
+makes the given angle with the axis. This determines it. To find
+the section of the pyramid by this plane there are two methods
+applicable: we find the sections of the plane either with the faces
+or with the edges of the pyramid. We use the latter.</p>
+
+<p>As the plane &alpha; is perpendicular to the vertical plane, the trace
+&alpha;&Prime; contains the projection of every figure in it; the points E<span class="su">2</span>, F<span class="su">2</span>,
+G<span class="su">2</span>, H<span class="su">2</span> where this trace cuts the elevations of the edges will therefore
+be the elevations of the points where the edges cut &alpha;. From these
+we find the plans E<span class="su">1</span>, F<span class="su">1</span>, G<span class="su">1</span>, H<span class="su">1</span>, and by joining them the plan
+of the section. If from E<span class="su">1</span>, F<span class="su">1</span> lines be drawn perpendicular to AB,
+these will determine the points E, F on the developed face in which
+the plane &alpha; cuts it; hence also the line EF. Similarly on the other
+faces. Of course BF must be the same length on BP and on BQ.
+If the plane &alpha; be rabatted to the plan, we get the real shape of the
+section as shown in the figure in EFGH. This is done easily by
+making F<span class="su">0</span>F = OF<span class="su">2</span>, &amp;c. If the figure representing the development
+of the pyramid, or better a copy of it, is cut out, and if the lateral
+faces be bent along the lines AB, BC, &amp;c., we get a model of the pyramid
+with the section marked on its faces. This may be placed on
+its plan ABCD and the plane of elevation bent about the axis x.
+The pyramid stands then in front of its elevations. If next the plane
+&alpha; with a hole cut out representing the true section be bent along the
+trace &alpha;&prime; till its edge coincides with &alpha;&Prime;, the edges of the hole ought to
+coincide with the lines EF, FG, &amp;c., on the faces.</p>
+
+<p>§ 18. Polyhedra like the pyramid in § 17 are represented by the
+projections of their edges and vertices. But solids bounded by
+curved surfaces, or surfaces themselves, cannot be thus represented.</p>
+
+<p>For a surface we may use, as in case of the plane, its traces&mdash;that
+is, the curves in which it cuts the planes of projection. We may
+also project points and curves on the surface. A ray cuts the
+surface generally in more than one point; hence it will happen
+that some of the rays touch the surface, if two of these points coincide.
+The points of contact of these rays will form some curve on the surface,
+and this will appear from the centre of projection as the boundary
+of the surface or of part of the surface. The outlines of all surfaces
+of solids which we see about us are formed by the points at which
+rays through our eye touch the surface. The projections of these
+contours are therefore best adapted to give an idea of the shape of a
+surface.</p>
+
+<table class="nobctr" style="clear: both;" summary="Illustration">
+<tr><td class="figcenter"><img style="width:481px; height:566px" src="images/img710b.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 47.</span></td></tr></table>
+
+<p>Thus the tangents drawn from any finite centre to a sphere form
+a right circular cone, and this will be cut by any plane in a conic.
+It is often called the projection of a sphere, but it is better called
+the contour-line of the sphere, as it is the boundary of the projections
+of all points on the sphere.</p>
+
+<p>If the centre is at infinity the tangent cone becomes a right
+circular cylinder touching the sphere along a great circle, and if
+the projection is, as in our case, orthographic, then the section of
+this cone by a plane of projection will be a circle equal to the great
+circle of the sphere. We get such a circle in the plan and another in
+the elevation, their centres being plan and elevation of the centre of
+the sphere.</p>
+
+<p>Similarly the rays touching a cone of the second order will lie
+in two planes which pass through the vertex of the cone, the contour-line
+of the projection of the cone consists therefore of two lines
+meeting in the projection of the vertex. These may, however,
+be invisible if no real tangent rays can be drawn from the centre of
+projection; and this happens when the ray projecting the centre
+of the vertex lies within the cone. In this case the traces of the
+cone are of importance. Thus in representing a cone of revolution
+with a vertical axis we get in the plan a circular trace of the surface
+whose centre is the plan of the vertex of the cone, and in the elevation
+the contour, consisting of a pair of lines intersecting in the elevation
+of the vertex of the cone. The circle in the plan and the pair of lines
+in the elevation do not determine the surface, for an infinite number
+of surfaces might be conceived which pass through the circular trace
+and touch two planes through the contour lines in the vertical plane.
+The surface becomes only completely defined if we write down to
+the figure that it shall represent a cone. The same holds for all
+<span class="pagenum"><a name="page711" id="page711"></a>711</span>
+surfaces. Even a plane is fully represented by its traces only under
+the silent understanding that the traces are those of a plane.</p>
+
+<p>§ 19. Some of the simpler problems connected with the representation
+of surfaces are the determination of plane sections and of
+the curves of intersection of two such surfaces. The former is
+constantly used in nearly all problems concerning surfaces. Its
+solution depends of course on the nature of the surface.</p>
+
+<p>To determine the curve of intersection of two surfaces, we take a
+plane and determine its section with each of the two surfaces,
+rabatting this plane if necessary. This gives two curves which lie
+in the same plane and whose intersections will give us points on
+both surfaces. It must here be remembered that two curves in
+space do not necessarily intersect, hence that the points in which
+their projections intersect are not necessarily the projections of
+points common to the two curves. This will, however, be the case
+if the two curves lie in a common plane. By taking then a number
+of plane sections of the surfaces we can get as many points on their
+curve of intersection as we like. These planes have, of course, to
+be selected in such a way that the sections are curves as simple as
+the case permits of, and such that they can be easily and accurately
+drawn. Thus when possible the sections should be straight lines
+or circles. This not only saves time in drawing but determines all
+points on the sections, and therefore also the points where the two
+curves meet, with equal accuracy.</p>
+
+<p>§ 20. We give a few examples how these sections have to be
+selected. A cone is cut by every plane through the vertex in lines,
+and if it is a cone of revolution by planes perpendicular to the
+axis in circles.</p>
+
+<p>A cylinder is cut by every plane parallel to the axis in lines, and
+if it is a cylinder of revolution by planes perpendicular to the axis
+in circles.</p>
+
+<p>A sphere is cut by every plane in a circle.</p>
+
+<p>Hence in case of two cones situated anywhere in space we take
+sections through both vertices. These will cut both cones in lines.
+Similarly in case of two cylinders we may take sections parallel to
+the axis of both. In case of a sphere and a cone of revolution with
+vertical axis, horizontal sections will cut both surfaces in circles
+whose plans are circles and whose elevations are lines, whilst vertical
+sections through the vertex of the cone cut the latter in lines and
+the sphere in circles. To avoid drawing the projections of these
+circles, which would in general be ellipses, we rabatt the plane and
+then draw the circles in their real shape. And so on in other cases.</p>
+
+<p>Special attention should in all cases be paid to those points in
+which the tangents to the projection of the curve of intersection are
+parallel or perpendicular to the axis x, or where these projections
+touch the contour of one of the surfaces.</p>
+</div>
+<div class="author">(O. H.)</div>
+
+<p class="pt2 center sc">IV. Analytical Geometry</p>
+
+<p>1. In the name <i>geometry</i> there is a lasting record that the
+science had its origin in the knowledge that two distances may
+be compared by measurement, and in the idea that measurement
+must be effectual in the dissociation of different directions as well
+as in the comparison of distances in the same direction. The
+distance from an observer&rsquo;s eye of an object seen would be
+specified as soon as it was ascertained that a rod, straight to the
+eye and of length taken as known, could be given the direction
+of the line of vision, and had to be moved along it a certain
+number of times through lengths equal to its own in order to
+reach the object from the eye. Moreover, if a field had for two
+of its boundaries lines straight to the eye, one running from south
+to north and the other from west to east, the position of a point
+in the field would be specified if the rod, when directed west,
+had to be shifted from the point one observed number of times
+westward to meet the former boundary, and also, when directed
+south, had to be shifted another observed number of times
+southward to meet the latter. Comparison by measurement,
+the beginning of geometry, involved counting, the basis of arithmetic;
+and the science of number was marked out from the
+first as of geometrical importance.</p>
+
+<p>But the arithmetic of the ancients was inadequate as a science
+of number. Though a length might be recognized as known
+when measurement certified that it was so many times a standard
+length, it was not every length which could be thus specified
+in terms of the same standard length, even by an arithmetic
+enriched with the notion of fractional number. The idea of
+possible incommensurability of lengths was introduced into
+Europe by Pythagoras; and the corresponding idea of irrationality
+of number was absent from a crude arithmetic, while there
+were great practical difficulties in the way of its introduction.
+Hence perhaps it arose that, till comparatively modern times,
+appeal to arithmetical aid in geometrical reasoning was in all
+possible ways restrained. Geometry figured rather as the helper
+of the more difficult science of arithmetic.</p>
+
+<p>2. It was reserved for algebra to remove the disabilities of
+arithmetic, and to restore the earliest ideas of the land-measurer
+to the position of controlling ideas in geometrical investigation.
+This unified science of pure number made comparatively little
+headway in the hands of the ancients, but began to receive
+due attention shortly after the revival of learning. It expresses
+whole classes of arithmetical facts in single statements, gives
+to arithmetical laws the form of equations involving symbols
+which may mean any known or sought numbers, and provides
+processes which enable us to analyse the information given by an
+equation and derive from that equation other equations, which
+express laws that are in effect consequences or causes of a law
+started from, but differ greatly from it in form. Above all, for
+present purposes, it deals not only with integral and fractional
+number, but with number regarded as capable of continuous
+growth, just as distance is capable of continuous growth. The
+difficulty of the arithmetical expression of irrational number,
+a difficulty considered by the modern school of analysts to have
+been at length surmounted (see <span class="sc"><a href="#artlinks">Function</a></span>), is not vital to it.
+It can call the ratio of the diagonal of a square to a side, for
+instance, or that of the circumference of a circle to a diameter,
+a number, and let a or x denote that number, just as properly
+as it may allow either letter to denote any rational number
+which may be greater or less than the ratio in question by a
+difference less than any minute one we choose to assign.</p>
+
+<p>Counting only, and not the counting of objects, is of the essence
+of arithmetic, and of algebra. But it is lawful to count objects,
+and in particular to count equal lengths by measure. The
+widened idea is that even when a or x is an irrational number
+we may speak of a or x unit lengths by measure. We may give
+concrete interpretation to an algebraical equation by allowing
+its terms all to mean numbers of times the same unit length,
+or the same unit area, or &amp;c. and in any equation lawfully
+derived from the first by algebraical processes we may do the
+same. Descartes in his <i>Géométrie</i> (1637) was the first to systematize
+the application of this principle to the inherent first
+notions of geometry; and the methods which he instituted have
+become the most potent methods of all in geometrical research.
+It is hardly too much to say that, when known facts as to a
+geometrical figure have once been expressed in algebraical
+terms, all strictly consequential facts as to the figure can be
+deduced by almost mechanical processes. Some may well be
+unexpected consequences; and in obtaining those of which
+there has been suggestion beforehand the often bewildering
+labour of constant attention to the figure is obviated. These
+are the methods of what is now called <i>analytical</i>, or sometimes
+<i>algebraical</i>, <i>geometry</i>.</p>
+
+<p>3. The modern use of the term &ldquo;analytical&rdquo; in geometry has
+obscured, but not made obsolete, an earlier use, one as old as
+Plato. There is nothing algebraical in this analysis, as distinguished
+from synthesis, of the Greeks, and of the expositors
+of pure geometry. It has reference to an order of ideas in
+demonstration, or, more frequently, in discovering means to
+effect the geometrical construction of a figure with an assigned
+special property. We have to suppose hypothetically that the
+construction has been performed, drawing a rough figure which
+exhibits it as nearly as is practicable. We then analyse or
+critically examine the figure, treated as correct, and ascertain
+other properties which it can only possess in association with
+the one in question. Presently one of these properties will often
+be found which is of such a character that the construction of
+a figure possessing it is simple. The means of effecting synthetically
+a construction such as was desired is thus brought to light by
+what Plato called <i>analysis</i>. Or again, being asked to prove a
+theorem A, we ascertain that it must be true if another theorem
+B is, that B must be if C is, and so on, thus eventually finding
+that the theorem A is the consequence, through a chain of intermediaries,
+of a theorem Z of which the establishment is easy.
+This geometrical analysis is not the subject of the present article;
+but in the reasoning from form to form of an equation or system
+<span class="pagenum"><a name="page712" id="page712"></a>712</span>
+of equations, with the object of basing the algebraical proof
+of a geometrical fact on other facts of a more obvious character,
+the same logic is utilized, and the name &ldquo;analytical geometry&rdquo;
+is thus in part explained.</p>
+
+<p>4. In algebra real positive number was alone at first dealt
+with, and in geometry actual signless distance. But in algebra
+it became of importance to say that every equation of the first
+degree has a root, and the notion of negative number was introduced.
+The negative unit had to be defined as what can be
+added to the positive unit and produce the sum zero. The
+corresponding notion was readily at hand in geometry, where
+it was clear that a unit distance can be measured to the left
+or down from the farther end of a unit distance already measured
+to the right or up from a point O, with the result of reaching O
+again. Thus, to give full interpretation in geometry to the
+algebraically negative, it was only necessary to associate distinctness
+of sign with oppositeness of direction. Later it was discovered
+that algebraical reasoning would be much facilitated, and that
+conclusions as to the real would retain all their soundness, if a pair
+of imaginary units ±&radic;&minus;1 of what might be called number were
+allowed to be contemplated, the pair being defined, though not
+separately, by the two properties of having the real sum 0 and
+the real product 1. Only in these two real combinations do they
+enter in conclusions as to the real. An advantage gained was
+that every quadratic equation, and not some quadratics only,
+could be spoken of as having two roots. These admissions of
+new units into algebra were final, as it admitted of proof that all
+equations of degrees higher than two have the full numbers of
+roots possible for their respective degrees in any case, and that
+every root has a value included in the form a + b &radic;&minus;1, with a, b,
+real. The corresponding enrichment could be given to geometry,
+with corresponding advantages and the same absence of danger,
+and this was done. On a line of measurement of distance we
+contemplate as existing, not only an infinite continuum of points
+at real distances from an origin of measurement O, but a doubly
+infinite continuum of points, all but the singly infinite continuum
+of real ones imaginary, and imaginary in conjugate pairs, a
+conjugate pair being at imaginary distances from O, which have
+a real arithmetic and a real geometric mean. To geometry
+enriched with this conception all algebra has its application.</p>
+
+<p>5. Actual geometry is one, two or three-dimensional, <i>i.e.</i>
+lineal, plane or solid. In one-dimensional geometry positions
+and measurements in a single line only are admitted. Now
+descriptive constructions for points in a line are impossible
+without going out of the line. It has therefore been held that
+there is a sense in which no science of geometry strictly confined
+to one dimension exists. But an algebra of one variable can be
+applied to the study of distances along a line measured from a
+chosen point on it, so that the idea of construction as distinct
+from measurement is not essential to a one-dimensional geometry
+aided by algebra. In geometry of two dimensions, the
+flat of the land-measurer, the passage from one point O to any
+other point, can be effected by two successive marches, one east
+or west and one north or south, and, as will be seen, an algebra
+of two variables suffices for geometrical exploitation. In
+geometry of three dimensions, that of space, any point can be
+reached from a chosen one by three marches, one east or west,
+one north or south, and one up or down; and we shall see that
+an algebra of three variables is all that is necessary. With
+three dimensions actual geometry stops; but algebra can supply
+any number of variables. Four or more variables have been
+used in ways analogous to those in which one, two and three
+variables are used for the purposes of one, two and three-dimensional
+geometry, and the results have been expressed in
+quasi-geometrical language on the supposition that a higher
+space can be conceived of, though not realized, in which four
+independent directions exist, such that no succession of marches
+along three of them can effect the same displacement of a point
+as a march along the fourth; and similarly for higher numbers
+than four. Thus analytical, though not actual, geometries exist
+for four and more dimensions. They are in fact algebras furnished
+with nomenclature of a geometrical cast, suggested by convenient
+forms of expression which actual geometry has, in return for
+benefits received, conferred on algebras of one, two and three
+variables.</p>
+
+<p>We will confine ourselves to the dimensions of actual geometry,
+and will devote no space to the one-dimensional, except incidentally
+as existing within the two-dimensional. The analytical
+method will now be explained for the cases of two and three
+dimensions in succession. The form of it originated by Descartes,
+and thence known as Cartesian, will alone be considered in much
+detail.</p>
+
+<div class="condensed">
+<p class="pt2 center">I. <i>Plane Analytical Geometry.</i></p>
+
+<table class="nobctr" style="clear: both;" summary="Illustration">
+<tr><td class="figcenter" colspan="2"><img style="width:520px; height:209px" src="images/img712.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 48.</span></td>
+<td class="caption"><span class="sc">Fig. 49.</span></td></tr></table>
+
+<p>6. <i>Coordinates.</i>&mdash;It is assumed that the points, lines and figures
+considered lie in one and the same plane, which plane therefore need
+not be in any way referred to. In the plane a point O, and two lines
+x&prime;Ox, y&prime;Oy, intersecting in O, are taken once for all, and regarded as
+fixed. O is called the origin, and x&prime;Ox, y&prime;Oy the axes of x and y
+respectively. Other positions in the plane are specified in relation
+to this fixed origin and these fixed axes. From any point P we
+suppose PM drawn parallel to the axis of y to meet the axis of x in
+M, and may also suppose PN drawn parallel to the axis of x to meet
+the axis of y in N, so that OMPN is a parallelogram. The position
+of P is determined when we know OM (= NP) and MP (= ON).
+If OM is x times the unit of a scale of measurement chosen at pleasure,
+and MP is y times the unit, so that x and y have numerical values,
+we call x and y the (Cartesian) coordinates of P. To distinguish
+them we often speak of y as the ordinate, and of x as the abscissa.</p>
+
+<p>It is necessary to attend to signs; x has one sign or the other
+according as the point P is on one side or the other of the axis of y,
+and y one sign or the other according as P is on one side or the other
+of the axis of x. Using the letters N, E, S, W, as in a map, and
+considering the plane as divided into four quadrants by the axes,
+the signs are usually taken to be:</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcc">x</td> <td class="tcc">y</td> <td class="tcc">For quadrant</td></tr>
+
+<tr><td class="tcc">+</td> <td class="tcc">+</td> <td class="tcc">N &ensp; E</td></tr>
+<tr><td class="tcc">+</td> <td class="tcc">&minus;</td> <td class="tcc">S &ensp; E</td></tr>
+<tr><td class="tcc">&minus;</td> <td class="tcc">+</td> <td class="tcc">N &ensp; W</td></tr>
+<tr><td class="tcc">&minus;</td> <td class="tcc">&minus;</td> <td class="tcc">S &ensp; W</td></tr>
+</table>
+
+<p class="noind">A point is referred to as the point (a, b), when its coordinates are
+x = a, y = b. A point may be fixed, or it may be variable, <i>i.e.</i> be
+regarded for the time being as free to move in the plane. The
+coordinates (x, y) of a variable point are algebraic variables, and are
+said to be &ldquo;current coordinates.&rdquo;</p>
+
+<p>The axes of x and y are usually (as in fig. 48) taken at right angles
+to one another, and we then speak of them as rectangular axes,
+and of x and y as &ldquo;rectangular coordinates&rdquo; of a point P; OMPN
+is then a rectangle. Sometimes, however, it is convenient to use
+axes which are oblique to one another, so that (as in fig. 49) the angle
+xOy between their positive directions is some known angle &omega;
+distinct from a right angle, and OMPN is always an oblique parallelogram
+with given angles; and we then speak of x and y as &ldquo;oblique
+coordinates.&rdquo; The coordinates are as a rule taken to be rectangular
+in what follows.</p>
+
+<p>7. <i>Equations and loci.</i> If (x, y) is the point P, and if we are
+given that x = 0, we are told that, in fig. 48 or fig. 49, the point M lies
+at O, whatever value y may have, <i>i.e.</i> we are told the one fact that
+P lies on the axis of y. Conversely, if P lies anywhere on the axis
+of y, we have always OM = 0, <i>i.e.</i> x = 0. Thus the equation x = 0 is
+one satisfied by the coordinates (x, y) of every point in the axis of y,
+and not by those of any other point. We say that x = 0 is the
+equation of the axis of y, and that the axis of y is the locus represented
+by the equation x = 0. Similarly y = 0 is the equation of the
+axis of x. An equation x = a, where a is a constant, expresses that
+P lies on a parallel to the axis of y through a point M on the axis
+of x such that OM = a. Every line parallel to the axis of y has an
+equation of this form. Similarly, every line parallel to the axis of x
+has an equation of the form y = b, where b is some definite constant.</p>
+
+<p>These are simple cases of the fact that a single equation in the
+current coordinates of a variable point (x, y) imposes one limitation
+on the freedom of that point to vary. The coordinates of a point
+<span class="pagenum"><a name="page713" id="page713"></a>713</span>
+taken at random in the plane will, as a rule, not satisfy the equation,
+but infinitely many points, and in most cases infinitely many real
+ones, have coordinates which do satisfy it, and these points are
+exactly those which lie upon some locus of one dimension, a straight
+line or more frequently a curve, which is said to be represented by
+the equation. Take, for instance, the equation y = mx, where m
+is a given constant. It is satisfied by the coordinates of every point
+P, which is such that, in fig. 48, the distance MP, with its proper sign,
+is m times the distance OM, with its proper sign, <i>i.e.</i> by the coordinates
+of every point in the straight line through O which we
+arrive at by making a line, originally coincident with x&prime;Ox, revolve
+about O in the direction opposite to that of the hands of a watch
+through an angle of which m is the tangent, and by those of no other
+points. That line is the locus which it represents. Take, more
+generally, the equation y = &phi;(x), where &phi;(x) is any given non-ambiguous
+function of x. Choosing any point M on x&prime;Ox in fig. 1, and
+giving to x the value of the numerical measure of OM, the equation
+determines a single corresponding y, and so determines a single
+point P on the line through M parallel to y&prime;Oy. This is one point
+whose coordinates satisfy the equation. Now let M move from the
+extreme left to the extreme right of the line x&prime;Ox, regarded as
+extended both ways as far as we like, <i>i.e.</i> let x take all real values
+from &minus;&infin; to &infin;. With every value goes a point P, as above, on
+the parallel to y&prime;Oy through the corresponding M; and we thus find
+that there is a path from the extreme left to the extreme right of
+the figure, all points P along which are distinguished from other
+points by the exceptional property of satisfying the equation by
+their coordinates. This path is a locus; and the equation y = &phi;(x)
+represents it. More generally still, take an equation f(x, y) = 0
+which involves both x and y under a functional form. Any particular
+value given to x in it produces from it an equation for the determination
+of a value or values of y, which go with that value of x in specifying
+a point or points (x, y), of which the coordinates satisfy the
+equation f(x, y) = 0. Here again, as x takes all values, the point or
+points describe a path or paths, which constitute a locus represented
+by the equation. Except when y enters to the first degree only in
+f(x, y), it is not to be expected that all the values of y, determined
+as going with a chosen value of x, will be necessarily real; indeed
+it is not uncommon for all to be imaginary for some ranges of values
+of x. The locus may largely consist of continua of imaginary
+points; but the real parts of it constitute a real curve or real curves.
+Note that we have to allow x to admit of all imaginary, as well as
+of all real, values, in order to obtain all imaginary parts of the
+locus.</p>
+
+<p>A locus or curve may be algebraically specified in another way;
+viz. we may be given two equations x = f(&theta;), y = F(&theta;), which express
+the coordinates of any point of it as two functions of the same
+variable parameter &theta; to which all values are open. As &theta; takes all
+values in turn, the point (x, y) traverses the curve.</p>
+
+<p>It is a good exercise to trace a number of curves, taken as defined
+by the equations which represent them. This, in simple cases, can
+be done approximately by plotting the values of y given by the
+equation of a curve as going with a considerable number of values
+of x, and connecting the various points (x, y) thus obtained. But
+methods exist for diminishing the labour of this tentative process.</p>
+
+<p>Another problem, which will be more attended to here, is that of
+determining the equations of curves of known interest, taken as
+defined by geometrical properties. It is not a matter for surprise
+that the curves which have been most and longest studied geometrically
+are among those represented by equations of the simplest
+character.</p>
+
+<table class="flt" style="float: right; width: 300px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:250px; height:245px" src="images/img713.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 50.</span></td></tr></table>
+
+<p>8. <i>The Straight Line.</i>&mdash;This is the simplest type of locus. Also
+the simplest type of equation in x and y is Ax + By + C = 0, one of
+the first degree. Here the coefficients A, B, C are constants. They
+are, like the current coordinates, x, y, numerical. But, in giving
+interpretation to such an equation, we must of course refer to
+numbers Ax, By, C of unit magnitudes of the same kind, of units
+of counting for instance, or unit lengths or unit squares. It will
+now be seen that every straight
+line has an equation of the first
+degree, and that every equation
+of the first degree represents a
+straight line.</p>
+
+<p>It has been seen (§ 7) that lines
+parallel to the axes have equations
+of the first degree, free
+from one of the variables. Take
+now a straight line ABC inclined
+to both axes. Let it make a
+given angle &alpha; with the positive
+direction of the axis of x, <i>i.e.</i> in
+fig. 50 let this be the angle
+through which Ax must be revolved
+counter-clockwise about
+A in order to be made coincident
+with the line. Let C, of
+coordinates (h, k), be a fixed point
+on the line, and P(x, y) any other point upon it. Draw the ordinates
+CD, PM of C and P, and let the parallel to the axis of x through C
+meet PM, produced if necessary, in R. The right-angled triangle
+CRP tells us that, with the signs appropriate to their directions
+attached to CR and RP,</p>
+
+<p class="center">RP = CR tan &alpha;, <i>i.e.</i> MP &minus; DC = (OM &minus; OD) tan &alpha;,</p>
+
+<p class="noind">and this gives that</p>
+
+<p class="center">y &minus; k = tan &alpha; (x &minus; h),</p>
+
+<p class="noind">an equation of the first degree satisfied by x and y. No point not
+on the line satisfies the same equation; for the line from C to any
+point off the line would make with CR some angle &beta; different from &alpha;,
+and the point in question would satisfy an equation y &minus; k = tan &beta;(x &minus; h),
+which is inconsistent with the above equation.</p>
+
+<p>The equation of the line may also be written y = mx + b, where
+m = tan &alpha;, and b = k &minus; h tan &alpha;. Here b is the value obtained for y
+from the equation when 0 is put for x, <i>i.e.</i> it is the numerical measure,
+with proper sign, of OB, the intercept made by the line on the axis
+of y, measured from the origin. For different straight lines, m and b
+may have any constant values we like.</p>
+
+<p>Now the general equation of the first degree Ax + By + C = 0 may
+be written y = &minus;(A/B)x &minus; C/B, unless B = 0, in which case it represents a
+line parallel to the axis of y; and &minus;A/B, &minus;C/B are values which
+can be given to m and b, so that every equation of the first degree
+represents a straight line. It is important to notice that the general
+equation, which in appearance contains three constants A, B, C, in
+effect depends on two only, the ratios of two of them to the third.
+In virtue of this last remark, we see that two distinct conditions
+suffice to determine a straight line. For instance, it is easy from the
+above to see that</p>
+
+<table class="math0" summary="math">
+<tr><td>x</td>
+<td rowspan="2">+</td> <td>y</td>
+<td rowspan="2">= 1</td></tr>
+<tr><td class="denom">a</td> <td class="denom">b</td></tr></table>
+
+<p class="noind">is the equation of a straight line determined by the two conditions
+that it makes intercepts OA, OB on the two axes, of which a and b
+are the numerical measures with proper signs: note that in fig. 50 a
+is negative. Again,</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">y &minus; y<span class="su">1</span> =</td> <td>y<span class="su">2</span> &minus; y<span class="su">1</span></td>
+<td rowspan="2">(x &minus; x<span class="su">1</span>),</td></tr>
+<tr><td class="denom">x<span class="su">2</span> &minus; x<span class="su">1</span></td></tr></table>
+
+<p class="noind"><i>i.e.</i></p>
+
+<p class="center">(y<span class="su">1</span> &minus; y<span class="su">2</span>) x &minus; (x<span class="su">1</span> &minus; x<span class="su">2</span>) y + x<span class="su">1</span>y<span class="su">2</span> &minus; x<span class="su">2</span>y<span class="su">1</span> = 0,</p>
+
+<p class="noind">represents the line determined by the data that it passes through
+two given points (x<span class="su">1</span>, y<span class="su">1</span>) and (x<span class="su">2</span>, y<span class="su">2</span>). To prove this find m in the
+equation y &minus; y<span class="su">1</span> = m(x &minus; x<span class="su">1</span>) of a line through (x<span class="su">1</span>, y<span class="su">1</span>), from the condition
+that (x<span class="su">2</span>, y<span class="su">2</span>) lies on the line.</p>
+
+<p>In this paragraph the coordinates have been assumed rectangular.
+Had they been oblique, the doctrine of similar triangles would have
+given the same results, except that in the forms of equation y &minus; k = m(x &minus; h),
+y = mx + b, we should not have had m = tan &alpha;.</p>
+
+<p>9. <i>The Circle.</i>&mdash;It is easy to write down the equation of a given
+circle. Let (h, k) be its given centre C, and &rho; the numerical measure
+of its given radius. Take P (x, y) any point on its circumference,
+and construct the triangle CRP, in fig. 50 as above. The fact that
+this is right-angled tells us that</p>
+
+<p class="center">CR² + RP² = CP²,</p>
+
+<p class="noind">and this at once gives the equation</p>
+
+<p class="center">(x &minus; h)² + (y &minus; k)² = &rho;².</p>
+
+<p class="noind">A point not upon the circumference of the particular circle is at some
+distance from (h, k) different from &rho;, and satisfies an equation
+inconsistent with this one; which accordingly represents the circumference,
+or, as we say, the circle.</p>
+
+<p>The equation is of the form</p>
+
+<p class="center">x² + y² + 2Ax + 2By + C = 0.</p>
+
+<p class="noind">Conversely every equation of this form represents a circle: we have
+only to take &minus;A, &minus;B, A² + B² &minus; C for h, k, &rho;² respectively, to obtain
+its centre and radius. But this statement must appear too unrestricted.
+Ought we not to require A² + B² &minus; C to be positive?
+Certainly, if by circle we are only to mean the visible round circumference
+of the geometrical definition. Yet, analytically, we
+contemplate altogether imaginary circles, for which &rho;² is negative,
+and circles, for which &rho; = 0, with all their reality condensed into
+their centres. Even when &rho;² is positive, so that a visible round
+circumference exists, we do not regard this as constituting the
+whole of the circle. Giving to x any value whatever in (x &minus; h)² + (y &minus; k)² = &rho;²,
+we obtain two values of y, real, coincident or imaginary,
+each of which goes with the abscissa x as the ordinate of a point,
+real or imaginary, on what is represented by the equation of the
+circle.</p>
+
+<p>The doctrine of the imaginary on a circle, and in geometry generally,
+is of purely algebraical inception; but it has been in its entirety
+accepted by modern pure geometers, and signal success has attended
+the efforts of those who, like K.G.C. von Staudt, have striven to
+base its conclusions on principles not at all algebraical in form,
+though of course cognate to those adopted in introducing the
+imaginary into algebra.</p>
+
+<p>A circle with its centre at the origin has an equation x² + y² = &rho;².</p>
+
+<p>In oblique coordinates the general equation of a circle is
+x² + 2xy cos &omega; + y² + 2Ax + 2By + C = 0.</p>
+
+<p>10. The conic sections are the next simplest loci; and it will be
+seen later that they are the loci represented by equations of the
+second degree. Circles are particular cases of conic sections; and
+<span class="pagenum"><a name="page714" id="page714"></a>714</span>
+they have just been seen to have for their equations a particular
+class of equations of the second degree. Another particular class
+of such equations is that included in the form (Ax + By + C)(A&prime;x +
+B&prime;y + C&prime;) = 0, which represents two straight lines, because the product
+on the left vanishes if, and only if, one of the two factors does, <i>i.e.</i>
+if, and only if, (x, y) lies on one or other of two straight lines. The
+condition that ax² + 2hxy + by² + 2gx + 2fy + c = 0, which is often
+written (a, b, c, f, g, h)(x, y, I)² = 0, takes this form is abc + 2fgh &minus; af² &minus;
+bg² &minus; ch² = 0. Note that the two lines may, in particular cases, be
+parallel or coincident.</p>
+
+<p>Any equation like F<span class="su">1</span>(x, y) F<span class="su">2</span>(x, y) ... F<span class="su">n</span>(x, y) = 0, of which
+the left-hand side breaks up into factors, represents all the loci
+separately represented by F<span class="su">1</span>(x, y) = 0, F<span class="su">2</span>(x, y) = 0, ... F<span class="su">n</span>(x, y) = 0.
+In particular an equation of degree n which is free from x represents
+n straight lines parallel to the axis of x, and one of degree n which
+is homogeneous in x and y, <i>i.e.</i> one which upon division by x<span class="sp">n</span>, becomes
+an equation in the ratio y/x, represents n straight lines through
+the origin.</p>
+
+<p>Curves represented by equations of the third degree are called
+cubic curves. The general equation of this degree will be written
+(*)(x, y, I)³ = 0.</p>
+
+<table class="flt" style="float: right; width: 330px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:281px; height:263px" src="images/img714a.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 51.</span></td></tr></table>
+
+<p>11. <i>Descriptive Geometry.</i>&mdash;A geometrical proposition is either
+descriptive or metrical: in the former case the statement of it is
+independent of the idea of magnitude (length, inclination, &amp;c.),
+and in the latter it has reference to this idea. The method of coordinates
+seems to be by its inception essentially metrical. Yet
+in dealing by this method with descriptive propositions we are
+eminently free from metrical considerations, because of our power to
+use general equations, and
+to avoid all assumption that
+measurements implied are
+any particular measurements.</p>
+
+<p>12. It is worth while to
+illustrate this by the instance
+of the well-known
+theorem of the radical centre
+of three circles. The theorem
+is that, given any three circles
+A, B, C (fig. 51), the common
+chords &alpha;&alpha;&prime;, &beta;&beta;&prime;, &gamma;&gamma;&prime; of the
+three pairs of circles meet in
+a point.</p>
+
+<p>The geometrical proof is
+metrical throughout:&mdash;</p>
+
+<p>Take O the point of intersection
+of &alpha;&alpha;&prime;, &beta;&beta;&prime;, and joining
+this with &gamma;&prime;, suppose that &gamma;&prime;O does not pass through &gamma;, but that it
+meets the circles A, B in two distinct points &gamma;<span class="su">2</span>, &gamma;<span class="su">1</span> respectively. We
+have then the known metrical property of intersecting chords of a
+circle; viz. in circle C, where &alpha;&alpha;&prime;, &beta;&beta;&prime;, are chords meeting at a point O,</p>
+
+<p class="center">O&alpha;·O&alpha;&prime; = O&beta;·O&beta;&prime;,</p>
+
+<p class="noind">where, as well as in what immediately follows, O&alpha;, &amp;c. denote, of
+course, <i>lengths</i> or <i>distances</i>.</p>
+
+<p>Similarly in circle A,</p>
+
+<p class="center">O&beta;·O&beta;&prime; = O&gamma;<span class="su">2</span>·O&gamma;&prime;,</p>
+
+<p class="noind">and in circle B,</p>
+
+<p class="center">O&alpha;·O&alpha;&prime; = O&gamma;<span class="su">1</span>·O&gamma;&prime;.</p>
+
+<p class="noind">Consequently O&gamma;<span class="su">1</span>·O&gamma;&prime; = O&gamma;<span class="su">2</span>·O&gamma;&prime;, that is, O&gamma;<span class="su">1</span> = O&gamma;<span class="su">2</span>, or the points
+&gamma;<span class="su">1</span> and &gamma;<span class="su">2</span> coincide; that is, they each coincide with &gamma;.</p>
+
+<p>We contrast this with the analytical method:&mdash;</p>
+
+<p>Here it only requires to be known that an equation Ax + By + C = 0
+represents a line, and an equation x² + y² + Ax + By + C = 0 represents
+a circle. A, B, C have, in the two cases respectively, metrical
+significations; but these we are not concerned with. Using S to
+denote the function x² + y² + Ax + By + C, the equation of a circle is
+S = o. Let the equation of any other circle be S&prime;, = x² + y² + A&prime;x + B&prime;y + C&prime; = 0;
+the equation S-S&prime; = 0 is a linear equation (S &minus; S&prime; is in
+fact = (A &minus; A&prime;)x + (B &minus; B&prime;)y + C-C), and it thus represents a line;
+this equation is satisfied by the coordinates of each of the points of
+intersection of the two circles (for at each of these points S = 0 and
+S&prime; = 0, therefore also S &minus; S&prime; = 0); hence the equation S &minus; S&prime; = 0 is
+that of the line joining the two points of intersection of the two circles,
+or say it is the equation of the common chord of the two circles.
+Considering then a third circle S&Prime;, = x² + y² + A&Prime;x + B&Prime;y + C&Prime; = 0, the
+equations of the common chords are S &minus; S&prime; = 0, S &minus; S&Prime; = 0, S&prime; &minus; S&Prime; = 0
+(each of these a linear equation); at the intersection of the first and
+second of these lines S = S&prime; and S = S&Prime;, therefore also S&prime; = S&Prime;, or the
+equation of the third line is satisfied by the coordinates of the point
+in question; that is, the three chords intersect in a point O, the coordinates
+of which are determined by the equations S = S&prime; = S&Prime;.</p>
+
+<p>It further appears that if the two circles S = 0, S&prime; = 0 do not intersect
+in any real points, they must be regarded as intersecting in two
+imaginary points, such that the line joining them is the real line
+represented by the equation S &minus; S&prime; = 0; or that two circles, whether
+their intersections be real or imaginary, have always a real common
+chord (or radical axis), and that for <i>any</i> three circles the common
+chords intersect in a point (of course real) which is the radical centre.
+And by this very theorem, given two circles with imaginary intersections,
+we can, by drawing circles which meet each of them in
+real points, construct the radical axis of the first-mentioned two
+circles.</p>
+
+<p>13. The principle employed in showing that the equation of the
+common chord of two circles is S &minus; S&prime; = 0 is one of very extensive
+application, and some more illustrations of it may be given.</p>
+
+<p>Suppose S = 0, S&prime; = 0 are lines (that is, let S, S&prime; now denote linear
+functions Ax + By + C, A&prime;x + B&prime;y + C&prime;), then S &minus; kS&prime; = 0 (k an arbitrary
+constant) is the equation of any line passing through the point
+of intersection of the two given lines. Such a line may be made to
+pass through any given point, say the point (x<span class="su">0</span>, y<span class="su">0</span>); if S<span class="su">0</span>, S&prime;<span class="su">0</span> are
+what S, S&prime; respectively become on writing for (x, y) the values (x<span class="su">0</span>, y<span class="su">0</span>),
+then the value of k is k = S<span class="su">0</span> ÷ S&prime;<span class="su">0</span>. The equation in fact is SS&prime;<span class="su">0</span> &minus; S<span class="su">0</span>S&prime; = 0;
+and starting from this equation we at once verify it <i>a posteriori</i>;
+the equation is a linear equation satisfied by the values of (x, y)
+which make S = 0, S&prime; = 0; and satisfied also by the values (x<span class="su">0</span>, y<span class="su">0</span>);
+and it is thus the equation of the line in question.</p>
+
+<p>If, as before, S = 0, S&prime; = 0 represent circles, then (k being arbitrary)
+S &minus; kS&prime; = 0 is the equation of any circle passing through the two
+points of intersection of the two circles; and to make this pass
+through a given point (x<span class="su">0</span>, y<span class="su">0</span>) we have again k = S<span class="su">0</span> ÷ S&prime;<span class="su">0</span>. In the
+particular case k = 1, the circle becomes the common chord (more
+accurately it becomes the common chord together with the line
+infinity; see § 23 below).</p>
+
+<p>If S denote the general quadric function,</p>
+
+<p class="center">S = ax<span class="sp">2</span> + 2hxy + by<span class="sp">2</span> + 2fy + 2gx + c,</p>
+
+<p class="noind">then the equation S = 0 represents a conic; assuming this, then, if
+S&prime; = 0 represents another conic, the equation S &minus; kS&prime; = 0 represents
+<i>any</i> conic through the four points of intersection of the two conics.</p>
+
+<table class="flt" style="float: right; width: 300px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:246px; height:143px" src="images/img714b.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 52.</span></td></tr></table>
+
+<p>14. The object still being to illustrate the mode of working with
+coordinates for descriptive purposes,
+we consider the theorem
+of the polar of a point in regard
+to a circle. Given a circle and
+a point O (fig. 52), we draw
+through O any two lines meeting
+the circle in the points A, A&prime; and
+B, B&prime; respectively, and then
+taking Q as the intersection of
+the lines AB&prime; and A&prime;B, the
+theorem is that the locus of the
+point Q is a right line depending
+only upon O and the circle, but independent of the
+particular lines OAA&prime; and OBB&prime;.</p>
+
+<p>Taking O as the origin, and for the axes any two lines through O
+at right angles to each other, the equation of the circle will be</p>
+
+<p class="center">x<span class="sp">2</span> + y<span class="sp">2</span> + 2Ax + 2By + C = 0;</p>
+
+<p class="noind">and if the equation of the line OAA&prime; is taken to be y = mx, then the
+points A, A&prime; are found as the intersections of the straight line with
+the circle; or to determine x we have</p>
+
+<p class="center">x<span class="sp">2</span> (1 + m<span class="sp">2</span>) + 2x (A + Bm) + C = 0.</p>
+
+<p class="noind">If(x<span class="su">1</span>, y<span class="su">1</span>) are the coordinates of A, and (x<span class="su">2</span>, y<span class="su">2</span>) of A&prime;, then the roots
+of this equation are x<span class="su">1</span>, x<span class="su">2</span>, whence easily</p>
+
+<table class="math0" summary="math">
+<tr><td>1</td>
+<td rowspan="2">+</td> <td>1</td>
+<td rowspan="2">= &minus;2</td> <td>A + Bm</td>
+<td rowspan="2">.</td></tr>
+<tr><td class="denom">x<span class="su">1</span></td> <td class="denom">x<span class="su">2</span></td>
+<td class="denom">C</td></tr></table>
+
+<p class="noind">And similarly, if the equation of the line OBB&prime; is taken to be y = m&prime;x<span class="su">1</span>
+and the coordinates of B, B&prime; to be (x<span class="su">3</span>, y<span class="su">3</span>) and (x<span class="su">4</span>, y<span class="su">4</span>) respectively,
+then</p>
+
+<table class="math0" summary="math">
+<tr><td>1</td>
+<td rowspan="2">+</td> <td>1</td>
+<td rowspan="2">= &minus;2</td> <td>A + Bm&prime;</td>
+<td rowspan="2">.</td></tr>
+<tr><td class="denom">x<span class="su">3</span></td> <td class="denom">x<span class="su">4</span></td>
+<td class="denom">C&prime;</td></tr></table>
+
+<p>We have then by § 8</p>
+
+<table class="reg" summary="poem"><tr><td> <div class="poemr">
+<p>x (y<span class="su">1</span> &minus; y<span class="su">4</span>) &minus; y (x<span class="su">1</span> &minus; x<span class="su">4</span>) + x<span class="su">1</span>y<span class="su">4</span> &minus; x<span class="su">4</span>y<span class="su">1</span> = 0,</p>
+
+<p>x (y<span class="su">2</span> &minus; y<span class="su">3</span>) &minus; y (x<span class="su">2</span> &minus; x<span class="su">3</span>) + x<span class="su">2</span>y<span class="su">3</span> &minus; x<span class="su">3</span>y<span class="su">2</span> = 0,</p>
+</div> </td></tr></table>
+
+<p class="noind">as the equations of the lines AB&prime; and A&prime;B respectively. Reducing
+by means of the relations y<span class="su">1</span> &minus; mx<span class="su">1</span> = 0, y<span class="su">2</span> &minus; mx<span class="su">2</span> = 0, y<span class="su">3</span> &minus; m&prime;x<span class="su">3</span> = 0,
+y<span class="su">4</span> &minus; m&prime;x<span class="su">4</span> = 0, the two equations become</p>
+
+<table class="reg" summary="poem"><tr><td> <div class="poemr">
+<p>x (mx<span class="su">1</span> &minus; m&prime;x<span class="su">4</span>) &minus; y (x<span class="su">1</span> &minus; x<span class="su">4</span>) + (m&prime; &minus; m) x<span class="su">1</span>x<span class="su">4</span> = 0,</p>
+
+<p>x (mx<span class="su">2</span> &minus; m&prime;x<span class="su">3</span>) &minus; y (x<span class="su">2</span> &minus; x<span class="su">3</span>) + (m&prime; &minus; m) x<span class="su">2</span>x<span class="su">3</span> = 0,</p>
+</div> </td></tr></table>
+
+<p class="noind">and if we divide the first of these equations by x<span class="su">1</span>x<span class="su">4</span>, and the second
+by x<span class="su">2</span>x<span class="su">3</span> and then add, we obtain</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">x <span class="f150">{</span> m <span class="f150">(</span></td> <td>1</td>
+<td rowspan="2">+</td> <td>1</td>
+<td rowspan="2"><span class="f150">)</span> &minus; m&prime; <span class="f150">(</span></td> <td>1</td>
+<td rowspan="2">+</td> <td>1</td>
+<td rowspan="2"><span class="f150">) }</span> &minus; y <span class="f150">{</span></td> <td>1</td>
+<td rowspan="2">+</td> <td>1</td>
+<td rowspan="2">&minus; <span class="f150">(</span></td> <td>1</td>
+<td rowspan="2">+</td> <td>1</td>
+<td rowspan="2"><span class="f150">) }</span> + 2m&prime; &minus; 2m = 0,</td></tr>
+<tr><td class="denom">x<span class="su">3</span></td> <td class="denom">x<span class="su">4</span></td>
+<td class="denom">x<span class="su">1</span></td> <td class="denom">x<span class="su">2</span></td>
+<td class="denom">x<span class="su">3</span></td> <td class="denom">x<span class="su">4</span></td>
+<td class="denom">x<span class="su">1</span></td> <td class="denom">x<span class="su">2</span></td></tr></table>
+
+<p class="noind">or, what is the same thing,</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2"><span class="f150">(</span></td> <td>1</td>
+<td rowspan="2">+</td> <td>1</td>
+<td rowspan="2"><span class="f150">)</span> (y &minus; m&prime;x) &minus; <span class="f150">(</span></td> <td>1</td>
+<td rowspan="2">+</td> <td>1</td>
+<td rowspan="2"><span class="f150">)</span> (y &minus; mx) + 2m&prime; &minus; 2m = 0,</td></tr>
+<tr><td class="denom">x<span class="su">1</span></td> <td class="denom">x<span class="su">2</span></td>
+<td class="denom">x<span class="su">3</span></td> <td class="denom">x<span class="su">4</span></td></tr></table>
+
+<p class="noind">which by what precedes is the equation of a line through the point Q.
+Substituting herein for 1/x<span class="su">1</span> + 1/x<span class="su">2</span>, 1/x<span class="su">3</span> + 1/x<span class="su">4</span> their foregoing values, the
+equation becomes</p>
+
+<p class="center">&minus;(A + Bm) (y &minus; m&prime;x) + (A + Bm&prime;) (y &minus; mx) + C (m&prime; &minus; m) = 0;</p>
+
+<p class="noind">that is,</p>
+
+<p class="center">(m &minus; m&prime;) (Ax + By + C) = 0;</p>
+
+<p><span class="pagenum"><a name="page715" id="page715"></a>715</span></p>
+
+<p class="noind">or finally it is Ax + By + C = 0, showing that the point Q lies in a line
+the position of which is independent of the particular lines OAA&prime;,
+OBB&prime; used in the construction. It is proper to notice that there is
+no correspondence to each other of the points A, A&prime; and B, B&prime;; the
+grouping might as well have been A, A&prime; and B&prime;, B; and it thence
+appears that the line Ax + By + C = 0 just obtained is in fact the line
+joining the point Q with the point R which is the intersection of
+AB and A&prime;B&prime;.</p>
+
+<p>15. In § 8 it has been seen that two conditions determine the
+equation of a straight line, because in Ax + By + C = 0 one of the
+coefficients may be divided out, leaving only two parameters to be
+determined. Similarly five conditions instead of six determine an
+equation of the second degree (a, b, c, f, g, h)(x, y, 1)² = 0, and nine
+instead of ten determine a cubic (*)(x, y, 1)³ = 0. It thus appears
+that a cubic can be made to pass through 9 given points, and that
+the cubic so passing through 9 given points is completely determined.
+There is, however, a remarkable exception. Considering two given
+cubic curves S = 0, S&prime; = 0, these intersect in 9 points, and through
+these 9 points we have the whole series of cubics S &minus; kS&prime; = 0, where
+k is an arbitrary constant: k may be determined so that the cubic
+shall pass through a given tenth point (k = S<span class="su">0</span> ÷ S&prime;<span class="su">0</span>, if the coordinates
+are (x<span class="su">0</span>, y<span class="su">0</span>), and S<span class="su">0</span>, S&prime;<span class="su">0</span> denote the corresponding values of S, S&prime;).
+The resulting curve SS&prime;<span class="su">0</span> &minus; S&prime;S<span class="su">0</span> = 0 may be regarded as the cubic
+determined by the conditions of passing through 8 of the 9 points
+and through the given point (x<span class="su">0</span>, y<span class="su">0</span>); and from the equation it
+thence appears that the curve passes through the remaining one of
+the 9 points. In other words, we thus have the theorem, any cubic
+curve which passes through 8 of the 9 intersections of two given
+cubic curves passes through the 9th intersection.</p>
+
+<p>The applications of this theorem are very numerous; for instance,
+we derive from it Pascal&rsquo;s theorem of the inscribed hexagon. Consider
+a hexagon inscribed in a conic. The three alternate sides
+constitute a cubic, and the other three alternate sides another cubic.
+The cubics intersect in 9 points, being the 6 vertices of the hexagon,
+and the 3 Pascalian points, or intersections of the pairs of opposite
+sides of the hexagon. Drawing a line through two of the Pascalian
+points, the conic and this line constitute a cubic passing through 8
+of the 9 points of intersection, and it therefore passes through the
+remaining point of intersection&mdash;that is, the third Pascalian point;
+and since obviously this does not lie on the conic, it must lie on the
+line&mdash;that is, we have the theorem that the three Pascalian points
+(or points of intersection of the pairs of opposite sides) lie on a
+line.</p>
+
+<p>16. <i>Metrical Theory resumed.</i> <i>Projections and Perpendiculars.</i>&mdash;It
+is a metrical fact of fundamental importance, already used in § 8,
+that, if a finite line PQ be projected on any other line OO&prime; by perpendiculars
+PP&prime;, QQ&prime; to OO&prime;, the length of the projection P&prime;Q&prime; is
+equal to that of PQ multiplied by the cosine of the acute angle
+between the two lines. Also the algebraical sum of the projections
+of the sides of any closed polygon upon any line is zero, because as a
+point goes round the polygon, from any vertex A to A again, the
+point which is its projection on the line passes from A&prime; the projection
+of A to A&prime; again, <i>i.e.</i> traverses equal distances along the line in
+positive and negative senses. If we consider the polygon as consisting
+of two broken lines, each extending from the same initial
+to the same terminal point, the sum of the projections of the lines
+which compose the one is equal, in sign and magnitude, to the sum
+of the projections of the lines composing the other. Observe that
+the projection on a line of a length perpendicular to the line is
+zero.</p>
+
+<p>Let us hence find the equation of a straight line such that the
+perpendicular OD on it from the origin is of length &rho; taken as
+positive, and is inclined to the axis of x at an angle xOD = &alpha;,
+measured counter-clockwise from Ox. Take any point P(x, y) on
+the line, and construct OM and MP as in fig. 48. The sum of the
+projections of OM and MP on OD is OD itself; and this gives the
+equation of the line</p>
+
+<p class="center">x cos &alpha; + y sin &alpha; = &rho;.</p>
+
+<p class="noind">Observe that cos &alpha; and sin &alpha; here are the sin &alpha; and &minus;cos &alpha;, or the
+&minus;sin &alpha; and cos &alpha; of § 8 according to circumstances.</p>
+
+<p>We can write down an expression for the perpendicular distance
+from this line of any point (x&prime;, y&prime;) which does not lie upon it. If the
+parallel through (x&prime;, y&prime;) to the line meet OD in E, we have x&prime; cos &alpha; + y&prime; sin &alpha; = OE,
+and the perpendicular distance required is OD &minus; OE,
+<i>i.e.</i> &rho; &minus; x&prime; cos &alpha; &minus; y&prime; sin &alpha;; it is the perpendicular distance taken
+positively or negatively according as (x&prime;, y&prime;) lies on the same side
+of the line as the origin or not.</p>
+
+<p>The general equation Ax + By + C = 0 may be given the form
+x cos &alpha; + y sin &alpha; &minus; &rho; = 0 by dividing it by &radic;(A² + B³). Thus (Ax&prime; +
+By&prime; + C) ÷ &radic;(A² + B²) is in absolute value the perpendicular distance
+of (x&prime;, y&prime;) from the line Ax + By + C = 0. Remember, however, that
+there is an essential ambiguity of sign attached to a square root.
+The expression found gives the distance taken positively when
+(x&prime;, y&prime;) is on the origin side of the line, if the sign of C is given to
+&radic;(A² + B²).</p>
+
+<p>17. <i>Transformation of Coordinates.</i>&mdash;We often need to adopt new
+axes of reference in place of old ones; and the above principle of
+projections readily expresses the old coordinates of any point in
+terms of the new.</p>
+
+<table class="flt" style="float: right; width: 310px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:264px; height:216px" src="images/img715.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 53.</span></td></tr></table>
+
+<p>Suppose, for instance, that we want to take for new origin the
+point O&prime; of old coordinates OA = h, AO&prime; = k, and for new axes of
+X and Y lines through O&prime; obtained by rotating parallels to the old
+axes of x and y through an angle &theta; counter-clockwise. Construct
+(fig. 53) the old and new coordinates
+of any point P. Expressing
+that the projections,
+first on the old axis of x and
+secondly on the old axis of y, of
+OP are equal to the sums of the
+projections, on those axes respectively,
+of the parts of the broken
+line OO&prime;M&prime;P, we obtain:</p>
+
+<p class="center">x = h + X cos &theta; + Y cos (&theta; + ½&pi;) =
+h + X cos &theta; &minus; Y sin &theta;,</p>
+
+<p class="noind">and</p>
+
+<p class="center">y = k + X cos (½&pi; &minus; &theta;) + Y cos &theta; =
+k + X sin &theta; + Y cos &theta;.</p>
+
+<p>Be careful to observe that these
+formulae do not apply to every
+conceivable change of reference from one set of rectangular axes to
+another. It might have been required to take O&prime;X, O&prime;Y&prime; for the
+positive directions of the new axes, so that the change of directions
+of the axes could not be effected by rotation. We must then write
+&minus;Y for Y in the above.</p>
+
+<p>Were the new axes oblique, making angles &alpha;, &beta; respectively with
+the old axis of x, and so inclined at the angle &beta; &minus; &alpha;, the same method
+would give the formulae</p>
+
+<p class="center">x = h + X cos &alpha; + Y cos &beta;, y = k + X sin &alpha; + Y sin &beta;.</p>
+
+<p>18. <i>The Conic Sections.</i>&mdash;The conics, as they are now called, were
+at first defined as curves of intersection of planes and a cone; but
+Apollonius substituted a definition free from reference to space of
+three dimensions. This, in effect, is that a conic is the locus of a
+point the distance of which from a given point, called the focus, has
+a given ratio to its distance from a given line, called the directrix
+(see <span class="sc"><a href="#artlinks">Conic Section</a></span>). If e : 1 is the ratio, e is called the eccentricity.
+The distances are considered signless.</p>
+
+<p>Take (h, k) for the focus, and x cos &alpha; + y sin &alpha; &minus; p = 0 for the
+directrix. The absolute values of &radic;{(x &minus; h)² + (y &minus; k)²} and p &minus; x cos &alpha; &minus;
+y sin &alpha; are to have the ratio e : 1; and this gives</p>
+
+<p class="center">(x &minus; h)² + (y &minus; k)² = e² (p &minus; x cos &alpha; &minus; y sin &alpha;)²</p>
+
+<p class="noind">as the general equation, in rectangular coordinates, of a conic.</p>
+
+<p>It is of the second degree, and is the general equation of that
+degree. If, in fact, we multiply it by an unknown &lambda;, we can, by
+solving six simultaneous equations in the six unknowns &lambda;, h, k, e, p, &alpha;,
+so choose values for these as to make the coefficients in the equation
+equal to those in any equation of the second degree which may be
+given. There is no failure of this statement in the special case
+when the given equation represents two straight lines, as in § 10,
+but there is speciality: if the two lines intersect, the intersection
+and either bisector of the angle between them are a focus and
+directrix; if they are united in one line, any point on the line and a
+perpendicular to it through the point are: if they are parallel,
+the case is a limiting one in which e and h² + k² have become infinite
+while e<span class="sp">&minus;2</span>(h² + k²) remains finite. In the case (§ 9) of an equation
+such as represents a circle there is another instance of proceeding
+to a limit: e has to become 0, while ep remains finite: moreover &alpha;
+is indeterminate. The centre of a circle is its focus, and its directrix
+has gone to infinity, having no special direction. This last fact
+illustrates the necessity, which is also forced on plane geometry by
+three-dimensional considerations, of treating all points at infinity
+in a plane as lying on a single straight line.</p>
+
+<p>Sometimes, in reducing an equation to the above focus and directrix
+form, we find for h, k, e, p, tan &alpha;, or some of them, only imaginary
+values, as quadratic equations have to be solved; and we have in
+fact to contemplate the existence of entirely imaginary conics.
+For instance, no real values of x and y satisfy x² + 2y² + 3 = 0. Even
+when the locus represented is real, we obtain, as a rule, four sets of
+values of h, k, e, p, of which two sets are imaginary; a real conic
+has, besides two real foci and corresponding directrices, two others
+that are imaginary.</p>
+
+<p>In oblique as well as rectangular coordinates equations of the
+second degree represent conics.</p>
+
+<p>19. <i>The three Species of Conics.</i>&mdash;A real conic, which does not
+degenerate into straight lines, is called an ellipse, parabola or hyperbola
+according as e &lt;, = , or &gt; 1. To trace the three forms it is
+best so to choose the axes of reference as to simplify their equations.</p>
+
+<p>In the case of a parabola, let 2c be the distance between the given
+focus and directrix, and take axes referred to which these are the
+point (c, 0) and the line x = &minus; c. The equation becomes (x &minus; c)² + y² =
+(x + c)², <i>i.e.</i> y² = 4cx.</p>
+
+<p>In the other cases, take a such that a(e ~ e<span class="sp">&minus;1</span>) is the distance of focus
+from directrix, and so choose axes that these are (ae, 0) and x = ae<span class="sp">-1</span>,
+thus getting the equation(x &minus; ae)² + y² = e²(x &minus; ae<span class="sp">-1</span>)², <i>i.e.</i> (1 &minus; e²)x² + y² =
+a²(1 &minus; e²). When e &lt; 1, <i>i.e.</i> in the case of an ellipse, this may be
+written x²/a² + y²/b² = 1, where b² = a²(1 &minus; e²); and when e &gt; 1, <i>i.e.</i>
+in the case of an hyperbola, x²/a² &minus; y²/b² = 1, where b² = a²(e² &minus; 1).
+<span class="pagenum"><a name="page716" id="page716"></a>716</span>
+The axes thus chosen for the ellipse and hyperbola are called the
+principal axes.</p>
+
+<p>In figs. 54, 55, 56 in order, conics of the three species, thus referred,
+are depicted.</p>
+
+<table class="nobctr" style="clear: both;" summary="Illustration">
+<tr><td class="figcenter" colspan="2"><img style="width:511px; height:227px" src="images/img716a.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 54</span></td>
+<td class="caption"><span class="sc">Fig. 55</span></td></tr></table>
+
+<table class="nobctr" style="clear: both;" summary="Illustration">
+<tr><td class="figcenter"><img style="width:331px; height:189px" src="images/img716b.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 56.</span></td></tr></table>
+
+<p>The oblique straight lines in fig. 56 are the <i>asymptotes</i> x/a = ±y/b
+of the hyperbola, lines to which the curve tends with unlimited
+closeness as it goes to infinity. The hyperbola would have an equation
+of the form xy = c if referred to its asymptotes as axes, the coordinates
+being then oblique, unless a = b, in which case the hyperbola
+is called rectangular. An ellipse has two imaginary asymptotes.
+In particular a circle x² + y² = a², a particular ellipse, has for asymptotes
+the imaginary lines x = ±y &radic;&minus;1. These run from the centre
+to the so-called circular points at infinity.</p>
+
+<p>20. <i>Tangents and Curvature.</i>&mdash;Let (x&prime;, y&prime;) and (x&prime; + h, y&prime; + k) be
+two neighbouring points P, P&prime; on a curve. The equation of the line
+on which both lie is h(y &minus; y&prime;) = k(x &minus; x&prime;). Now keep P fixed, and let
+P&prime; move towards coincidence with it along the curve. The connecting
+line will tend towards a limiting position, to which it can
+never attain as long as P and P&prime; are distinct. The line which
+occupies this limiting position is the tangent at P. Now if we subtract
+the equation of the curve, with (x&prime;, y&prime;) for the coordinates in it,
+from the like equation in (x&prime; + h, y&prime; + k), we obtain a relation in h
+and k, which will, as a rule, be of the form 0 = Ah + Bk + terms of
+higher degrees in h and k, where A, B and the other coefficients
+involve x&prime; and y&prime;. This gives k/h = &minus;A/B + terms which tend to
+vanish as h and k do, so that &minus;A : B is the limiting value tended to
+by k : h. Hence the equation of the tangent is B(y &minus; y&prime;) + A(x &minus; x&prime;) = 0.</p>
+
+<p>The <i>normal</i> at (x&prime;, y&prime;) is the line through it at right angles to the
+tangent, and its equation is A(y &minus; y&prime;) &minus; B(x &minus; x&prime;) = 0.</p>
+
+<p>In the case of the conic (a, b, c, f, g, h) (x, y, 1)² = 0 we find that
+A/B = (ax&prime; + hy&prime; + g)/(hx&prime; + by&prime; + f).</p>
+
+<p>We can obtain the coordinates of Q, the intersection of the normals
+QP, QP&prime; at (x&prime;, y&prime;) and (x&prime; + h, y&prime; + k), and then, using the limiting
+value of k : h, deduce those of its limiting position as P&prime; moves up
+to P. This is the <i>centre of curvature</i> of the curve at P (x&prime;, y&prime;), and
+is so called because it is the centre of the circle of closest contact
+with the curve at that point. That it is so follows from the facts
+that the closest circle is the limit tended to by the circle which touches
+the curve at P and passes through P&prime;, and that the arc from P to P&prime;
+of this circle lies between the circles of centre Q and radii QP, QP&prime;,
+which circles tend, not to different limits as P&prime; moves up to P, but
+to one. The distance from P to the centre of curvature is the <i>radius
+of curvature</i>.</p>
+
+<p>21. <i>Differential Plane Geometry.</i>&mdash;The language and notation of the
+differential calculus are very useful in the study of tangents and
+curvature. Denoting by (&xi;, &eta;) the current coordinates, we find,
+as above, that the tangent at a point (x, y) of a curve is &eta; &minus; y =
+(&xi; &minus; x)dy/dx, where dy/dx is found from the equation of the curve. If
+this be f(x, y) = 0 the tangent is (&xi; &minus; x) (&part;f/&part;x) + (&eta; &minus; y) (&part;f/&part;y) = 0. If &rho;
+and (&alpha;, &beta;) are the radius and centre of curvature at (x, y), we find that
+q(&alpha; &minus; x) = &minus;p(1 + p²), q(&beta; &minus; y) = 1 + p², q²&rho;² = (1 + p²)³, where p, q denote
+dy/dx, d²y/dx² respectively. (See <span class="sc"><a href="#artlinks">Infinitesimal Calculus</a></span>.)</p>
+
+<p>In any given case we can, at all events in theory, eliminate x, y
+between the above equations for &alpha; &minus; x and &beta; &minus; y, and the equation
+of the curve. The resulting equation in (&alpha;, &beta;) represents the locus
+of the centre of curvature. This is the <i>evolute</i> of the curve.</p>
+
+<p>22. <i>Polar Coordinates.</i>&mdash;In plane geometry the distance of any
+point P from a fixed origin (or pole) O, and the inclination xOP of OP
+to a fixed line Ox, determine the point: r, the numerical measure
+of OP, the <i>radius vector</i>, and &theta;, the circular measure of xOP, the
+<i>inclination</i>, are called polar coordinates of P. The formulae x =
+r cos &theta;, y = r sin &theta; connect Cartesian and polar coordinates, and make
+transition from either system to the other easy. In polar coordinates
+the equations of a circle through O, and of a conic with O as focus,
+take the simple forms r = 2a cos (&theta; &minus; &alpha;), r{1 &minus; e cos (&theta; &minus; &alpha;)} = l. The
+use of polar coordinates is very convenient in discussing curves
+which have properties of symmetry akin to that of a regular polygon,
+such curves for instance as r = a cos m &theta;, with m integral, and also the
+curves called spirals, which have equations giving r as functions of
+&theta; itself, and not merely of sin &theta; and cos &theta;. In the geometry of
+motion under central forces the advantage of working with polar
+coordinates is great.</p>
+
+<p>23. <i>Trilinear and Areal Coordinates.</i>&mdash;Consider a fixed triangle
+ABC, and regard its sides as produced without limit. Denote, as
+in trigonometry, by a, b, c the positive numbers of units of a chosen
+scale contained in the lengths BC, CA, AB, by A, B, C the angles,
+and by &Delta; the area, of the triangle. We might, as in § 6, take CA,
+CB as axes of x and y, inclined at an angle C. Any point P (x, y)
+in the plane is at perpendicular distances y sin C and x sin C from
+CA and CB. Call these &beta; and &alpha; respectively. The signs of &beta; and &alpha;
+are those of y and x, <i>i.e.</i> &beta; is positive or negative according as P lies
+on the same side of CA as B does or the opposite, and similarly for &alpha;.
+An equation in (x, y) of any degree may, upon replacing in it x and y
+by &alpha; cosec C and &beta; cosec C, be written as one of the same degree in
+(&alpha;, &beta;). Now let &gamma; be the perpendicular distance of P from the third
+side AB, taken as positive or negative as P is on the C side of AB or
+not. The geometry of the figure tells us that a&alpha; + b&beta; + c&gamma; = 2&Delta;.
+By means of this relation in &alpha;, &beta;, &gamma; we can give an equation considered
+countless other forms, involving two or all of &alpha;, &beta;, &gamma;. In
+particular we may make it <i>homogeneous</i> in &alpha;, &beta;, &gamma;: to do this we
+have only to multiply the terms of every degree less than the highest
+present in the equation by a power of (a&alpha; + b&beta; + c&gamma;)/2&Delta; just sufficient
+to raise them, in each case, to the highest degree.</p>
+
+<p>We call (&alpha;, &beta;, &gamma;) <i>trilinear coordinates</i>, and an equation in them
+the trilinear equation of the locus represented. Trilinear equations
+are, as a rule, dealt with in their homogeneous forms. An advantage
+thus gained is that we need not mean by (&alpha;, &beta;, &gamma;) the actual measures
+of the perpendicular distances, but any properly signed numbers
+which have the same ratio two and two as these distances.</p>
+
+<p>In place of &alpha;, &beta;, &gamma; it is lawful to use, as coordinates specifying
+the position of a point in the plane of a triangle of reference ABC,
+any given multiples of these. For instance, we may use x = a&alpha;/2&Delta;,
+y = b&beta;/2&Delta;, z = c&gamma;/2&Delta;, the properly signed ratios of the triangular
+areas PBC, PCA, PAB to the triangular area ABC. These are called
+the <i>areal</i> coordinates of P. In areal coordinates the relation which
+enables us to make any equation homogeneous takes the simple
+form x + y + z = 1; and, as before, we need mean by x, y, z, in a
+homogeneous equation, only signed numbers in the right ratios.</p>
+
+<p>Straight lines and conics are represented in trilinear and in areal,
+because in Cartesian, coordinates by equations of the first and
+second degrees respectively, and these degrees are preserved when
+the equations are made homogeneous. What must be said about
+points infinitely far off in order to make universal the statement,
+to which there is no exception as long as finite distances alone are
+considered, that <i>every</i> homogeneous equation of the first degree
+represents a straight line? Let the point of areal coordinates
+(x&prime;, y&prime;, z&prime;) move infinitely far off, and mean by x, y, z finite quantities
+in the ratios which x&prime;, y&prime;, z&prime; tend to assume as they become infinite.
+The relation x&prime; + y&prime; + z&prime; = 1 gives that the limiting state of things
+tended to is expressed by x + y + z = 0. This particular equation of
+the first degree is satisfied by no point at a finite distance; but we
+see the propriety of saying that it has to be taken as satisfied by
+all the points conceived of as actually at infinity. Accordingly the
+special property of these points is expressed by saying that they lie
+on a special straight line, of which the areal equation is x + y + z = 0.
+In trilinear coordinates this <i>line at infinity</i> has for equation a&alpha; + b&beta; +
+c&gamma; = 0.</p>
+
+<p>On the one special line at infinity parallel lines are treated as
+meeting. There are on it two special (imaginary) points, the circular
+points at infinity of § 19, through which all circles pass in the same
+sense. In fact if S = O be one circle, in areal coordinates,
+S + (x + y + z)(lx + my + nz) = 0 may, by proper choice of l, m, n, be
+made any other; since the added terms are once lx + my + nz, and
+have the generality of any expression like a&prime;x + b&prime;y + c&prime; in Cartesian
+coordinates. Now these two circles intersect in the two points where
+either meets x + y + z = 0 as well as in two points on the radical axis
+lx + my + nz = 0.</p>
+
+<p>24. Let us consider the perpendicular distance of a point (&alpha;&prime;, &beta;&prime;, &gamma;&prime;)
+from a line l&alpha; + m&beta; + n&gamma;. We can take rectangular axes of Cartesian
+coordinates (for clearness as to equalities of angle it is best to
+choose an origin inside ABC), and refer to them, by putting expressions
+p &minus; x cos &theta; &minus; y sin &theta;, &amp;c., for &alpha; &amp;c.; we can then apply § 16 to
+get the perpendicular distance; and finally revert to the trilinear
+notation. The result is to find that the required distance is</p>
+
+<p class="center">(l&alpha;&prime; + m&beta;&prime; + n&gamma;&prime;) / {l, m, n},</p>
+
+<p class="noind">where {l, m, n}² = l² + m² + n² &minus; 2mn cos A &minus; 2nl cos B &minus; 2lm cos C.</p>
+
+<p>In areal coordinates the perpendicular distance from (x&prime;, y&prime;, z&prime;)
+<span class="pagenum"><a name="page717" id="page717"></a>717</span>
+to lx + my + nz = 0 is 2&Delta;(lx&prime; + my&prime; + nz&prime;)/{al, bm, cn}. In both cases
+the coordinates are of course actual values.</p>
+
+<p>Now let &xi;, &eta;, &zeta; be the perpendiculars on the line from the vertices
+A, B, C, <i>i.e.</i> the points (1, 0, 0), (0, 1, 0), (0, 0, 1), with signs in
+accord with a convention that oppositeness of sign implies distinction
+between one side of the line and the other. Three applications
+of the result above give</p>
+
+<p class="center">&xi;/l = 2&Delta; / {al, bm, cn} = &eta;/m = &zeta;/n;</p>
+
+<p class="noind">and we thus have the important fact that &xi;x&prime; + &eta;y&prime; + &zeta;z&prime; is the
+perpendicular distance between a point of areal coordinates (x&prime;y&prime;z&prime;)
+and a line on which the perpendiculars from A, B, C are &xi;, &eta;, &zeta;
+respectively. We have also that &xi;x + &eta;y + &zeta;z = 0 is the areal equation
+of the line on which the perpendiculars are &xi;, &eta;, &zeta;; and, by equating
+the two expressions for the perpendiculars from (x&prime;, y&prime;, z&prime;) on the
+line, that in all cases {a&xi;, b&eta;, c&zeta;}² = 4&Delta;².</p>
+
+<p>25. <i>Line-coordinates.</i> <i>Duality.</i>&mdash;A quite different order of ideas
+may be followed in applying analysis to geometry. The notion of a
+straight line specified may precede that of a point, and points may
+be dealt with as the intersections of lines. The specification of
+a line may be by means of coordinates, and that of a point by an
+equation, satisfied by the coordinates of lines which pass through it.
+Systems of <i>line-coordinates</i> will here be only briefly considered.
+Every such system is allied to some system of point-coordinates;
+and space will be saved by giving prominence to this fact, and not
+recommencing <i>ab initio</i>.</p>
+
+<p>Suppose that any particular system of point-coordinates, in which
+lx + my + nz = 0 may represent any straight line, is before us: notice
+that not only are trilinear and areal coordinates such systems, but
+Cartesian coordinates also, since we may write x/z, y/z for the
+Cartesian x, y, and multiply through by z. The line is exactly
+assigned if l, m, n, or their mutual ratios, are known. Call (l, m, n)
+the <i>coordinates</i> of the line. Now keep x, y, z constant, and let the
+coordinates of the line vary, but always so as to satisfy the equation.
+This equation, which we now write xl + ym + zn = 0, is satisfied by
+the coordinates of every line through a certain fixed point, and by
+those of no other line; it is the equation of that point in the line-coordinates
+l, m, n.</p>
+
+<p>Line-coordinates are also called <i>tangential</i> coordinates. A curve
+is the envelope of lines which touch it, as well as the locus of points
+which lie on it. A homogeneous equation of degree above the first
+in l, m, n is a relation connecting the coordinates of every line which
+touches some curve, and represents that curve, regarded as an
+envelope. For instance, the condition that the line of coordinates
+(l, m, n), <i>i.e.</i> the line of which the allied point-coordinate equation
+is lx + my + nz = 0, may touch a conic (a, b, c, f, g, h) (x, y, z)² = 0,
+is readily found to be of the form (A, B, C, F, G, H) (l, m, n)² = 0,
+<i>i.e.</i> to be of the second degree in the line-coordinates. It is not hard
+to show that the <i>general</i> equation of the second degree in l, m, n
+thus represents a conic; but the degenerate conics of line-coordinates
+are not line-pairs, as in point-coordinates, but point-pairs.</p>
+
+<p>The degree of the point-coordinate equation of a curve is the
+<i>order</i> of the curve, the number of points in which it cuts a straight
+line. That of the line-coordinate equation is its <i>class</i>, the number
+of tangents to it from a point. The order and class of a curve are
+generally different when either exceeds two.</p>
+
+<p>26. The system of line-coordinates allied to the areal system of
+point-coordinates has special interest.</p>
+
+<p>The l, m, n of this system are the perpendiculars &xi;, &eta;, &zeta; of § 24;
+and x&prime;&xi; + y&prime;&eta; + z&prime;&zeta; = 0 is the equation of the point of areal coordinates
+(x&prime;, y&prime;, z&prime;), <i>i.e.</i> is a relation which the perpendiculars from the vertices
+of the triangle of reference on every line through the point, but no
+other line, satisfy. Notice that a non-homogeneous equation of the
+first degree in &xi;, &eta;, &zeta; does not, as a homogeneous one does, represent
+a point, but a circle. In fact x&prime;&xi; + y&prime;&eta; + z&prime;&zeta; = R expresses the constancy
+of the perpendicular distance of the fixed point x&prime;&xi; + y&prime;&eta; +
+z&prime;&zeta; = 0 from the variable line (&xi;, &eta;, &zeta;), <i>i.e.</i> the fact that (&xi;, &eta;, &zeta;) touches
+a circle with the fixed point for centre. The relation in any &xi;, &eta;, &zeta;
+which enables us to make an equation homogeneous is not linear,
+as in point-coordinates, but quadratic, viz. it is the relation {a&xi;, b&eta;,
+c&zeta;}² = 4&Delta;² of § 24. Accordingly the homogeneous equation of the
+above circle is</p>
+
+<p class="center">4&Delta;² (x&prime;&xi; + y&prime;&eta; + z&prime;&zeta;)² = R² {a&xi;, b&eta;, c&zeta;}².</p>
+
+<p>Every circle has an equation of this form in the present system of
+line-coordinates. Notice that the equation of any circle is satisfied
+by those coordinates of lines which satisfy both x&prime;&xi; + y&prime;&eta; + z&prime;&zeta; = 0,
+the equation of its centre, and {a&xi;, b&eta;, c&zeta;}² = 0. This last equation,
+of which the left-hand side satisfies the condition for breaking up
+into two factors, represents the two imaginary circular points at
+infinity, through which all circles and their asymptotes pass.</p>
+
+<p>There is strict duality in descriptive geometry between point-line-locus
+and line-point-envelope theorems. But in metrical geometry
+duality is encumbered by the fact that there is in a plane one special
+line only, associated with distance, while of special points, associated
+with direction, there are two: moreover the line is real, and the
+points both imaginary.</p>
+
+<p class="pt2 center">II. <i>Solid Analytical Geometry.</i></p>
+
+<p>27. Any point in space may be specified by three coordinates.
+We consider three fixed planes of reference, and generally, as in all
+that follows, three which are at right angles two and two. They
+intersect, two and two, in lines x&prime;Ox, y&prime;Oy, z&prime;Oz, called the axes
+of x, y, z respectively, and divide all space into eight parts called
+octants. If from any point P in space we draw PN parallel to
+zOz&prime; to meet the plane xOy in N, and then from N draw NM parallel
+to yOy&prime; to meet x&prime;Ox in M, the coordinates (x, y, z) of P are the
+numerical measures of OM, MN, NP; in the case of rectangular
+coordinates these are the perpendicular distances of P from the three
+planes of reference. The sign of each coordinate is positive or
+negative as P lies on one side or the other of the corresponding
+plane. In the octant delineated the signs are taken all positive.</p>
+
+<table class="nobctr" style="clear: both;" summary="Illustration">
+<tr><td class="figcenter" colspan="2"><img style="width:513px; height:254px" src="images/img717a.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 57.</span></td>
+<td class="caption"><span class="sc">Fig. 58.</span></td></tr></table>
+
+<p>In fig. 57 the delineation is on a plane of the paper taken parallel
+to the plane zOx, the points of a solid figure being projected on that
+plane by parallels to some chosen line through O in the positive
+octant. Sometimes it is clearer to delineate, as in fig. 58, by projection
+parallel to that line in the octant which is equally inclined to
+Ox, Oy, Oz upon a plane of the paper perpendicular to it. It is
+possible by parallel projection to delineate equal scales along Ox,
+Oy, Oz by scales having any ratios we like along lines in a plane
+having any mutual inclinations we like.</p>
+
+<table class="flt" style="float: right; width: 375px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:325px; height:293px" src="images/img717b.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 59.</span></td></tr></table>
+
+<p>For the delineation of a surface of simple form it frequently
+suffices to delineate the sections by the coordinate planes; and, in
+particular, when the surface has symmetry about each coordinate
+plane, to delineate the
+quarter-sections belonging
+to a single
+octant. Thus fig. 59
+conveniently represents
+an octant of the
+wave surface, which
+cuts each coordinate
+plane in a circle and
+an ellipse. Or we may
+delineate a series of
+contour lines, <i>i.e.</i> sections
+by planes parallel
+to xOy, or some other
+chosen plane; of course
+other sections may be
+indicated too for
+greater clearness. For
+the delineation of a
+curve a good method
+is to represent, as
+above, a series of points
+P thereof, each accompanied by its ordinate PN, which serves to
+refer it to the plane of xy. The employment of stereographic
+projection is also interesting.</p>
+
+<p>28. In plane geometry, reckoning the line as a curve of the first
+order, we have only the point and the curve. In solid geometry,
+reckoning a line as a curve of the first order, and the plane as a surface
+of the first order, we have the point, the curve and the surface;
+but the increase of complexity is far greater than would hence at
+first sight appear. In plane geometry a curve is considered in
+connexion with lines (its tangents); but in solid geometry the curve
+is considered in connexion with lines and planes (its tangents and
+osculating planes), and the surface also in connexion with lines and
+planes (its tangent lines and tangent planes); there are surfaces
+arising out of the line&mdash;cones, skew surfaces, developables, doubly
+and triply infinite systems of lines, and whole classes of theories
+which have nothing analogous to them in plane geometry: it is thus
+a very small part indeed of the subject which can be even referred
+to in the present article.</p>
+
+<p>In the case of a surface we have between the coordinates (x, y, z)
+a single, or say a onefold relation, which can be represented by a
+single relation &fnof;(x, y, z) = 0; or we may consider the coordinates
+expressed each of them as a given function of two variable parameters
+p, q; the form z = &fnof;(x, y) is a particular case of each of these
+modes of representation; in other words, we have in the first mode
+&fnof;(x, y, z) = z &minus; &fnof;(x, y), and in the second mode x = p, y = q for the
+expression of two of the coordinates in terms of the parameters.</p>
+
+<p><span class="pagenum"><a name="page718" id="page718"></a>718</span></p>
+
+<p>In the case of a curve we have between the coordinates (x, y, z) a
+twofold relation: two equations &fnof;(x, y, z) = 0, &phi;(x, y, z) = 0 give
+such a relation; <i>i.e.</i> the curve is here considered as the intersection
+of two surfaces (but the curve is not always the complete intersection
+of two surfaces, and there are hence difficulties); or, again, the coordinates
+may be given each of them as a function of a single variable
+parameter. The form y = &phi;(x), z = &psi;(x), where two of the coordinates
+are given in terms of the third, is a particular case of each of these
+modes of representation.</p>
+
+<p>29. The remarks under plane geometry as to descriptive and
+metrical propositions, and as to the non-metrical character of the
+method of coordinates when used for the proof of a descriptive
+proposition, apply also to solid geometry; and they might be
+illustrated in like manner by the instance of the theorem of the radical
+centre of four spheres. The proof is obtained from the consideration
+that S and S&prime; being each of them a function of the form x² + y² + z² +
+ax + by + cz + d, the difference S-S&prime; is a mere linear function of the
+coordinates, and consequently that S-S&prime; = 0 is the equation of the
+plane containing the circle of intersection of the two spheres S = 0
+and S&prime; = 0.</p>
+
+<table class="flt" style="float: left; width: 200px;" summary="Illustration">
+<tr><td class="figleft1"><img style="width:155px; height:298px" src="images/img718.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 60.</span></td></tr></table>
+
+<p>30. <i>Metrical Theory.</i>&mdash;The foundation in solid geometry of the
+metrical theory is in fact the before-mentioned theorem that if a
+finite right line PQ be projected upon any other line OO&prime; by lines
+perpendicular to OO&prime;, then the length of the
+projection P&prime;Q&prime; is equal to the length of PQ
+into the cosine of its inclination to P&prime;Q&prime;&mdash;or
+(in the form in which it is now convenient
+to state the theorem) the perpendicular
+distance P&prime;Q&prime; of two parallel planes is equal
+to the inclined distance PQ into the cosine
+of the inclination. The principle of § 16,
+that the algebraical sum of the projections of
+the sides of any closed polygon on any line is
+zero, or that the two sets of sides of the
+polygon which connect a vertex A and a
+vertex B have the same sum of projections
+on the line, in sign and magnitude, as we pass
+from A to B, is applicable when the sides do
+not all lie in one plane.</p>
+
+<p>31. Consider the skew quadrilateral QMNP,
+the sides QM, MN, NP being respectively
+parallel to the three rectangular axes Ox,
+Oy, Oz; let the lengths of these sides be
+&xi;, &eta;, &zeta;, and that of the side QP be = &rho;; and
+let the cosines of the inclinations (or say the cosine-inclinations) of
+&rho; to the three axes be &alpha;, &beta;, &gamma;; then projecting successively on
+the three sides and on QP we have</p>
+
+<p class="center">&xi;, &eta;, &zeta; = &rho;&alpha;, &rho;&beta;, &rho;&gamma;,</p>
+
+<p class="noind">and</p>
+
+<p class="center">&rho; = &alpha;&xi; + &beta;&eta; + &gamma;&zeta;,</p>
+
+<p class="noind">whence &rho;² = &xi;² + &eta;² + &zeta;², which is the relation between a distance &rho;
+and its projections &xi;, &eta;, &zeta; upon three rectangular axes. And from
+the same equations we obtain &alpha;² + &beta;² + &gamma;² = 1, which is a relation connecting
+the cosine-inclinations of a line to three rectangular axes.</p>
+
+<p>Suppose we have through Q any other line QT, and let the cosine-inclinations
+of this to the axes be &alpha;&prime;, &beta;&prime;, &gamma;&prime;, and &delta; be its cosine-inclination
+to QP; also let &rho; be the length of the projection of QP
+upon QT; then projecting on QT we have</p>
+
+<p class="center">&rho; = &alpha;&prime;&xi; + &beta;&prime;&eta; + &gamma;&prime;&zeta; = &rho;&delta;.</p>
+
+<p>And in the last equation substituting for &xi;, &eta;, &zeta; their values &rho;&alpha;,
+&rho;&beta;, &rho;&gamma; we find</p>
+
+<p class="center">&delta; = &alpha;&alpha;&prime; + &beta;&beta;&prime; + &gamma;&gamma;&prime;,</p>
+
+<p class="noind">which is an expression for the mutual cosine-inclination of two
+lines, the cosine-inclinations of which to the axes are &alpha;, &beta;, &gamma; and
+&alpha;&prime;, &beta;&prime;, &gamma;&prime; respectively. We have of course &alpha;² + &beta;² + &gamma;² = 1 and
+&alpha;&prime;² + &beta;&prime;² + &gamma;&prime;² = 1; and hence also</p>
+
+<p class="center">1 &minus; &delta;² = (&alpha;² + &beta;² + &gamma;²)(&alpha;&prime;² + &beta;&prime;² + &gamma;&prime;²) &minus; (&alpha;&alpha;&prime; + &beta;&beta;&prime; + &gamma;&gamma;&prime;)²,<br />
+
+= (&beta;&gamma;&prime; &minus; &beta;&prime;&gamma;)² + (&gamma;&alpha;&prime; &minus; &gamma;&prime;&alpha;)² + (&alpha;&beta;&prime; &minus; &alpha;&prime;&beta;)²;</p>
+
+<p class="noind">so that the sine of the inclination can only be expressed as a square
+root. These formulae are the foundation of spherical trigonometry.</p>
+
+<p>32. <i>Straight Lines, Planes and Spheres.</i>&mdash;The foregoing formulae
+give at once the equations of these loci.</p>
+
+<p>For first, taking Q to be a fixed point, coordinates (a, b, c), and
+the cosine-inclinations (&alpha;, &beta;, &gamma;) to be constant, then P will be a
+point in the line through Q in the direction thus determined; or,
+taking (x, y, z) for its coordinates, these will be the current coordinates
+of a point in the line. The values of &xi;, &eta;, &zeta; then are
+x &minus; a, y &minus; b, z &minus; c, and we thus have</p>
+
+<table class="math0" summary="math">
+<tr><td>x &minus; a</td>
+<td rowspan="2">=</td> <td>y &minus; b</td>
+<td rowspan="2">=</td> <td>z &minus; c</td>
+<td rowspan="2">(= &rho;),</td></tr>
+<tr><td class="denom">&alpha;</td> <td class="denom">&beta;</td>
+<td class="denom">&gamma;</td></tr></table>
+
+<p class="noind">which (omitting the last equation, = &rho;) are the equations of the line
+through the point (a, b, c), the cosine-inclinations to the axes being
+&alpha;, &beta;, &gamma;, and these quantities being connected by the relation
+&alpha;² + &beta;² + &gamma;² = 1. This equation may be omitted, and then &alpha;, &beta;, &gamma;,
+instead of being equal, will only be proportional, to the cosine-inclinations.</p>
+
+<p>Using the last equation, and writing</p>
+
+<p class="center">x, y, z = a + &alpha;&rho;, b + &beta;&rho;, c + &gamma;&rho;,</p>
+
+<p class="noind">these are expressions for the current coordinates in terms of a
+parameter &rho;, which is in fact the distance from the fixed point
+(a, b, c).</p>
+
+<p>It is easy to see that, if the coordinates (x, y, z) are connected by
+any two linear equations, these equations can always be brought
+into the foregoing form, and hence that the two linear equations
+represent a line.</p>
+
+<p>Secondly, taking for greater simplicity the point Q to be coincident
+with the origin, and &alpha;&prime;, &beta;&prime;, &gamma;&prime;, p to be constant, then p is the perpendicular
+distance of a plane from the origin, and &alpha;&prime;, &beta;&prime;, &gamma;&prime; are the cosine-inclinations
+of this distance to the axes (&alpha;&prime;² + &beta;&prime;² + &gamma;&prime;² = 1). P is
+any point in this plane, and taking its coordinates to be (x, y, z) then
+(&xi;, &eta;, &zeta;) are = (x, y, z), and the foregoing equation p = &alpha;&prime;&xi; + &beta;&prime;&eta; + &gamma;&prime;&zeta;
+becomes</p>
+
+<p class="center">&alpha;&prime;x + &beta;&prime;y + &gamma;&prime;z = p,</p>
+
+<p class="noind">which is the equation of the plane in question.</p>
+
+<p>If, more generally, Q is not coincident with the origin, then,
+taking its coordinates to be (a, b, c), and writing p<span class="su">1</span> instead of p, the
+equation is</p>
+
+<p class="center">&alpha;&prime; (x &minus; a) + &beta;&prime; (y &minus; b) + &gamma;&prime; (z &minus; c) = p<span class="su">1</span>;</p>
+
+<p class="noind">and we thence have p<span class="su">1</span> = p &minus; (a&alpha;&prime; + b&beta;&prime; + c&gamma;&prime;), which is an expression
+for the perpendicular distance of the point (a, b, c) from the plane
+in question.</p>
+
+<p>It is obvious that any linear equation Ax + By + Cz + D = O between
+the coordinates can always be brought into the foregoing form,
+and hence that such an equation represents a plane.</p>
+
+<p>Thirdly, supposing Q to be a fixed point, coordinates (a, b, c),
+and the distance QP = &rho;, to be constant, say this is = d, then, as
+before, the values of &xi;, &eta;, &zeta; are x &minus; a, y &minus; b, z &minus; c, and the equation
+&xi;² + &eta;² + &zeta;² = &rho;² becomes</p>
+
+<p class="center">(x &minus; a)² + (y &minus; b)² + (z &minus; c)² = d²,</p>
+
+<p class="noind">which is the equation of the sphere, coordinates of the centre = (a, b, c),
+and radius = d.</p>
+
+<p>A quadric equation wherein the terms of the second order are
+x² + y² + z², viz. an equation</p>
+
+<p class="center">x² + y² + z² + Ax + By + Cz + D = 0,</p>
+
+<p class="noind">can always, it is clear, be brought into the foregoing form; and it
+thus appears that this is the equation of a sphere, coordinates of
+the centre &minus;½A, &minus;½B, &minus;½C, and squared radius = ¼(A² + B² + C²) &minus; D.</p>
+
+<p>33. <i>Cylinders, Cones, ruled Surfaces.</i>&mdash;If the two equations of a
+straight line involve a parameter to which any value may be given,
+we have a singly infinite system of lines. They cover a surface, and
+the equation of the surface is obtained by eliminating the parameter
+between the two equations.</p>
+
+<p>If the lines all pass through a given point, then the surface is a
+cone; and, in particular, if the lines are all parallel to a given line,
+then the surface is a cylinder.</p>
+
+<p>Beginning with this last case, suppose the lines are parallel to
+the line x = mz, y = nz, the equations of a line of the system are
+x = mz + a, y = nz + b,&mdash;where a, b are supposed to be functions of
+the variable parameter, or, what is the same thing, there is between
+them a relation &fnof;(a, b) = 0: we have a = x &minus; mz, b = y &minus; nz, and the
+result of the elimination of the parameter therefore is &fnof;(x &minus; mz,
+y &minus; nz) = 0, which is thus the general equation of the cylinder the
+generating lines whereof are parallel to the line x = mz, y = nz. The
+equation of the section by the plane z = 0 is &fnof;(x, y) = 0, and conversely
+if the cylinder be determined by means of its curve of intersection
+with the plane z = 0, then, taking the equation of this curve to be
+&fnof;(x, y) = 0, the equation of the cylinder is &fnof;(x &minus; mz, y &minus; nz) = 0. Thus,
+if the curve of intersection be the circle (x &minus; &alpha;)² + (y &minus; &beta;)² = &gamma;², we
+have (x &minus; mz &minus; &alpha;)² + (y &minus; nz &minus; &beta;)² = &gamma;² as the equation of an oblique
+cylinder on this base, and thus also (x &minus; &alpha;)² + (y &minus; &beta;)² = &gamma;² as the
+equation of the right cylinder.</p>
+
+<p>If the lines all pass through a given point (a, b, c), then the equations
+of a line are x &minus; a = &alpha;(z &minus; c), y &minus; b = &beta;(z &minus; c), where &alpha;, &beta; are
+functions of the variable parameter, or, what is the same thing,
+there exists between them an equation &fnof;(&alpha;, &beta;) = 0; the elimination
+of the parameter gives, therefore, &fnof;[(x &minus; a)/(x &minus; c&prime;), (y &minus; b)/(z &minus; c)] = 0; and this
+equation, or, what is the same thing, any homogeneous equation
+&fnof;(x &minus; a, y &minus; b, z &minus; c) = 0, or, taking f to be a rational and integral
+function of the order n, say (*)(x &minus; a, y &minus; b, z &minus; c)<span class="sp">n</span> = 0, is the general
+equation of the cone having the point (a, b, c) for its vertex. Taking
+the vertex to be at the origin, the equation is (*)(x, y, z)<span class="sp">n</span> = 0; and,
+in particular, (*)(x, y, z)² = 0 is the equation of a cone of the second
+order, or quadricone, having the origin for its vertex.</p>
+
+<p>34. In the general case of a singly infinite system of lines, the
+locus is a ruled surface (or <i>regulus</i>). Now, when a line is changing
+its position in space, it may be looked upon as in a state of turning
+about some point in itself, while that point is, as a rule, in a state of
+moving out of the plane in which the turning takes place. If instantaneously
+it is only in a state of turning, it is usual, though not
+strictly accurate, to say that it intersects its consecutive position.
+A regulus such that consecutive lines on it do not intersect, in this
+sense, is called a skew surface, or <i>scroll</i>; one on which they do is
+called a developable surface or <i>torse</i>.</p>
+
+<p>Suppose, for instance, that the equations of a line (depending on
+<span class="pagenum"><a name="page719" id="page719"></a>719</span>
+the variable parameter &theta;) are x/a + y/c = &theta;
+(1 + y/b), x/a &minus; z/c = (1/&theta;)(1 &minus; y/b);
+then, eliminating &theta; we have x²/a² &minus; z²/c² = 1 &minus; y²/b², or say, x²/a² + y²/b² &minus; z²/c² = 1,
+the equation of a quadric surface, afterwards called the hyperboloid
+of one sheet; this surface is consequently a scroll. It is to be remarked
+that we have upon the surface a second singly infinite
+series of lines; the equations of a line of this second system (depending
+on the variable parameter &phi;) are</p>
+
+<table class="math0" summary="math">
+<tr><td>x</td>
+<td rowspan="2">+</td> <td>z</td>
+<td rowspan="2">= &phi; <span class="f150">(</span> 1 &minus;</td> <td>y</td>
+<td rowspan="2"><span class="f150">)</span>, &emsp;</td> <td>x</td>
+<td rowspan="2">&minus;</td> <td>z</td>
+<td rowspan="2">=</td> <td>1</td>
+<td rowspan="2"><span class="f150">(</span> 1 +</td> <td>y</td>
+<td rowspan="2"><span class="f150">)</span>.</td></tr>
+<tr><td class="denom">a</td> <td class="denom">c</td>
+<td class="denom">b</td> <td class="denom">a</td>
+<td class="denom">c</td> <td class="denom">&phi;</td>
+<td class="denom">b</td></tr></table>
+
+<p class="noind">It is easily shown that any line of the one system intersects every
+line of the other system.</p>
+
+<p>Considering any curve (of double curvature) whatever, the tangent
+lines of the curve form a singly infinite system of lines, each line
+intersecting the consecutive line of the system,&mdash;that is, they form
+a developable, or torse; the curve and torse are thus inseparably
+connected together, forming a single geometrical figure. An osculating
+plane of the curve (see § 38 below) is a tangent plane of the torse
+all along a generating line.</p>
+
+<p>35. <i>Transformation of Coordinates.</i>&mdash;There is no difficulty in
+changing the origin, and it is for brevity assumed that the origin
+remains unaltered. We have, then, two sets of rectangular axes,
+Ox, Oy, Oz, and Ox<span class="su">1</span>, Oy<span class="su">1</span>, Ozx<span class="su">1</span>, the mutual cosine-inclinations being
+shown by the diagram&mdash;</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcc bb">&nbsp;</td> <td class="tcc lb rb">x</td> <td class="tcc rb">y</td> <td class="tcc rb">z</td></tr>
+
+<tr><td class="tcc bb">x<span class="su">1</span></td> <td class="tcc allb">&alpha;</td> <td class="tcc allb">&beta;</td> <td class="tcc allb">&gamma;</td></tr>
+
+<tr><td class="tcc bb">y<span class="su">1</span></td> <td class="tcc allb">&alpha;</td> <td class="tcc allb">&beta;&prime;</td> <td class="tcc allb">&gamma;&prime;</td></tr>
+
+<tr><td class="tcc bb">z<span class="su">1</span></td> <td class="tcc allb">&alpha;&Prime;</td> <td class="tcc allb">&beta;&Prime;</td> <td class="tcc allb">&gamma;&Prime;</td></tr>
+
+</table>
+
+<p class="noind">that is, &alpha;, &beta;, &gamma; are the cosine-inclinations of
+Ox<span class="su">1</span> to Ox, Oy, Oz;
+&alpha;&prime;, &beta;&prime;, &gamma;&prime; those of Oy<span class="su">1</span>, &amp;c.</p>
+
+<p>And this diagram gives also the linear expressions of the coordinates
+(x<span class="su">1</span>, y<span class="su">1</span>, z<span class="su">1</span>) or (x, y, z) of either set in terms of those of the
+other set; we thus have</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcl">x<span class="su">1</span> = &alpha; x + &beta; y + &gamma; z,</td> <td class="tcl">x = &alpha;x<span class="su">1</span> + &alpha;&prime;y<span class="su">1</span> + &alpha;&Prime;z<span class="su">1</span>,</td></tr>
+
+<tr><td class="tcl">y<span class="su">1</span> = &alpha;&prime;x + &beta;&prime;y + &gamma;&prime;z,</td> <td class="tcl">y = &beta;x<span class="su">1</span> + &beta;&prime;y<span class="su">1</span> + &beta;&Prime;z<span class="su">1</span>,</td></tr>
+
+<tr><td class="tcl">z<span class="su">1</span> = &alpha;&Prime;x + &beta;&Prime;y + &gamma;&Prime;z,</td> <td class="tcl">z = &gamma;x<span class="su">1</span> + &gamma;&prime;y<span class="su">1</span> + &gamma;&Prime;z<span class="su">1</span>,</td></tr>
+</table>
+
+<p class="noind">which are obtained by projection, as above explained. Each of
+these equations is, in fact, nothing else than the before-mentioned
+equation p = &alpha;&prime;&xi; + &beta;&prime;&eta; + &gamma;&prime;&zeta;, adapted to the problem in hand.</p>
+
+<p>But we have to consider the relations between the nine coefficients.
+By what precedes, or by the consideration that we must have
+identically x² + y² + z² = x<span class="su">1</span>² + y<span class="su">1</span>² + z<span class="su">1</span>², it appears that these satisfy
+the relations&mdash;</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcl">&alpha;²</td> <td class="tcl">+ &beta;²</td> <td class="tcl">+ &gamma;²</td> <td class="tcl">= 1,&emsp;&emsp;</td> <td class="tcl">&alpha;² +</td> <td class="tcl">&alpha;&prime;²</td> <td class="tcl">+ &alpha;&Prime;²</td> <td class="tcl">= 1,</td></tr>
+<tr><td class="tcl">&alpha;&prime;²</td> <td class="tcl">+ &beta;&prime;²</td> <td class="tcl">+ &gamma;&prime;²</td> <td class="tcl">= 1,&emsp;&emsp;</td> <td class="tcl">&beta;²</td> <td class="tcl">+ &beta;&prime;²</td> <td class="tcl">+ &beta;&Prime;²</td> <td class="tcl">= 1,</td></tr>
+<tr><td class="tcl">&alpha;&Prime;²</td> <td class="tcl">+ &beta;&Prime;²</td> <td class="tcl">+ &gamma;&Prime;²</td> <td class="tcl">= 1,&emsp;&emsp;</td> <td class="tcl">&gamma;²</td> <td class="tcl">+ &gamma;&prime;²</td> <td class="tcl">+ &gamma;&Prime;²</td> <td class="tcl">= 1,</td></tr>
+<tr><td class="tcl">&alpha;&prime;a&Prime;</td> <td class="tcl">+ &beta;&prime;&beta;&Prime;</td> <td class="tcl">+ &gamma;&prime;&gamma;&Prime;</td> <td class="tcl">= 0,&emsp;&emsp;</td> <td class="tcl">&beta;&gamma;</td> <td class="tcl">+&beta;&prime;&gamma;&prime;</td> <td class="tcl">+ &beta;&Prime;&gamma;&Prime;</td> <td class="tcl">= 0,</td></tr>
+<tr><td class="tcl">&alpha;&Prime;&alpha;</td> <td class="tcl">+ &beta;&Prime;&beta;</td> <td class="tcl">+ &gamma;&Prime;&gamma;</td> <td class="tcl">= 0,&emsp;&emsp;</td> <td class="tcl">&gamma;&alpha;</td> <td class="tcl">+ &gamma;&prime;&alpha;&prime;</td> <td class="tcl">+ &gamma;&Prime;&alpha;&Prime;</td> <td class="tcl">= 0,</td></tr>
+<tr><td class="tcl">&alpha;&alpha;&prime;</td> <td class="tcl">+ &beta;&beta;&prime;</td> <td class="tcl">+ &gamma;&gamma;&prime;</td> <td class="tcl">= 0,&emsp;&emsp;</td> <td class="tcl">&alpha;&beta;</td> <td class="tcl">+&alpha;&prime;&beta;&prime;</td> <td class="tcl">+ &alpha;&Prime;&beta;&Prime;</td> <td class="tcl">= 0,</td></tr>
+</table>
+
+<p class="noind">either set of six equations being implied in the other set.</p>
+
+<p>It follows that the square of the determinant</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcc lb">&alpha;,</td> <td class="tcc">&beta;,</td> <td class="tcc rb">&gamma;</td></tr>
+
+<tr><td class="tcc lb">&alpha;&prime;,</td> <td class="tcc">&beta;&prime;,</td> <td class="tcc rb">&gamma;&prime;</td></tr>
+
+<tr><td class="tcc lb">&alpha;&Prime;,</td> <td class="tcc">&beta;&Prime;,</td> <td class="tcc rb">&gamma;&Prime;</td></tr>
+</table>
+
+<p class="noind">is = 1; and hence that the determinant itself is = ±1. The distinction
+of the two cases is an important one: if the determinant is
+= + 1, then the axes Ox<span class="su">1</span>, Oy<span class="su">1</span>, Oz<span class="su">1</span> are such that they can by a
+rotation about O be brought to coincide with Ox, Oy, Oz respectively;
+if it is = &minus;1, then they cannot. But in the latter case, by
+measuring x<span class="su">1</span>, y<span class="su">1</span>, z<span class="su">1</span> in the opposite directions we change the signs of
+all the coefficients and so make the determinant to be = + 1; hence
+the former case need alone be considered, and it is accordingly
+assumed that the determinant is = +1. This being so, it is found
+that we have the equality &alpha; = &beta;&prime;&gamma;&Prime; &minus; &beta;&Prime;&gamma;&prime;, and eight like ones,
+obtained from this by cyclical interchanges of the letters &alpha;, &beta;, &gamma;,
+and of unaccented, singly and doubly accented letters.</p>
+
+<p>36. The nine cosine-inclinations above are, as has been seen,
+connected by six equations. It ought then to be possible to express
+them all in terms of three parameters. An elegant means of doing
+this has been given by Rodrigues, who has shown that the tabular
+expression of the formulae of transformation may be written</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcc bb">&nbsp;</td> <td class="tcc lb rb">x</td> <td class="tcc rb">y</td> <td class="tcc rb">z</td></tr>
+
+<tr><td class="tcc bb">x<span class="su">1</span></td> <td class="tcc allb">1 + &lambda;² &minus; &mu;² &minus; &nu;²</td> <td class="tcc allb">2(&lambda;&mu; &minus; &nu;)</td> <td class="tcc allb">2(&nu;&lambda; + &mu;)</td></tr>
+
+<tr><td class="tcc bb">y<span class="su">1</span></td> <td class="tcc allb">2(&lambda;&mu; + &nu;)</td> <td class="tcc allb">1 &minus; &lambda;² + &mu;² &minus; &nu;²</td> <td class="tcc allb">2(&mu;&nu; + &lambda;)</td></tr>
+
+<tr><td class="tcc bb">z<span class="su">1</span></td> <td class="tcc allb">2(&nu;&lambda; &minus; &mu;)</td> <td class="tcc allb"> 2(&mu;&nu; + &lambda;)</td> <td class="tcc allb">1 &minus; &lambda;² &minus; &mu;² + &nu;²</td></tr>
+
+<tr><td class="tcc" colspan="4">÷ (1 + &lambda;² + &mu;² + &nu;²),</td></tr>
+</table>
+
+<p class="noind">the meaning being that the coefficients in the transformation are
+fractions, with numerators expressed as in the table, and the common
+denominator.</p>
+
+<p>37. <i>The Species of Quadric Surfaces</i>.&mdash;Surfaces represented by
+equations of the second degree are called <i>quadric</i> surfaces. Quadric
+surfaces are either <i>proper</i> or <i>special</i>. The special ones arise when the
+coefficients in the general equation are limited to satisfy certain
+special equations; they comprise (1) plane-pairs, including in
+particular one plane twice repeated, and (2) cones, including in
+particular cylinders; there is but one form of cone, but cylinders
+may be elliptic, parabolic or hyperbolic.</p>
+
+<p>A discussion of the general equation of the second degree shows
+that the <i>proper</i> quadric surfaces are of five kinds, represented
+respectively, when referred to the most convenient axes of reference,
+by equations of the five types (a and b positive):</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcl">(1)&emsp;&emsp;&emsp;</td> <td class="tcl">z = x²/2a + y²/2b, elliptic paraboloid.</td></tr>
+
+<tr><td class="tcl">(2)&emsp;&emsp;&emsp;</td> <td class="tcl">z = x²/2a &minus; y²/2b, hyperbolic paraboloid.</td></tr>
+
+<tr><td class="tcl">(3)&emsp;&emsp;&emsp;</td> <td class="tcl">x²/a² + y²/b² + z²/c² = 1, ellipsoid.</td></tr>
+
+<tr><td class="tcl">(4)&emsp;&emsp;&emsp;</td> <td class="tcl">x²/a² + y²/b² &minus; z²/c² = 1, hyperboloid of one sheet.</td></tr>
+
+<tr><td class="tcl">(5)&emsp;&emsp;&emsp;</td> <td class="tcl">x²/a² + y²/b² &minus; z²/c² = &minus;1, hyperboloid of two sheets.</td></tr>
+</table>
+
+<table class="flt" style="float: right; width: 270px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:221px; height:217px" src="images/img719a.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig</span>. 61.</td></tr></table>
+
+<p>It is at once seen that these are distinct surfaces; and the equations
+also show very readily the
+general form and mode of generation
+of the several surfaces.</p>
+
+<p>In the elliptic paraboloid (fig. 61)
+the sections by the planes of zx and
+zy are the parabolas</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">z =</td> <td>x²</td>
+<td rowspan="2">,&emsp; z =</td> <td>y²</td>
+<td rowspan="2">,</td></tr>
+<tr><td class="denom">2a</td> <td class="denom">2b</td></tr></table>
+
+<p class="noind">having the common axes Oz; and
+the section by any plane z = &gamma;
+parallel to that of xy is the ellipse</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&gamma; =</td> <td>x²</td>
+<td rowspan="2">+</td> <td>y²</td>
+<td rowspan="2">;</td></tr>
+<tr><td class="denom">2a</td> <td class="denom">2b</td></tr></table>
+
+<p class="noind">so that the surface is generated by
+a variable ellipse moving parallel to itself along the parabolas as
+directrices.</p>
+
+<table class="nobctr" style="clear: both;" summary="Illustration">
+<tr><td class="figcenter" colspan="2"><img style="width:515px; height:232px" src="images/img719b.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig</span>. 62.</td>
+<td class="caption"><span class="sc">Fig</span>. 63.</td></tr></table>
+
+<table class="flt" style="float: right; width: 350px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:300px; height:249px" src="images/img719c.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig</span>. 64.</td></tr></table>
+
+<p>In the hyperbolic paraboloid (figs. 62 and 63) the sections by the
+planes of zx, zy are the parabolas z = x²/2a, z = &minus; y²/2b, having the opposite
+axes Oz, Oz&prime;, and the section by a plane z = &gamma; parallel to that of
+xy is the hyperbola &gamma; = x²/2a &minus; y²/2b, which has its transverse axis parallel
+to Ox or Oy according as &gamma; is positive or negative. The surface is thus
+generated by a variable hyperbola moving parallel to itself along
+the parabolas as directrices. The form is best seen from fig. 63,
+which represents the sections
+by planes parallel to
+the plane of xy, or say the
+contour lines; the continuous
+lines are the sections
+above the plane of
+xy, and the dotted lines
+the sections below this
+plane. The form is, in
+fact, that of a saddle.</p>
+
+<p>In the ellipsoid (fig. 64)
+the sections by the planes
+of zx, zy, and xy are each
+of them an ellipse, and the
+section by any parallel
+plane is also an ellipse.
+The surface may be considered
+as generated by
+an ellipse moving parallel to itself along two ellipses as directrices.</p>
+
+<p><span class="pagenum"><a name="page720" id="page720"></a>720</span></p>
+
+<p>In the hyperboloid of one sheet (fig. 65), the sections by the planes
+of zx, zy are the hyperbolas</p>
+
+<table class="math0" summary="math">
+<tr><td>x²</td>
+<td rowspan="2">&minus;</td> <td>z²</td>
+<td rowspan="2">= 1,&emsp;</td> <td>y²</td>
+<td rowspan="2">&minus;</td> <td>z²</td>
+<td rowspan="2">= 1,</td></tr>
+<tr><td class="denom">c²</td> <td class="denom">c²</td>
+<td class="denom">b²</td> <td class="denom">c²</td></tr></table>
+
+<p class="noind">having a common conjugate axis zOz&prime;; the section by the plane of
+x, y, and that by any parallel plane, is an ellipse; and the surface
+may be considered as generated by a variable ellipse moving parallel
+to itself along the two hyperbolas as directrices. If we imagine two
+equal and parallel circular disks, their points connected by strings
+of equal lengths, so that these are the generators of a right circular
+cylinder, and if we turn one of the disks about its centre through an
+angle in its plane, the strings in their new positions will be one
+system of generators of a hyperboloid of one sheet, for which a = b;
+and if we turn it through the same angle in the opposite direction,
+we get in like manner the generators of the other system; there will
+be the same general configuration when a &#8800; b. The hyperbolic
+paraboloid is also covered by two systems of rectilinear generators
+as a method like that used in § 34 establishes without difficulty.
+The figures should be studied to see how they can lie.</p>
+
+<table class="nobctr" style="clear: both;" summary="Illustration">
+<tr><td class="figcenter" colspan="2"><img style="width:518px; height:328px" src="images/img720.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 65.</span></td>
+<td class="caption"><span class="sc">Fig. 66.</span></td></tr></table>
+
+<p>In the hyperboloid of two sheets (fig. 66) the sections by the planes
+of zx and zy are the hyperbolas</p>
+
+<table class="math0" summary="math">
+<tr><td>z²</td>
+<td rowspan="2">&minus;</td> <td>x²</td>
+<td rowspan="2">= 1,&emsp;</td> <td>z²</td>
+<td rowspan="2">&minus;</td> <td>y²</td>
+<td rowspan="2">= 1,</td></tr>
+<tr><td class="denom">c²</td> <td class="denom">a²</td>
+<td class="denom">c²</td> <td class="denom">b²</td></tr></table>
+
+<p class="noind">having a common transverse axis along z&prime;Oz; the section by any
+plane z = ±&gamma; parallel to that of xy is the ellipse</p>
+
+<table class="math0" summary="math">
+<tr><td>x²</td>
+<td rowspan="2">+</td> <td>y²</td>
+<td rowspan="2">=</td> <td>&gamma;²</td>
+<td rowspan="2">&minus; 1,</td></tr>
+<tr><td class="denom">a²</td> <td class="denom">b²</td>
+<td class="denom">c²</td></tr></table>
+
+<p class="noind">provided &gamma;² &gt; c², and the surface, consisting of two distinct portions
+or sheets, may be considered as generated by a variable ellipse
+moving parallel to itself along the hyperbolas as directrices.</p>
+
+<p>38. <i>Differential Geometry of Curves.</i>&mdash;For convenience consider the
+coordinates (x, y, z) of a point on a curve in space to be given as
+functions of a variable parameter &theta;, which may in particular be one
+of themselves. Use the notation x&prime;, x&Prime; for dx/d&theta;, d²x/d&theta;², and similarly
+as to y and z. Only a few formulae will be given. Call the
+current coordinates (&xi;, &eta;, &zeta;).</p>
+
+<p>The <i>tangent</i> at (x, y, z) is the line tended to as a limit by the
+connector of (x, y, z) and a neighbouring point of the curve when the
+latter moves up to the former: its equations are</p>
+
+<p class="center">(&xi; &minus; x)/x&prime; = (&eta; &minus; y)/y&prime; = (&zeta; &minus; z)/z&prime;.</p>
+
+<p>The <i>osculating plane</i> at (x, y, z) is the plane tended to as a limit by
+that through (x, y, z) and two neighbouring points of the curve as
+these, remaining distinct, both move up to (x, y, z): its one equation
+is</p>
+
+<p class="center">(&xi; &minus; x) (y&prime;z&Prime; &minus; y&Prime;z&prime;) + (&eta; &minus; y) (z&prime;x&Prime; &minus; z&Prime;x&prime;) + (&zeta; &minus; z) (x&prime;y&Prime; &minus; x&Prime;y&prime;) = 0.</p>
+
+<p>The <i>normal plane</i> is the plane through (x, y, z) at right angles to the
+tangent line, <i>i.e.</i> the plane</p>
+
+<p class="center">x&prime;(&xi; &minus; x) + y&prime; (&eta; &minus; y) + z&prime; (&zeta; &minus; z) = 0.</p>
+
+<p class="noind">It cuts the osculating plane in a line called the <i>principal normal</i>.
+Every line through (x, y, z) in the normal plane is a normal. The
+normal perpendicular to the osculating plane is called the <i>binormal</i>.
+A tangent, principal normal, and binormal are a convenient set of
+rectangular axes to use as those of reference, when the nature of a
+curve near a point on it is to be discussed.</p>
+
+<p>Through (x, y, z) and three neighbouring points, all on the curve,
+passes a single sphere; and as the three points all move up to (x, y, z)
+continuing distinct, the sphere tends to a limiting size and position.
+The limit tended to is the sphere of closest contact with the curve at
+(x, y, z); its centre and radius are called the centre and radius of
+<i>spherical curvature</i>. It cuts the osculating plane in a circle, called the
+<i>circle of absolute curvature</i>; and the centre and radius of this circle
+are the centre and radius of absolute curvature. The centre of
+absolute curvature is the limiting position of the point where the
+principal normal at (x, y, z) is cut by the normal plane at a neighbouring
+point, as that point moves up to (x, y, z).</p>
+
+<p>39. <i>Differential Geometry of Surfaces.</i>&mdash;Let (x, y, z) be any chosen
+point on a surface &fnof;(x, y, z) = 0. As a second point of the surface
+moves up to (x, y, z), its connector with (x, y, z) tends to a limiting
+position, a tangent line to the surface at (x, y, z). All these tangent
+lines at (x, y, z), obtained by approaching (x, y, z) from different
+directions on a surface, lie in one plane</p>
+
+<table class="math0" summary="math">
+<tr><td>&part;&fnof;</td>
+<td rowspan="2">(&xi; &minus; x) +</td> <td>&part;&fnof;</td>
+<td rowspan="2">(&eta; &minus; y) +</td> <td>&part;&fnof;</td>
+<td rowspan="2">(&zeta; &minus; z) = 0.</td></tr>
+<tr><td class="denom">&part;x</td> <td class="denom">&part;y</td>
+<td class="denom">&part;z</td></tr></table>
+
+<p class="noind">This plane is called the <i>tangent plane</i> at (x, y, z). One line through
+(x, y, z) is at right angles to the tangent plane. This is the normal</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">(&xi; &minus; x) <span class="f150">/</span></td> <td>&part;&fnof;</td>
+<td rowspan="2">= (&eta; &minus; y) <span class="f150">/</span></td> <td>&part;&fnof;</td>
+<td rowspan="2">= (&zeta; &minus; z) <span class="f150">/</span></td> <td>&part;&fnof;</td>
+<td rowspan="2">.</td></tr>
+<tr><td class="denom">&part;x</td> <td class="denom">&part;y</td>
+<td class="denom">&part;z</td></tr></table>
+
+<p>The tangent plane is cut by the surface in a curve, real or imaginary,
+with a node or double point at (x, y, z). Two of the tangent lines
+touch this curve at the node. They are called the &ldquo;chief tangents&rdquo;
+(<i>Haupt-tangenten</i>) at (x, y, z); they have closer contact with the
+surface than any other tangents.</p>
+
+<p>In the case of a quadric surface the curve of intersection of a
+tangent and the surface is of the second order and has a node,
+it must therefore consist of two straight lines. Consequently a
+quadric surface is covered by two sets of straight lines, a pair through
+every point on it; these are imaginary for the ellipsoid, hyperboloid
+of two sheets, and elliptic paraboloid.</p>
+
+<p>A surface of any order is covered by two singly infinite systems
+of curves, a pair through every point, the tangents to which are all
+chief tangents at their respective points of contact. These are
+called <i>chief-tangent curves</i>; on a quadric surface they are the above
+straight lines.</p>
+
+<p>40. The tangents at a point of a surface which bisect the angles
+between the chief tangents are called the <i>principal tangents</i> at the
+point. They are at right angles, and together with the normal
+constitute a convenient set of rectangular axes to which to refer the
+surface when its properties near the point are under discussion.
+At a special point which is such that the chief tangents there run
+to the circular points at infinity in the tangent plane, the principal
+tangents are indeterminate; such a special point is called an umbilic
+of the surface.</p>
+
+<p>There are two singly infinite systems of curves on a surface, a
+pair cutting one another at right angles through every point upon it,
+all tangents to which are principal tangents of the surface at their
+respective points of contact. These are called <i>lines of curvature</i>,
+because of a property next to be mentioned.</p>
+
+<p>As a point Q moves in an arbitrary direction on a surface from
+coincidence with a chosen point P, the normal at it, as a rule, at
+once fails to meet the normal at P; but, if it takes the direction of a
+line of curvature through P, this is instantaneously not the case.
+We have thus on the normal two centres of curvature, and the
+distances of these from the point on the surface are the two <i>principal
+radii of curvature</i> of the surface at that point; these are also the radii
+of curvature of the sections of the surface by planes through the
+normal and the two principal tangents respectively; or say they are
+the radii of curvature of the normal sections through the two principal
+tangents respectively. Take at the point the axis of z in the direction
+of the normal, and those of x and y in the directions of the principal
+tangents respectively, then, if the radii of curvature be a, b (the signs
+being such that the coordinates of the two centres of curvature are
+z = a and z = b respectively), the surface has in the neighbourhood
+of the point the form of the paraboloid</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">z =</td> <td>x²</td>
+<td rowspan="2">+</td> <td>y²</td>
+<td rowspan="2">,</td></tr>
+<tr><td class="denom">2a</td> <td class="denom">2b</td></tr></table>
+
+<p class="noind">and the chief-tangents are determined by the equation 0 = x²/2a + y²/2b.
+The two centres of curvature may be on the same side of the point
+or on opposite sides; in the former case a and b have the same sign,
+the paraboloid is elliptic, and the chief-tangents are imaginary;
+in the latter case a and b have opposite signs, the paraboloid is
+hyperbolic, and the chief-tangents are real.</p>
+
+<p>The normal sections of the surface and the paraboloid by the same
+plane have the same radius of curvature; and it thence readily
+follows that the radius of curvature of a normal section of the surface
+by a plane inclined at an angle &theta; to that of zx is given by the equation</p>
+
+<table class="math0" summary="math">
+<tr><td>1</td>
+<td rowspan="2">=</td> <td>cos² &theta;</td>
+<td rowspan="2">+</td> <td>sin² &theta;</td>
+<td rowspan="2">.</td></tr>
+<tr><td class="denom">&rho;</td> <td class="denom">a</td>
+<td class="denom">b</td></tr></table>
+
+<p>The section in question is that by a plane through the normal
+and a line in the tangent plane inclined at an angle &theta; to the principal
+tangent along the axis of x. To complete the theory, consider the
+section by a plane having the same trace upon the tangent plane,
+but inclined to the normal at an angle &phi;; then it is shown without
+difficulty (Meunier&rsquo;s theorem) that the radius of curvature of this
+inclined section of the surface is = &rho; cos &phi;.</p>
+
+<p><span class="sc">Authorities.</span>&mdash;The above article is largely based on that by
+Arthur Cayley in the 9th edition of this work. Of early and important
+recent publications on analytical geometry, special mention
+<span class="pagenum"><a name="page721" id="page721"></a>721</span>
+is to be made of R. Descartes, <i>Géométrie</i> (Leyden, 1637); John
+Wallis, <i>Tractatus de sectionibus conicis nova methodo expositis</i> (1655,
+<i>Opera mathematica</i>, i., Oxford, 1695); de l&rsquo;Hospital, <i>Traité analytique
+des sections coniques</i> (Paris, 1720); Leonhard Euler, <i>Introductio in
+analysin infinitorum</i>, ii. (Lausanne, 1748); Gaspard Monge, &ldquo;Application
+d&rsquo;algčbre ą la géométrie&rdquo; (<i>Journ. École Polytech.</i>, 1801);
+Julius Plücker, <i>Analytisch-geometrische Entwickelungen</i>, 3 Bde.
+(Essen, 1828-1831); <i>System der analytischen Geometrie</i> (Berlin,
+1835); G. Salmon, <i>A Treatise on Conic Sections</i> (Dublin, 1848;
+6th ed., London, 1879); Ch. Briot and J. Bouquet, <i>Leēons de géométrie
+analytique</i> (Paris, 1851; 16th ed., 1897); M. Chasles, <i>Traité
+de géométrie supérieure</i> (Paris, 1852); Wilhelm Fiedler, <i>Analytische
+Geometrie der Kegelschnitte</i> nach G. Salmon frei bearbeitet (Leipzig,
+5te Aufl., 1887-1888); N.M. Ferrers, <i>An Elementary Treatise on
+Trilinear Coordinates</i> (London, 1861); Otto Hesse, <i>Vorlesungen
+aus der analytischen Geometrie</i> (Leipzig, 1865, 1881); W.A. Whitworth,
+<i>Trilinear Coordinates and other Methods of Modern Analytical
+Geometry</i> (Cambridge, 1866); J. Booth, <i>A Treatise on Some New
+Geometrical Methods</i> (London, i., 1873; ii., 1877); A. Clebsch-F.
+Lindemann, <i>Vorlesungen über Geometrie</i>, Bd. i. (Leipzig, 1876,
+2te Aufl., 1891); R. Baltser, <i>Analytische Geometrie</i> (Leipzig, 1882);
+Charlotte A. Scott, <i>Modern Methods of Analytical Geometry</i> (London,
+1894); G. Salmon, <i>A Treatise on the Analytical Geometry of three
+Dimensions</i> (Dublin, 1862; 4th ed., 1882); Salmon-Fiedler, <i>Analytische
+Geometrie des Raumes</i> (Leipzig, 1863; 4te Aufl., 1898); P.
+Frost, <i>Solid Geometry</i> (London, 3rd ed., 1886; 1st ed., Frost and
+J. Wolstenholme). See also E. Pascal, <i>Repertorio di matematiche
+superiori, II. Geometria</i> (Milan, 1900), and articles now appearing
+in the <i>Encyklopädie der mathematischen Wissenschaften</i>, Bd. iii. 1, 2.</p>
+</div>
+<div class="author">(E. B. El.)</div>
+
+<p class="pt2 center sc">V. Line Geometry</p>
+
+<p>Line geometry is the name applied to those geometrical
+investigations in which the straight line replaces the point as
+element. Just as ordinary geometry deals primarily with points
+and systems of points, this theory deals in the first instance
+with straight lines and systems of straight lines. In two dimensions
+there is no necessity for a special line geometry, inasmuch
+as the straight line and the point are interchangeable by the
+principle of duality; but in three dimensions the straight line
+is its own reciprocal, and for the better discussion of systems
+of lines we require some new apparatus, <i>e.g.</i>, a system of coordinates
+applicable to straight lines rather than to points.
+The essential features of the subject are most easily elucidated
+by analytical methods: we shall therefore begin with the notion
+of line coordinates, and in order to emphasize the merits of the
+system of coordinates ultimately adopted, we first notice a
+system without these advantages, but often useful in special
+investigations.</p>
+
+<div class="condensed">
+<p>In ordinary Cartesian coordinates the two equations of a straight
+line may be reduced to the form y = rx + s, z = tx + u, and r, s, t, u
+may be regarded as the four coordinates of the line. These coordinates
+lack symmetry: moreover, in changing from one base of
+reference to another the transformation is not linear, so that the
+degree of an equation is deprived of real significance. For purposes
+of the general theory we employ homogeneous coordinates; if
+x<span class="su">1</span>y<span class="su">1</span>z<span class="su">1</span>w<span class="su">1</span> and x<span class="su">2</span>y<span class="su">2</span>z<span class="su">2</span>w<span class="su">2</span> are two points on the line, it is easily verified
+that the six determinants of the array</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcc lb rb">x<span class="su">1</span>y<span class="su">1</span>z<span class="su">1</span>w<span class="su">1</span></td></tr>
+
+<tr><td class="tcc lb rb">x<span class="su">2</span>y<span class="su">2</span>z<span class="su">2</span>w<span class="su">2</span></td></tr>
+</table>
+
+<p class="noind">are in the same ratios for all point-pairs on the line, and further,
+that when the point coordinates undergo a linear transformation
+so also do these six determinants. We therefore adopt these six
+determinants for the coordinates of the line, and express them by the
+symbols l, &lambda;, m, &mu;, n, &nu; where l = x<span class="su">1</span>w<span class="su">2</span> &minus; x<span class="su">2</span>w<span class="su">1</span>, &lambda; = y<span class="su">1</span>z<span class="su">2</span> &minus; y<span class="su">2</span>z<span class="su">1</span>, &amp;c.
+There is the further advantage that if a<span class="su">1</span>b<span class="su">1</span>c<span class="su">1</span>d<span class="su">1</span> and a<span class="su">2</span>b<span class="su">2</span>c<span class="su">2</span>d<span class="su">2</span> be two
+planes through the line, the six determinants</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcc lb rb">a<span class="su">1</span>b<span class="su">1</span>c<span class="su">1</span>d<span class="su">1</span></td></tr>
+
+<tr><td class="tcc lb rb">a<span class="su">2</span>b<span class="su">2</span>c<span class="su">2</span>d<span class="su">2</span></td></tr>
+</table>
+
+<p class="noind">are in the same ratios as the foregoing, so that except as regards a
+factor of proportionality we have &lambda; = b<span class="su">1</span>c<span class="su">2</span> &minus; b<span class="su">2</span>c<span class="su">1</span>, l = c<span class="su">1</span>d<span class="su">2</span> &minus; c<span class="su">2</span>d<span class="su">1</span>, &amp;c.
+The identical relation l&lambda; + m&mu; + n&nu; = o reduces the number of independent
+constants in the six coordinates to four, for we are only
+concerned with their mutual ratios; and the quadratic character
+of this relation marks an essential difference between point geometry
+and line geometry. The condition of intersection of two lines is</p>
+
+<p class="center">l&lambda;&prime; + l&prime;&lambda; + m&mu;&prime; + m&prime;&mu; + n&nu;&prime; + n&prime;&nu; = 0</p>
+
+<p class="noind">where the accented letters refer to the second line. If the coordinates
+are Cartesian and l, m, n are direction cosines, the quantity on the
+left is the mutual moment of the two lines.</p>
+
+<p>Since a line depends on four constants, there are three distinct types
+of configurations arising in line geometry&mdash;those containing a triply-infinite,
+a doubly-infinite and a singly-infinite number of lines; they
+are called Complexes, Congruences, and Ruled Surfaces or Skews
+respectively. A <i>Complex</i> is thus a system of lines satisfying one
+condition&mdash;that is, the coordinates are connected by a single relation;
+and the degree of the complex is the degree of this equation supposing
+it to be algebraic. The lines of a complex of the nth degree which
+pass through any point lie on a cone of the nth degree, those which
+lie in any plane envelop a curve of the nth class and there are n lines
+of the complex in any plane pencil; the last statement combines
+the former two, for it shows that the cone is of the nth degree and
+the curve is of the nth class. To find the lines common to four
+complexes of degrees n<span class="su">1</span>, n<span class="su">2</span>, n<span class="su">3</span>, n<span class="su">4</span>, we have to solve five equations, viz.
+the four complex equations together with the quadratic equation
+connecting the line coordinates, therefore the number of common
+lines is 2n<span class="su">1</span>n<span class="su">2</span>n<span class="su">3</span>n<span class="su">4</span>. As an example of complexes we have the lines
+meeting a twisted curve of the nth degree, which form a complex
+of the nth degree.</p>
+
+<p>A <i>Congruence</i> is the set of lines satisfying two conditions: thus
+a finite number m of the lines pass through any point, and a finite
+number n lie in any plane; these numbers are called the degree
+and class respectively, and the congruence is symbolically written
+(m, n).</p>
+
+<p>The simplest example of a congruence is the system of lines
+constituted by all those that pass through m points and those that
+lie in n planes; through any other point there pass m of these lines,
+and in any other plane there lie n, therefore the congruence is of
+degree m and class n. It has been shown by G.H. Halphen that the
+number of lines common to two congruences is mm&prime; + nn&prime;, which may
+be verified by taking one of them to be of this simple type. The
+lines meeting two fixed lines form the general (1, 1) congruence;
+and the chords of a twisted cubic form the general type of a (1, 3)
+congruence; Halphen&rsquo;s result shows that two twisted cubics have
+in general ten common chords. As regards the analytical treatment,
+the difficulty is of the same nature as that arising in the theory of
+curves in space, for a congruence is not in general the complete
+intersection of two complexes.</p>
+
+<p>A <i>Ruled Surface</i>, <i>Regulus</i> or <i>Skew</i> is a configuration of lines
+which satisfy three conditions, and therefore depend on only one
+parameter. Such lines all lie on a surface, for we cannot draw one
+through an arbitrary point; only one line passes through a point of
+the surface; the simplest example, that of a quadric surface, is
+really two skews on the same surface.</p>
+
+<p>The degree of a ruled surface <i>qua</i> line geometry is the number of
+its generating lines contained in a linear complex. Now the number
+which meets a given line is the degree of the surface <i>qua</i> point geometry,
+and as the lines meeting a given line form a particular case
+of linear complex, it follows that the degree is the same from whichever
+point of view we regard it. The lines common to three complexes
+of degrees, n<span class="su">1</span>n<span class="su">2</span>n<span class="su">3</span>, form a ruled surface of degree 2n<span class="su">1</span>n<span class="su">2</span>n<span class="su">3</span>;
+but not every ruled surface is the complete intersection of three
+complexes.</p>
+
+<p>In the case of a complex of the first degree (or linear complex)
+the lines through a fixed point lie in a plane called the polar plane
+or nul-plane of that point, and those lying in a fixed plane
+pass through a point called the nul-point or pole of the
+<span class="sidenote">Linear complex.</span>
+plane. If the nul-plane of A pass through B, then the
+nul-plane of B will pass through A; the nul-planes of all points on
+one line l<span class="su">1</span> pass through another line l<span class="su">2</span>. The relation between l<span class="su">1</span> and
+l<span class="su">2</span> is reciprocal; any line of the complex that meets one will also
+meet the other, and every line meeting both belongs to the complex.
+They are called conjugate or polar lines with respect to the complex.
+On these principles can be founded a theory of reciprocation with
+respect to a linear complex.</p>
+
+<p>This may be aptly illustrated by an elegant example due to A.
+Voss. Since a twisted cubic can be made to satisfy twelve conditions,
+it might be supposed that a finite number could be drawn to touch
+four given lines, but this is not the case. For, suppose one such can
+be drawn, then its reciprocal with respect to any linear complex
+containing the four lines is a curve of the third class, <i>i.e.</i> another
+twisted cubic, touching the same four lines, which are unaltered
+in the process of reciprocation; as there is an infinite number of
+complexes containing the four lines, there is an infinite number of
+cubics touching the four lines, and the problem is poristic.</p>
+
+<p>The following are some geometrical constructions relating to the
+unique linear complex that can be drawn to contain five arbitrary
+lines:</p>
+
+<p>To construct the nul-plane of any point O, we observe that the
+two lines which meet any four of the given five are conjugate lines
+of the complex, and the line drawn through O to meet them is
+therefore a ray of the complex; similarly, by choosing another
+four we can find another ray through O: these rays lie in the nul-plane,
+and there is clearly a result involved that the five lines so
+obtained all lie in one plane. A reciprocal construction will enable
+us to find the nul-point of any plane. Proceeding now to the metrical
+properties and the statical and dynamical applications, we remark
+that there is just one line such that the nul-plane of any point on it
+is perpendicular to it. This is called the central axis; if d be the
+shortest distance, &theta; the angle between it and a ray of the complex,
+then d tan &theta; = p, where p is a constant called the pitch or parameter.
+Any system of forces can be reduced to a force R along a certain line,
+and a couple G perpendicular to that line; the lines of nul-moment
+<span class="pagenum"><a name="page722" id="page722"></a>722</span>
+for the system form a linear complex of which the given line is the
+central axis and the quotient G/R is the pitch. Any motion of a
+rigid body can be reduced to a screw motion about a certain line,
+<i>i.e.</i> to an angular velocity &omega; about that line combined with a linear
+velocity u along the line. The plane drawn through any point
+perpendicular to the direction of its motion is its nul-plane with
+respect to a linear complex having this line for central axis, and the
+quotient u/&omega; for pitch (cf. Sir R.S. Ball, <i>Theory of Screws</i>).</p>
+
+<p>The following are some properties of a configuration of two linear
+complexes:</p>
+
+<p>The lines common to the two-complexes also belong to an infinite
+number of linear complexes, of which two reduce to single straight
+lines. These two lines are conjugate lines with respect to each of
+the complexes, but they may coincide, and then some simple modifications
+are required. The locus of the central axis of this system
+of complexes is a surface of the third degree called the cylindroid,
+which plays a leading part in the theory of screws as developed
+synthetically by Ball. Since a linear complex has an invariant of
+the second degree in its coefficients, it follows that two linear complexes
+have a lineo-linear invariant. This invariant is fundamental:
+if the complexes be both straight lines, its vanishing is the condition
+of their intersection as given above; if only one of them be a straight
+line, its vanishing is the condition that this line should belong to the
+other complex. When it vanishes for any two complexes they
+are said to be in <i>involution</i> or <i>apolar</i>; the nul-points P, Q of any
+plane then divide harmonically the points in which the plane meets
+the common conjugate lines, and each complex is its own reciprocal
+with respect to the other. As regards a configuration of these
+linear complexes, the common lines from one system of generators
+of a quadric, and the doubly infinite system of complexes containing
+the common lines, include an infinite number of straight lines which
+form the other system of generators of the same quadric.</p>
+
+<p>If the equation of a linear complex is Al + Bm + Cn + D&lambda; + E&mu; +
+F&nu; = 0, then for a line not belonging to the complex we may regard
+the expression on the left-hand side as a multiple of the
+moment of the line with respect to the complex, the word
+<span class="sidenote">General line coordinates.</span>
+moment being used in the statical sense; and we infer
+that when the coordinates are replaced by linear functions
+of themselves the new coordinates are multiples of the moments
+of the line with respect to six fixed complexes. The essential features
+of this coordinate system are the same as those of the original one,
+viz. there are six coordinates connected by a quadratic equation,
+but this relation has in general a different form. By suitable choice
+of the six fundamental complexes, as they may be called, this connecting
+relation may be brought into other simple forms of which
+we mention two: (i.) When the six are mutually in involution it can
+be reduced to x<span class="su">1</span>² + x<span class="su">2</span>² + x<span class="su">3</span>² + x<span class="su">4</span>² + x<span class="su">5</span>² + x<span class="su">6</span>² = 0; (ii.) When the first
+four are in involution and the other two are the lines common to
+the first four it is x<span class="su">1</span>² + x<span class="su">2</span>² + x<span class="su">3</span>² + x<span class="su">4</span>² &minus; 2x<span class="su">5</span>x<span class="su">6</span> = 0. These generalized
+coordinates might be explained without reference to actual magnitude,
+just as homogeneous point coordinates can be; the essential
+remark is that the equation of any coordinate to zero represents a
+linear complex, a point of view which includes our original system,
+for the equation of a coordinate to zero represents all the lines
+meeting an edge of the fundamental tetrahedron.</p>
+
+<p>The system of coordinates referred to six complexes mutually
+in involution was introduced by Felix Klein, and in many cases is
+more useful than that derived directly from point coordinates; <i>e.g.</i>
+in the discussion of quadratic complexes: by means of it Klein has
+developed an analogy between line geometry and the geometry of
+spheres as treated by G. Darboux and others. In fact, in that
+geometry a point is represented by <i>five</i> coordinates, connected by a
+relation of the same type as the one just mentioned when the five
+fundamental spheres are mutually at right angles and the equation
+of a sphere is of the first degree. Extending this to four dimensions
+of space, we obtain an exact analogue of line geometry, in which
+(i.) a point corresponds to a line; (ii.) a linear complex to a hypersphere;
+(iii.) two linear complexes in involution to two orthogonal
+hyperspheres; (iv.) a linear complex and two conjugate lines to
+a hypersphere and two inverse points. Many results may be obtained
+by this principle, and more still are suggested by trying to extend
+the properties of circles to spheres in three and four dimensions.
+Thus the elementary theorem, that, given four lines, the circles
+circumscribed to the four triangles formed by them are concurrent,
+may be extended to six hyperplanes in four dimensions; and then
+we can derive a result in line geometry by translating the inverse
+of this theorem. Again, just as there is an infinite number of spheres
+touching a surface at a given point, two of them having contact of a
+closer nature, so there is an infinite number of linear complexes
+touching a non-linear complex at a given line, and <i>three</i> of these
+have contact of a closer nature (cf. Klein, <i>Math. Ann.</i> v.).</p>
+
+<p>Sophus Lie has pointed out a different analogy with sphere
+geometry. Suppose, in fact, that the equation of a sphere of radius
+r is</p>
+
+<p class="center">x² + y² + z² + 2ax + 2by + 2cz + d = 0,</p>
+
+<p class="noind">so that r² = a² + b² + c² &minus; d; then introducing the quantity e to make
+this equation homogeneous, we may regard the sphere as given by
+the six coordinates a, b, c, d, e, r connected by the equation a² +
+b² + c² &minus; r² &minus; de = 0, and it is easy to see that two spheres touch, if
+the polar form 2aa<span class="su">1</span> + 2bb<span class="su">1</span> + 2cc<span class="su">1</span> &minus; 2rr<span class="su">1</span> &minus; de<span class="su">1</span> &minus; d<span class="su">1</span>e vanishes. Comparing
+this with the equation x<span class="su">1</span>² + x<span class="su">2</span>² + x<span class="su">3</span>² + x<span class="su">4</span>² &minus; 2x<span class="su">5</span>x<span class="su">6</span> = 0 given
+above, it appears that this sphere geometry and line geometry are
+identical, for we may write a = x<span class="su">1</span>, b = x<span class="su">2</span>, c = x<span class="su">3</span>, r = x<span class="su">4</span><span class="ov">&delta; &minus; 1</span>, d = x<span class="su">5</span>,
+e = ½x<span class="su">6</span>; but it is to be noticed that a sphere is really replaced by two
+lines whose coordinates only differ in the sign of x<span class="su">4</span>, so that they are
+polar lines with respect to the complex x<span class="su">4</span> = 0. Two spheres which
+touch correspond to two lines which intersect, or more accurately
+to two pairs of lines (p, p&prime;) and (q, q&prime;), of which the pairs (p, q) and
+(p&prime;, q&prime;) both intersect. By this means the problem of describing a
+sphere to touch four given spheres is reduced to that of drawing a
+pair of lines (t, t&prime;) (of which t intersects one line of the four pairs
+(pp&prime;), (qq&prime;), (rr&prime;), (ss&prime;), and t&prime; intersects the remaining four). We
+may, however, ignore the accented letters in translating theorems,
+for a configuration of lines and its polar with respect to a linear
+complex have the same projective properties. In Lie&rsquo;s transformation
+a linear complex corresponds to the totality of spheres cutting a
+given sphere at a given angle. A most remarkable result is that lines
+of curvature in the sphere geometry become asymptotic lines in
+the line geometry.</p>
+
+<p>Some of the principles of line geometry may be brought into
+clearer light by admitting the ideas of space of four and five
+dimensions.</p>
+
+<p>Thus, regarding the coordinates of a line as homogeneous coordinates
+in five dimensions, we may say that line geometry is
+equivalent to geometry on a quadric surface in five dimensions.
+A linear complex is represented by a hyperplane section; and if
+two such complexes are in involution, the corresponding hyperplanes
+are conjugate with respect to the fundamental quadric. By projecting
+this quadric stereographically into space of four dimensions
+we obtain Klein&rsquo;s analogy. In the same way geometry in a linear
+complex is equivalent to geometry on a quadric in four dimensions;
+when two lines intersect the representative points are on the same
+generator of this quadric. Stereographic projection, therefore,
+converts a curve in a linear complex, <i>i.e.</i> one whose tangents all
+belong to the complex, into one whose tangents intersect a fixed
+conic: when this conic is the imaginary circle at infinity the curve
+is what Lie calls a minimal curve. Curves in a linear complex have
+been extensively studied. The osculating plane at any point of such
+a curve is the nul-plane of the point with respect to the complex,
+and points of superosculation always coincide in pairs at the points
+of contact of stationary tangents. When a point of such a curve is
+given, the osculating plane is determined, hence all the curves through
+a given point with the same tangent have the same torsion.</p>
+
+<p>The lines through a given point that belong to a complex of the
+nth degree lie on a cone of the nth degree: if this cone has a double
+line the point is said to be a singular point. Similarly,
+<span class="sidenote">Non-linear complexes.</span>
+a plane is said to be singular when the envelope of the
+lines in it has a double tangent. It is very remarkable
+that the same surface is the locus of the singular points
+and the envelope of the singular planes: this surface is called the
+singular surface, and both its degree and class are in general 2n(n &minus; 1)²,
+which is equal to four for the quadratic complex.</p>
+
+<p>The singular lines of a complex F = 0 are the lines common to F
+and the complex</p>
+
+<table class="math0" summary="math">
+<tr><td>&delta;F</td>
+<td rowspan="2">&nbsp;</td> <td>&delta;F</td>
+<td rowspan="2">+</td> <td>&delta;F</td>
+<td rowspan="2">&nbsp;</td> <td>&delta;F</td>
+<td rowspan="2">+</td> <td>&delta;F</td>
+<td rowspan="2">&nbsp;</td> <td>&delta;F</td>
+<td rowspan="2">= 0.</td></tr>
+<tr><td class="denom">&delta;l</td> <td class="denom">&delta;&lambda;</td>
+<td class="denom">&delta;m</td> <td class="denom">&delta;&mu;</td>
+<td class="denom">&delta;n</td> <td class="denom">&delta;&nu;</td></tr></table>
+
+<p class="noind">As already mentioned, at each line l of a complex there is an infinite
+number of tangent linear complexes, and they all contain the lines
+adjacent to l. If now l be a singular line, these complexes all reduce
+to straight lines which form a plane pencil containing the line l.
+Suppose the vertex of the pencil is A, its plane a, and one of its lines
+&xi;, then l&prime; being a complex line near l, meets &xi;, or more accurately
+the mutual moment of l&prime;, and is of the second order of small quantities.
+If P be a point on l, a line through P quite near l in the plane
+a will meet &xi; and is therefore a line of the complex; hence the
+complex-cones of all points on l touch a and the complex-curves
+of all planes through l touch l at A. It follows that l is a double
+line of the complex-cone of A, and a double tangent of the complex-curve
+of a. Conversely, a double line of a cone or curve is a singular
+line, and a singular line clearly touches the curves of all planes
+through it in the same point. Suppose now that the consecutive
+line l&prime; is also a singular line, A&prime; being the allied singular point, a&prime;
+the singular plane and &xi;&prime; any line of the pencil (A&prime;, a&prime;) so that &xi;&prime; is
+a tangent line at l&prime; to the complex: the mutual moments of the
+pairs l&prime;, &xi; and l, &xi; are each of the second order; hence the plane a&prime;
+meets the lines l and &xi;&prime; in two points very near A. This being true
+for all singular planes, near a the point of contact of a with its
+envelope is in A, <i>i.e.</i> the locus of singular points is the same as the
+envelope of singular planes. Further, when a line touches a complex
+it touches the singular surface, for it belongs to a plane pencil like
+(Aa), and thus in Klein&rsquo;s analogy the analogue of a focus of a hyper-surface
+being a bitangent line of the complex is also a bitangent line
+of the singular surface. The theory of cosingular complexes is thus
+brought into line with that of confocal surfaces in four dimensions,
+and guided by these principles the existence of cosingular quadratic
+complexes can easily be established, the analysis required being
+almost the same as that invented for confocal cyclides by Darboux
+<span class="pagenum"><a name="page723" id="page723"></a>723</span>
+and others. Of cosingular complexes of higher degree nothing is
+known.</p>
+
+<p>Following J. Plücker, we give an account of the lines of a quadratic
+complex that meet a given line.</p>
+
+<p>The cones whose vertices are on the given line all pass through
+eight fixed points and envelop a surface of the fourth degree; the
+conics whose planes contain the given line all lie on a surface of the
+fourth class and touch eight fixed planes. It is easy to see by elementary
+geometry that these two surfaces are identical. Further,
+the given line contains four singular points A<span class="su">1</span>, A<span class="su">2</span>, A<span class="su">3</span>, A<span class="su">4</span>, and the
+planes into which their cones degenerate are the eight common
+tangent planes mentioned above; similarly, there are four singular
+planes, a<span class="su">1</span>, a<span class="su">2</span>, a<span class="su">3</span>, a<span class="su">4</span>, through the line, and the eight points into
+which their conics degenerate are the eight common points above.
+The locus of the pole of the line with respect to all the conics in
+planes through it is a straight line called the <i>polar line</i> of the given
+one; and through this line passes the polar plane of the given line
+with respect to each of the cones. The name polar is applied in the
+ordinary analytical sense; any line has an infinite number of polar
+complexes with respect to the given complex, for the equation of the
+latter can be written in an infinite number of ways; one of these
+polars is a straight line, and is the polar line already introduced.
+The surface on which lie all the conics through a line l is called the
+Plücker surface of that line: from the known properties of (2, 2)
+correspondences it can be shown that the Plücker surface of l cuts l<span class="su">1</span>
+in a range of the same cross ratio as that of the range in which the
+Plücker surface of l<span class="su">1</span> cuts l. Applying this to the case in which l<span class="su">1</span>
+is the polar of l, we find that the cross ratios of (A<span class="su">1</span>, A<span class="su">2</span>,
+A<span class="su">3</span>, A<span class="su">4</span>) and (a<span class="su">1</span>, a<span class="su">2</span>, a<span class="su">3</span>, a<span class="su">4</span>) are equal. The identity of the locus of the A&prime;s with the
+envelope of the a&prime;s follows at once; moreover, a line meets the
+singular surface in four points having the same cross ratio as that
+of the four tangent planes drawn through the line to touch the surface.
+The Plücker surface has eight nodes, eight singular tangent
+planes, and is a double line. The relation between a line and its
+polar line is not a reciprocal one with respect to the complex; but
+W. Stahl has pointed out that the relation is reciprocal as far as the
+singular surface is concerned.</p>
+
+<p>To facilitate the discussion of the general quadratic complex we
+<span class="sidenote">Quadratic complexes.</span>
+introduce Klein&rsquo;s canonical form. We have, in fact, to
+deal with two quadratic equations in six variables; and by
+suitable linear transformations these can be reduced to the
+form</p>
+
+<table class="ws" style="clear: both;" summary="Contents">
+<tr><td class="tcl">a<span class="su">1</span>x<span class="su">1</span><span class="sp">2</span></td> <td class="tcl">+ a<span class="su">2</span>x<span class="su">2</span><span class="sp">2</span></td> <td class="tcl">+ a<span class="su">3</span>x<span class="su">3</span><span class="sp">2</span></td> <td class="tcl">+ a<span class="su">4</span>x<span class="su">4</span><span class="sp">2</span></td> <td class="tcl">+ a<span class="su">5</span>x<span class="su">5</span><span class="sp">2</span></td> <td class="tcl">+  a<span class="su">6</span>x<span class="su">6</span><span class="sp">2</span></td> <td class="tcl">= 0</td></tr>
+<tr><td class="tcl">x<span class="su">1</span><span class="sp">2</span></td> <td class="tcl">+  x<span class="su">2</span><span class="sp">2</span></td> <td class="tcl">+  x<span class="su">3</span><span class="sp">2</span></td> <td class="tcl">+  x<span class="su">4</span><span class="sp">2</span></td> <td class="tcl">+  x<span class="su">5</span><span class="sp">2</span></td> <td class="tcl">+   x<span class="su">6</span><span class="sp">2</span></td> <td class="tcl">= 0</td></tr>
+</table>
+
+<p class="noind">subject to certain exceptions, which will be mentioned later.</p>
+
+<p>Taking the first equation to be that of the complex, we remark
+that both equations are unaltered by changing the sign of any coordinate;
+the geometrical meaning of this is, that the quadratic
+complex is its own reciprocal with respect to each of the six fundamental
+complexes, for changing the sign of a coordinate is equivalent
+to taking the polar of a line with respect to the corresponding
+fundamental complex. It is easy to establish the existence of
+six systems of bitangent linear complexes, for the complex
+l<span class="su">1</span>x<span class="su">1</span> + l<span class="su">2</span>x<span class="su">2</span> + l<span class="su">3</span>x<span class="su">3</span> + l<span class="su">4</span>x<span class="su">4</span> + l<span class="su">5</span>x<span class="su">5</span> + l<span class="su">6</span>x<span class="su">6</span> = 0 is a bitangent when</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">l<span class="su">1</span> = 0, and</td> <td>l<span class="su">2</span>²</td>
+<td rowspan="2">+</td> <td>l<span class="su">3</span>²</td>
+<td rowspan="2">+</td> <td>l<span class="su">4</span>²</td>
+<td rowspan="2">+</td> <td>l<span class="su">5</span>²</td>
+<td rowspan="2">+</td> <td>l<span class="su">6</span>²</td>
+<td rowspan="2">= 0,</td></tr>
+<tr><td class="denom">a<span class="su">2</span> &minus; a<span class="su">1</span></td> <td class="denom">a<span class="su">3</span> &minus; a<span class="su">1</span></td>
+<td class="denom">a<span class="su">4</span> &minus; a<span class="su">1</span></td> <td class="denom">a<span class="su">5</span> &minus; a<span class="su">1</span></td>
+<td class="denom">a<span class="su">6</span> &minus; a<span class="su">1</span></td></tr></table>
+
+<p class="noind">and its lines of contact are conjugate lines with respect to the first
+fundamental complex. We therefore infer the existence of six systems
+of bitangent lines of the complex, of which the first is given by</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">x<span class="su">1</span> = 0,</td> <td>x<span class="su">2</span>²</td>
+<td rowspan="2">+</td> <td>x<span class="su">3</span>²</td>
+<td rowspan="2">+</td> <td>x<span class="su">4</span>²</td>
+<td rowspan="2">+</td> <td>x<span class="su">5</span>²</td>
+<td rowspan="2">+</td> <td>x<span class="su">6</span>²</td>
+<td rowspan="2">= 0,</td></tr>
+<tr><td class="denom">a<span class="su">2</span> &minus; a<span class="su">1</span></td> <td class="denom">a<span class="su">3</span> &minus; a<span class="su">1</span></td>
+<td class="denom">a<span class="su">4</span> &minus; a<span class="su">1</span></td> <td class="denom">a<span class="su">5</span> &minus; a<span class="su">1</span></td>
+<td class="denom">a<span class="su">6</span> &minus; a<span class="su">1</span></td></tr></table>
+
+<p class="noind">Each of these lines is a bitangent of the singular surface, which is
+therefore completely determined as being the focal surface of the
+(2, 2) congruence above. It is thence easy to verify that the two
+complexes &Sigma;ax<span class="sp">2</span> = 0 and &Sigma;bx<span class="sp">2</span> = 0 are cosingular if b<span class="su">r</span> = a<span class="su">r</span>&lambda; + &mu;/a<span class="su">r</span>&nu; + &rho;.</p>
+
+<p>The singular surface of the general quadratic complex is the
+famous quartic, with sixteen nodes and sixteen singular tangent
+planes, first discovered by E.E. Kümmer.</p>
+
+<p>We cannot give a full account of its properties here, but we deduce
+at once from the above that its bitangents break up into six (2, 2)
+congruences, and the six linear complexes containing these are
+mutually in involution. The nodes of the singular surface are points
+whose complex cones are coincident planes, and the complex conic
+in a singular tangent plane consists of two coincident points. This
+configuration of sixteen points and planes has many interesting
+properties; thus each plane contains six points which lie on a conic,
+while through each point there pass six planes which touch a quadric
+cone. In many respects the Kümmer quartic plays a part in three
+dimensions analogous to the general quartic curve in two; it further
+gives a natural representation of certain relations between hyperelliptic
+functions (cf. R.W.H.T. Hudson, <i>Kümmer&rsquo;s Quartic</i>, 1905).</p>
+
+<p>As might be expected from the magnitude of a form in six variables,
+the number of projectivally distinct varieties of quadratic complexes
+is very great; and in fact Adolf Weiler, by whom the
+<span class="sidenote">Classification of quadratic complexes.</span>
+question was first systematically studied on lines indicated
+by Klein, enumerated no fewer than forty-nine different
+types. But the principle of the classification is so important,
+and withal so simple, that we give a brief sketch
+which indicates its essential features.</p>
+
+<p>We have practically to study the intersection of two quadrics
+F and F&prime; in six variables, and to classify the different cases arising
+we make use of the results of Karl Weierstrass on the equivalence
+conditions of two pairs of quadratics. As far as at present required,
+they are as follows: Suppose that the factorized form of the determinantal
+equation Disct (F + &lambda;F&prime;) = 0 is</p>
+
+<p class="center">(&lambda; &minus; &alpha;)<span class="sp">s<span class="su">1</span> + s<span class="su">2</span> + s<span class="su">3</span> ...</span> (&lambda; &minus; &beta;)<span class="sp">t<span class="su">1</span> + t<span class="su">2</span> + t<span class="su">3</span> + ...</span> ...</p>
+
+<p class="noind">where the root &alpha; occurs s<span class="su">1</span> + s<span class="su">2</span> + s<span class="su">3</span> ... times in the determinant,
+s<span class="su">2</span> + s<span class="su">3</span> ... times in every first minor, s<span class="su">3</span> + ... times in every second
+minor, and so on; the meaning of each exponent is then perfectly
+definite. Every factor of the type (&lambda; &minus; &alpha;)<span class="sp">s</span> is called an <i>elementartheil</i>
+(elementary divisor) of the determinant, and the condition of equivalence
+of two pairs of quadratics is simply that their determinants have
+the same elementary divisors. We write the pair of forms symbolically
+thus [(s<span class="su">1</span>s<span class="su">2</span> ...), (t<span class="su">1</span>t<span class="su">2</span> ...), ...], letters in the inner brackets
+referring to the same factor. Returning now to the two quadratics
+representing the complex, the sum of the exponents will be six,
+and two complexes are put in the same class if they have the same
+symbolical expression; <i>i.e.</i> the actual values of the roots of the
+determinantal equation need not be the same for both, but their
+manner of occurrence, as far as here indicated, must be identical in
+the two. The enumeration of all possible cases is thus reduced
+to a simple question in combinatorial analysis, and the actual study
+of any particular case is much facilitated by a useful rule of Klein&rsquo;s
+for writing down in a simple form two quadratics belonging to a
+given class&mdash;one of which, of course, represents the equation connecting
+line coordinates, and the other the equation of the complex.
+The general complex is naturally [111111]; the complex of tangents
+to a quadric is [(111), (111)] and that of lines meeting a conic is
+[(222)]. Full information will be found in Weiler&rsquo;s memoir, <i>Math.
+Ann.</i> vol. vii.</p>
+
+<p>The detailed study of each variety of complex opens up a vast
+subject; we only mention two special cases, the harmonic complex
+and the tetrahedral complex.</p>
+
+<p>The harmonic complex, first studied by Battaglini, is generated
+in an infinite number of ways by the lines cutting two quadrics
+harmonically. Taking the most general case, and referring the
+quadrics to their common self-conjugate tetrahedron, we can find its
+equation in a simple form, and verify that this complex really
+depends only on seventeen constants, so that it is not the most
+general quadratic complex. It belongs to the general type in so far
+as it is discussed above, but the roots of the determinant are in involution.
+The singular surface is the &ldquo;tetrahedroid&rdquo; discussed by
+Cayley. As a particular case, from a metrical point of view, we have
+L.F. Painvin&rsquo;s complex generated by the lines of intersection of
+perpendicular tangent planes of a quadric, the singular surface now
+being Fresnel&rsquo;s wave surface. The tetrahedral or Reye complex is
+the simplest and best known of proper quadratic complexes. It is
+generated by the lines which cut the faces of a tetrahedron in a
+constant cross ratio, and therefore by those subtending the same
+cross ratio at the four vertices. The singular surface is made up of
+the faces or the vertices of the fundamental tetrahedron, and each
+edge of this tetrahedron is a double line of the complex. The
+complex was first discussed by K.T. Reye as the assemblage of lines
+joining corresponding points in a homographic transformation of
+space, and this point of view leads to many important and elegant
+properties. A (metrically) particular case of great interest is the
+complex generated by the normals to a family of confocal quadrics,
+and for many investigations it is convenient to deal with this complex
+referred to the principal axes. For example, Lie has developed
+the theory of curves in a Reye complex (<i>i.e.</i> curves whose tangents
+belong to the complex) as solutions of a differential equation of the
+form (b &minus; c)xdydz + (c &minus; a)ydzdx + (a &minus; b)zdxdy = 0, and we can simplify
+this equation by a logarithmic transformation. Many theorems
+connecting complexes with differential equations have been given
+by Lie and his school. A line complex, in fact, corresponds to a
+Mongian equation having &infin;<span class="sp">3</span> line integrals.</p>
+
+<p>As the coordinates of a line belonging to a congruence are functions
+of two independent parameters, the theory of congruences is analogous
+to that of surfaces, and we may regard it as a fundamental
+inquiry to find the simplest form of surface into which
+<span class="sidenote">Congruences.</span>
+a given congruence can be transformed. Most of those
+whose properties have been extensively discussed can be represented
+on a plane by a birational transformation. But in addition to the
+difficulties of the theory of algebraic surfaces, a subject still in its
+infancy, the theory of congruences has other difficulties in that a
+congruence is seldom completely represented, even by two equations.</p>
+
+<p>A fundamental theorem is that the lines of a congruence are in
+general bitangents of a surface; in fact, since the condition of intersection
+of two consecutive straight lines is ld&lambda; + dmd&mu; + dnd&nu; = 0, a
+line l of the congruence meets two adjacent lines, say l<span class="su">1</span> and l<span class="su">2</span>.
+Suppose l, l<span class="su">1</span> lie in the plane pencil (A<span class="su">1</span>a<span class="su">1</span>) and l, l<span class="su">2</span> in the plane pencil
+(A<span class="su">2</span>a<span class="su">2</span>), then the locus of the A&prime;s is the same as the envelope of the
+a&prime;s, but a<span class="su">2</span> is the tangent plane at A<span class="su">1</span> and a<span class="su">1</span> at A<span class="su">2</span>. This surface is
+called the focal surface of the congruence, and to it all the lines l
+are bitangent. The distinctive property of the points A is that two
+of the congruence lines through them coincide, and in like manner
+the planes a each contain two coincident lines. The focal surface
+consists of two sheets, but one or both may degenerate into curves;
+<span class="pagenum"><a name="page724" id="page724"></a>724</span>
+thus, for example, the normals to a surface are bitangents of the
+surface of centres, and in the case of Dupin&rsquo;s cyclide this surface
+degenerates into two conics.</p>
+
+<p>In the discussion of congruences it soon becomes necessary to
+introduce another number r, called the rank, which expresses the
+number of plane pencils each of which contains an arbitrary line
+and two lines of the congruence. The order of the focal surface is
+2m(n &minus; 1) &minus; 2r, and its class is m(m &minus; 1) &minus; 2r. Our knowledge of
+congruences is almost exclusively confined to those in which either
+m or n does not exceed two. We give a brief account of those of
+the second order without singular lines, those of order unity not
+being especially interesting. A congruence generally has singular
+points through which an infinite number of lines pass; a singular
+point is said to be of order r when the lines through it lie on a cone
+of the rth degree. By means of formulae connecting the number of
+singular points and their orders with the class m of quadratic congruence
+Kümmer proved that the class cannot exceed seven. The
+focal surface is of degree four and class 2m; this kind of quartic
+surface has been extensively studied by Kümmer, Cayley, Rohn and
+others. The varieties (2, 2), (2, 3), (2, 4), (2, 5) all belong to at
+least one Reye complex; and so also does the most important class
+of (2, 6) congruences which includes all the above as special cases.
+The congruence (2, 2) belongs to a linear complex and forty different
+Reye complexes; as above remarked, the singular surface is
+Kümmer&rsquo;s sixteen-nodal quartic, and the same surface is focal for
+six different congruences of this variety. The theory of (2, 2)
+congruences is completely analogous to that of the surfaces called
+cyclides in three dimensions. Further particulars regarding quadratic
+congruences will be found in Kümmer&rsquo;s memoir of 1866, and
+the second volume of Sturm&rsquo;s treatise. The properties of quadratic
+congruences having singular lines, <i>i.e.</i> degenerate focal surfaces, are
+not so interesting as those of the above class; they have been
+discussed by Kümmer, Sturm and others.</p>
+
+<p>Since a ruled surface contains only &infin;¹ elements, this theory is
+practically the same as that of curves. If a linear complex contains
+more than n generators of a ruled surface of the nth degree,
+it contains all the generators, hence for n = 2 there are
+<span class="sidenote">Ruled surfaces.</span>
+three linearly independent complexes, containing all the
+generators, and this is a well-known property of quadric surfaces.
+In ruled cubics the generators all meet two lines which may or may
+not coincide; these two cases correspond to the two main classes of
+cubics discussed by Cayley and Cremona. As regards ruled quartics,
+the generators must lie in one and may lie in two linear complexes.
+The first class is equivalent to a quartic in four dimensions and is
+always rational, but the latter class has to be subdivided into the
+elliptic and the rational, just like twisted quartic curves. A quintic
+skew may not lie in a linear complex, and then it is unicursal, while of
+sextics we have two classes not in a linear complex, viz. the elliptic
+variety, having thirty-six places where a linear complex contains
+six consecutive generators, and the rational, having six such
+places.</p>
+
+<p>The general theory of skews in two linear complexes is identical
+with that of curves on a quadric in three dimensions and is known.
+But for skews lying in only one linear complex there are difficulties;
+the curve now lies in four dimensions, and we represent it in three by
+stereographic projection as a curve meeting a given plane in n points
+on a conic. To find the maximum deficiency for a given degree would
+probably be difficult, but as far as degree eight the space-curve
+theory of Halphen and Nöther can be translated into line geometry
+at once. When the skew does not lie in a linear complex at all the
+theory is more difficult still, and the general theory clearly cannot
+advance until further progress is made in the study of twisted
+curves.</p>
+
+<p><span class="sc">References</span>.&mdash;The earliest works of a general nature are Plücker,
+<i>Neue Geometrie des Raumes</i> (Leipzig, 1868); and Kümmer, &ldquo;Über
+die algebraischen Strahlensysteme,&rdquo; <i>Berlin Academy</i> (1866). Systematic
+development on purely synthetic lines will be found in the
+three volumes of Sturm, <i>Liniengeometrie</i> (Leipzig, 1892, 1893, 1896);
+vol. i. deals with the linear and Reye complexes, vols. ii. and iii.
+with quadratic congruences and complexes respectively. For a
+highly suggestive review by Gino Loria see <i>Bulletin des sciences
+mathématiques</i> (1893, 1897). A shorter treatise, giving a very
+interesting account of Klein&rsquo;s coordinates, is the work of Koenigs,
+<i>La Géométrie réglée et ses applications</i> (Paris, 1898). English treatises
+are C.M. Jessop, <i>Treatise on the Line Complex</i> (1903); R.W.H.T.
+Hudson, <i>Kümmer&rsquo;s Quartic</i> (1905). Many references to memoirs on
+line geometry will be found in Hagen, <i>Synopsis der höheren Mathematik</i>,
+ii. (Berlin, 1894); Loria, <i>Il passato ed il presente delle principali
+teorie geometriche</i> (Milan, 1897); a clear résumé of the principal
+results is contained in the very elegant volume of Pascal, <i>Repertorio
+di mathematiche superiori</i>, ii. (Milan, 1900). Another treatise dealing
+extensively with line geometry is Lie, <i>Geometrie der Berührungstransformationen</i>
+(Leipzig, 1896). Many memoirs on the subject have
+appeared in the <i>Mathematische Annalen</i>; a full list of these will be
+found in the index to the first fifty volumes, p. 115. Perhaps the
+two memoirs which have left most impression on the subsequent
+development of the subject are Klein, &ldquo;Zur Theorie der Liniencomplexe
+des ersten und zweiten Grades,&rdquo; <i>Math. Ann.</i> ii.; and Lie,
+&ldquo;Über Complexe, insbesondere Linien- und Kugelcomplexe,&rdquo;
+<i>Math. Ann.</i> v.</p>
+</div>
+<div class="author">(J. H. Gr.)</div>
+
+<p class="pt2 center sc">VI. Non-Euclidean Geometry</p>
+
+<p>The various metrical geometries are concerned with the
+properties of the various types of congruence-groups, which are
+defined in the study of the <i>axioms</i> of <i>geometry</i> and of their
+immediate consequences. But this point of view of the subject
+is the outcome of recent research, and historically the subject
+has a different origin. Non-Euclidean geometry arose from the
+discussion, extending from the Greek period to the present day,
+of the various assumptions which are implicit in the traditional
+Euclidean system of geometry. In the course of these investigations
+it became evident that metrical geometries, each internally
+consistent but inconsistent in many respects with each other
+and with the Euclidean system, could be developed. A short
+historical sketch will explain this origin of the subject, and
+describe the famous and interesting progress of thought on the
+subject. But previously a description of the chief characteristic
+properties of elliptic and of hyperbolic geometries will be given,
+assuming the standpoint arrived at below under VII. <i>Axioms
+of Geometry</i>.</p>
+
+<p>First assume the equation to the absolute (cf. <i>loc. cit.</i>) to
+be w² &minus; x² &minus; y² &minus; z² = 0. The absolute is then real, and the
+geometry is hyberbolic.</p>
+
+<div class="condensed">
+<p>The distance (d<span class="su">12</span>) between the two points (x<span class="su">1</span>, y<span class="su">1</span>, z<span class="su">1</span>, w<span class="su">1</span>) and (x<span class="su">2</span>, y<span class="su">2</span>,
+z<span class="su">2</span>, w<span class="su">2</span>) is given by</p>
+
+<p class="center">cosh (d<span class="su">12</span>/&gamma;) = (w<span class="su">1</span>w<span class="su">2</span> &minus; x<span class="su">1</span>x<span class="su">2</span> &minus; y<span class="su">1</span>y<span class="su">2</span> &minus; z<span class="su">1</span>z<span class="su">2</span>) / {(w<span class="su">1</span>² &minus; x<span class="su">1</span>² &minus; y<span class="su">1</span>² &minus; z<span class="su">1</span>²)
+(w<span class="su">2</span>² &minus; x<span class="su">2</span>² &minus; y<span class="su">2</span>² &minus; z<span class="su">2</span>²)}<span class="sp">1/2</span></p>
+<div class="aut">(1)</div>
+
+<p class="noind">The only points to which the metrical geometry applies are those
+within the region enclosed by the quadric; the other points are
+&ldquo;improper ideal points.&rdquo; The angle (&theta;<span class="su">12</span>) between two planes,
+l<span class="su">1</span>x + m<span class="su">1</span>y + n<span class="su">1</span>z + r<span class="su">1</span>w = 0 and l<span class="su">2</span>x + m<span class="su">2</span>y + n<span class="su">2</span>z + r<span class="su">2</span>w = 0, is given by</p>
+
+<p class="center">cos &theta;<span class="su">12</span> = (l<span class="su">1</span>l<span class="su">2</span> + m<span class="su">1</span>m<span class="su">2</span> + n<span class="su">1</span>n<span class="su">2</span> &minus; r<span class="su">1</span>r<span class="su">2</span>) / {(l<span class="su">1</span>² + m<span class="su">1</span>² + n<span class="su">1</span>² &minus; r<span class="su">1</span>²)
+(l<span class="su">2</span>² + m<span class="su">2</span>² + n<span class="su">2</span>² &minus; r<span class="su">2</span>²)}<span class="sp">1/2</span></p>
+<div class="aut">(2)</div>
+
+<p class="noind">These planes only have a real angle of inclination if they possess a
+line of intersection within the actual space, <i>i.e.</i> if they intersect.
+Planes which do not intersect possess a shortest distance along a line
+which is perpendicular to both of them. If this shortest distance is
+&delta;<span class="su">12</span>, we have</p>
+
+<p class="center">cosh (&delta;<span class="su">12</span>/&gamma;) = (l<span class="su">1</span>l<span class="su">2</span> + m<span class="su">1</span>m<span class="su">2</span> + n<span class="su">1</span>n<span class="su">2</span> &minus; r<span class="su">1</span>r<span class="su">2</span>) / {(l<span class="su">1</span>² + m<span class="su">1</span>² + n<span class="su">1</span>² &minus; r<span class="su">1</span>²)
+(l<span class="su">2</span>² + m<span class="su">2</span>² + n<span class="su">2</span>² &minus; r<span class="su">2</span>²)}<span class="sp">1/2</span></p>
+<div class="aut">(3)</div>
+
+<table class="flt" style="float: right; width: 300px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:248px; height:220px" src="images/img724a.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig</span>. 67.</td></tr></table>
+
+<p>Thus in the case of the two planes one and only one of the two, &theta;12
+and &delta;<span class="su">12</span>, is real. The same considerations hold for coplanar straight
+lines (see VII. <i>Axioms of Geometry</i>). Let O (fig. 67) be the point
+(0, 0, 0, 1), OX the line y = 0,
+z = 0, OY the line z = 0, x = 0, and
+OZ the line x = 0, y = 0. These are
+the coordinate axes and are at
+right angles to each other. Let
+P be any point, and let &rho; be the
+distance OP, &theta; the angle POZ, and
+&phi; the angle between the planes
+ZOX and ZOP. Then the coordinates
+of P can be taken to be</p>
+
+<p class="center">sinh (&rho;/&gamma;) sin &theta; cos &phi;, sinh (&rho;/&gamma;) sin &theta;
+sin &phi;, sinh (&rho;/&gamma;) cos &theta;, cosh (&rho;/&gamma;).</p>
+
+<p>If ABC is a triangle, and the
+sides and angles are named according
+to the usual convention, we have</p>
+
+<p class="center">sinh (a/&gamma;) / sin A = sinh (b/&gamma;) / sin B = sinh (c/&gamma;) / sin C,</p>
+<div class="aut">(4)</div>
+
+<p class="noind">and also</p>
+
+<p class="center">cosh (a/&gamma;) = cosh (b/&gamma;) cosh (c/&gamma;) &minus; sinh (b/&gamma;) sinh (c/&gamma;) cos A,</p>
+<div class="aut">(5)</div>
+
+<table class="flt" style="float: left; width: 260px;" summary="Illustration">
+<tr><td class="figleft1"><img style="width:211px; height:88px" src="images/img724b.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig</span>. 68.</td></tr></table>
+
+<p class="noind">with two similar equations. The sum of the three angles of a triangle
+is always less than two right angles. The area of the triangle ABC
+is &lambda;²(&pi; &minus; A &minus; B &minus; C). If the base BC of a triangle is kept fixed
+and the vertex A moves in the fixed plane ABC so that the area
+ABC is constant, then the locus of A is a line of equal distance from
+BC. This locus is not a straight line. The whole theory of similarity
+is inapplicable; two triangles are either congruent, or their angles
+are not equal two by two. Thus the elements of a triangle are
+determined when its three angles are
+given. By keeping A and B and the
+line BC fixed, but by making C move
+off to infinity along BC, the lines BC
+and AC become parallel, and the sides
+a and b become infinite. Hence from
+equation (5) above, it follows that two
+parallel lines (cf. Section VII. <i>Axioms of
+Geometry</i>) must be considered as making a zero angle with each
+other. Also if B be a right angle, from the equation (5), remembering
+that, in the limit,</p>
+
+<p class="center">cosh (a/&gamma;) / cosh (b/&gamma;) = cosh (a/&gamma;) / sinh (b/&gamma;) = 1,</p>
+
+<p><span class="pagenum"><a name="page725" id="page725"></a>725</span></p>
+
+<p class="noind">we have</p>
+
+<p class="center">cos A = tanh (c/2&gamma;)</p>
+<div class="aut">(6).</div>
+
+<p class="noind">The angle A is called by N.I. Lobatchewsky the &ldquo;angle of parallelism.&rdquo;</p>
+
+<p>The whole theory of lines and planes at right angles to each other
+is simply the theory of conjugate elements with respect to the
+absolute, where ideal lines and planes are introduced.</p>
+
+<p>Thus if l and l&prime; be any two conjugate lines with respect to the
+absolute (of which one of the two must be improper, say l&prime;), then
+any plane through l&prime; and containing proper points is perpendicular
+to l. Also if p is any plane containing proper points, and P is its
+pole, which is necessarily improper, then the lines through P are
+the normals to P. The equation of the sphere, centre (x<span class="su">1</span>, y<span class="su">1</span>, z<span class="su">1</span>, w<span class="su">1</span>)
+and radius &rho;, is</p>
+
+<p class="center">(w<span class="su">1</span>² &minus; x<span class="su">1</span>² &minus; y<span class="su">1</span>² &minus; z<span class="su">1</span>²) (w² &minus; x² &minus; y² &minus; z²) cosh² (&rho;/&gamma;) =
+(w<span class="su">1</span>w &minus; x<span class="su">1</span>x &minus; y<span class="su">1</span>y &minus; z<span class="su">1</span>z)²</p>
+<div class="aut">(7).</div>
+
+<p class="noind">The equation of the surface of equal distance (&sigma;) from the plane
+lx + my + nz + rw = 0 is</p>
+
+<p class="center">(l² + m² + n² &minus; r²) (w² &minus; x² &minus; y² &minus; z²) sinh² (&sigma;/&gamma;) =
+(rw + lx + my + nz)²</p>
+<div class="aut">(8).</div>
+
+<p class="noind">A surface of equal distance is a sphere whose centre is improper;
+and both types of surface are included in the family</p>
+
+<p class="center">k² (w² &minus; x² &minus; y² &minus; z²) = (ax + by + cz + dw)²</p>
+<div class="aut">(9).</div>
+
+<p>But this family also includes a third type of surfaces, which can
+be looked on either as the limits of spheres whose centres have
+approached the absolute, or as the limits of surfaces of equal distance
+whose central planes have approached a position tangential to the
+absolute. These surfaces are called limit-surfaces. Thus (9) denotes
+a limit-surface, if d² &minus; a² &minus; b² &minus; c² = 0. Two limit-surfaces only
+differ in position. Thus the two limit-surfaces which touch the plane
+YOZ at O, but have their concavities turned in opposite directions,
+have as their equations</p>
+
+<p class="center">w² &minus; x² &minus; y² &minus; z² = (w ± x)².</p>
+
+<p>The geodesic geometry of a sphere is elliptic, that of a surface of
+equal distance is hyperbolic, and that of a limit-surface is parabolic
+(<i>i.e.</i> <i>Euclidean</i>). The equation of the surface (cylinder) of equal
+distance (&delta;) from the line OX is</p>
+
+<p class="center">(w² &minus; x²) tanh² (&delta;/&gamma;) &minus; y² &minus; z² = 0.</p>
+
+<p class="noind">This is not a ruled surface. Hence in this geometry it is not possible
+for two straight lines to be at a constant distance from each other.</p>
+
+<p>Secondly, let the equation of the absolute be x² + y² + z² +
+w² = 0. The absolute is now imaginary and the geometry is
+elliptic.</p>
+
+<p>The distance (d<span class="su">12</span>) between the two points (x<span class="su">1</span>, y<span class="su">1</span>, z<span class="su">1</span>, w<span class="su">1</span>) and
+(x<span class="su">2</span>, y<span class="su">2</span>, z<span class="su">2</span>, w<span class="su">2</span>) is given by</p>
+
+<p class="center">cos (d<span class="su">12</span>/&gamma;) = ± (x<span class="su">1</span>x<span class="su">2</span> + y<span class="su">1</span>y<span class="su">2</span> + z<span class="su">1</span>z<span class="su">2</span> + w<span class="su">1</span>w<span class="su">2</span>) /
+{(x<span class="su">1</span>² + y<span class="su">1</span>² + z<span class="su">1</span>² +  w<span class="su">1</span>²)
+(x<span class="su">2</span>² + y<span class="su">2</span>² + z<span class="su">2</span>² + w<span class="su">2</span>²)}<span class="sp">1/2</span></p>
+<div class="aut">(10).</div>
+
+<p class="noind">Thus there are two distances between the points, and if one is d<span class="su">12</span>,
+the other is &pi;&gamma;-d<span class="su">12</span>. Every straight line returns into itself, forming
+a closed series. Thus there are two segments between any two
+points, together forming the whole line which contains them; one
+distance is associated with one segment, and the other distance with
+the other segment. The complete length of every straight line is
+&pi;&gamma;.</p>
+
+<p>The angle between the two planes l<span class="su">1</span>x + m<span class="su">1</span>y + n<span class="su">1</span>z + r + <span class="su">1</span>w = 0 and
+l<span class="su">2</span>x + m<span class="su">2</span>y + n<span class="su">2</span>z + r<span class="su">2</span>w = 0 is</p>
+
+<p class="center">cos &theta;<span class="su">12</span> = (l<span class="su">1</span>l<span class="su">2</span> + m<span class="su">1</span>m<span class="su">2</span> + n<span class="su">1</span>n<span class="su">2</span> + r<span class="su">1</span>r<span class="su">2</span>) /
+{(l<span class="su">1</span>² + m<span class="su">1</span>² + n<span class="su">1</span>² +r<span class="su">1</span>²)
+(l<span class="su">2</span>² + m<span class="su">2</span>² + n<span class="su">2</span>² + r<span class="su">2</span>²)}<span class="sp">1/2</span></p>
+<div class="aut">(11).</div>
+
+<p class="noind">The polar plane with respect to the absolute of the point (x<span class="su">1</span>, y<span class="su">1</span>, z<span class="su">1</span>, w<span class="su">1</span>)
+is the real plane x<span class="su">1</span>x + y<span class="su">1</span>y + z<span class="su">1</span>z + w<span class="su">1</span>w = 0, and the pole of the plane
+l<span class="su">1</span>x + m<span class="su">1</span>y + n<span class="su">1</span>z + r<span class="su">1</span>w = 0 is the point (l<span class="su">1</span>, m<span class="su">1</span>, n<span class="su">1</span>, r<span class="su">1</span>). Thus (from
+equations 10 and 11) it follows that the angle between the polar
+planes of the points (x<span class="su">1</span>, ...) and (x<span class="su">2</span>, ...) is d<span class="su">12</span>/&gamma;, and that the
+distance between the poles of the planes (l<span class="su">1</span>, ...) and (l<span class="su">2</span>, ...) is
+&gamma;&theta;<span class="su">12</span>. Thus there is complete reciprocity between points and planes
+in respect to all properties. This complete reign of the principle
+of duality is one of the great beauties of this geometry. The theory
+of lines and planes at right angles is simply the theory of conjugate
+elements with respect to the absolute. A tetrahedron self-conjugate
+with respect to the absolute has all its intersecting elements (edges
+and planes) at right angles. If l and l&prime; are two conjugate lines, the
+planes through one are the planes perpendicular to the other. If
+P is the pole of the plane p, the lines through P are the normals to
+the plane p. The distance from P to p is ½&pi;&gamma;. Thus every sphere
+is also a surface of equal distance from the polar of its centre, and
+conversely. A plane does not divide space; for the line joining any
+two points P and Q only cuts the plane once, in L say, then it is
+always possible to go from P to Q by the segment of the line PQ
+which does not contain L. But P and Q may be said to be separated
+by a plane p, if the point in which PQ cuts p lies on the shortest
+segment between P and Q. With this sense of &ldquo;separation,&rdquo; it is
+possible<a name="fa2d" id="fa2d" href="#ft2d"><span class="sp">2</span></a> to find three points P, Q, R such that P and Q are separated
+by the plane p, but P and R are not separated by p, nor are Q
+and R.</p>
+
+<p>Let A, B, C be any three non-collinear points, then four triangles
+are defined by these points. Thus if a, b, c and A, B, C are the
+elements of any one triangle, then the four triangles have as their
+elements:</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcc">(1)</td> <td class="tcc">a,</td> <td class="tcc">b,</td> <td class="tcc">c,</td> <td class="tcc">A,</td> <td class="tcc">B,</td> <td class="tcc">C.</td></tr>
+
+<tr><td class="tcc">(2)</td> <td class="tcc">a,</td> <td class="tcc">&pi;&gamma; &minus; b,</td> <td class="tcc">&pi;&gamma; &minus; c,</td> <td class="tcc">A,</td> <td class="tcc">&pi; &minus; B,</td> <td class="tcc">&pi; &minus; C.</td></tr>
+
+<tr><td class="tcc">(3)</td> <td class="tcc">&pi;&gamma; &minus; a,</td> <td class="tcc">b,</td> <td class="tcc">&pi;&gamma; &minus; c,</td> <td class="tcc">&pi; &minus; A,</td> <td class="tcc">B,</td> <td class="tcc">&pi; &minus; C.</td></tr>
+
+<tr><td class="tcc">(4)</td> <td class="tcc">&pi;&gamma; &minus; a,</td> <td class="tcc">&pi;&gamma; &minus; b,</td> <td class="tcc">c,</td> <td class="tcc">&pi; &minus; A,</td> <td class="tcc">&pi; &minus; B,</td> <td class="tcc">C.</td></tr>
+</table>
+
+<p class="noind">The formulae connecting the elements are</p>
+
+<p class="center">sin A/sin (a/&gamma;) = sin B/sin (b/&gamma;) = sin C/sin (c/&gamma;),</p>
+<div class="aut">(12)</div>
+
+<p class="noind">and</p>
+
+<p class="center">cos (a/&gamma;) = cos (b/&gamma;) cos (c/&gamma;) + sin (b/&gamma;) sin (c/&gamma;) cos A,</p>
+<div class="aut">(13)</div>
+
+<p class="noind">with two similar equations.</p>
+
+<p>Two cases arise, namely (I.) according as one of the four triangles
+has as its sides the shortest segments between the angular points,
+or (II.) according as this is not the case. When case I. holds there
+is said to be a &ldquo;principal triangle.&rdquo;<a name="fa3d" id="fa3d" href="#ft3d"><span class="sp">3</span></a> If all the figures considered lie
+within a sphere of radius ¼&pi;&gamma; only case I. can hold, and the principal
+triangle is the triangle wholly within this sphere, also the peculiarities
+in respect to the separation of points by a plane cannot then arise.
+The sum of the three angles of a triangle ABC is always greater than
+two right angles, and the area of the triangle is &gamma;²(A + B + C &minus; &pi;).
+Thus as in hyperbolic geometry the theory of similarity does not
+hold, and the elements of a triangle are determined when its three
+angles are given. The coordinates of a point can be written in the
+form</p>
+
+<p class="center">sin (&rho;/&gamma;) sin &Phi; cos &phi;, sin (&rho;/&gamma;) sin &Phi; sin &phi;, sin (&rho;/&gamma;) cos &Phi;, cos (&rho;/&gamma;),</p>
+
+<p class="noind">where &rho;, &Phi; and &phi; have the same meanings as in the corresponding
+formulae in hyperbolic geometry. Again, suppose a watch is laid
+on the plane OXY, face upwards with its centre at O, and the line
+12 to 6 (as marked on dial) along the line YOY. Let the watch be
+continually pushed along the plane along the line OX, that is, in
+the direction 9 to 3. Then the line XOX being of finite length, the
+watch will return to O, but at its first return it will be found to be
+face downwards on the other side of the plane, with the line 12 to 6
+reversed in direction along the line YOY. This peculiarity was first
+pointed out by Felix Klein. The theory of parallels as it exists in
+hyperbolic space has no application in elliptic geometry. But
+another property of Euclidean parallel lines holds in elliptic geometry,
+and by the use of it parallel lines are defined. For the equation
+of the surface (cylinder) of equal distance (&delta;) from the line
+XOX is</p>
+
+<p class="center">(x² + w²) tan² (&delta;/&gamma;) &minus; (y² + z²) = 0.</p>
+
+<p class="noind">This is also the surface of equal distance, ½&pi;&gamma;-&delta;, from the line
+conjugate to XOX. Now from the form of the above equation this
+is a ruled surface, and through every point of it two generators pass.
+But these generators are lines of equal distance from XOX. Thus
+throughout every point of space two lines can be drawn which are
+lines of equal distance from a given line l. This property was discovered
+by W.K. Clifford. The two lines are called Clifford&rsquo;s right
+and left parallels to l through the point. This property of parallelism
+is reciprocal, so that if m is a left parallel to l, then l is a left
+parallel to m. Note also that two parallel lines l and m are not
+coplanar. Many of those properties of Euclidean parallels, which do
+not hold for Lobatchewsky&rsquo;s parallels in hyperbolic geometry, do
+hold for Clifford&rsquo;s parallels in elliptic geometry. The geodesic
+geometry of spheres is elliptic, the geodesic geometry of surfaces of
+equal distance from lines (cylinders) is Euclidean, and surfaces of
+revolution can be found<a name="fa4d" id="fa4d" href="#ft4d"><span class="sp">4</span></a> of which the geodesic geometry is hyperbolic.
+But it is to be noticed that the connectivity of these surfaces
+is different to that of a Euclidean plane. For instance there are only
+&infin;² congruence transformations of the cylindrical surfaces of equal
+distance into themselves, instead of the &infin;³ for the ordinary plane.
+It would obviously be possible to state &ldquo;axioms&rdquo; which these
+geodesics satisfy, and thus to define independently, and not as loci,
+quasi-spaces of these peculiar types. The existence of such Euclidean
+quasi-geometries was first pointed out by Clifford.<a name="fa5d" id="fa5d" href="#ft5d"><span class="sp">5</span></a></p>
+</div>
+
+<p>In both elliptic and hyperbolic geometry the spherical
+geometry, <i>i.e.</i> the relations between the angles formed by lines
+and planes passing through the same point, is the same as the
+&ldquo;spherical trigonometry&rdquo; in Euclidean geometry. The constant
+&gamma;, which appears in the formulae both of hyperbolic and elliptic
+geometry, does not by its variation produce different types of
+geometry. There is only one type of elliptic geometry and one
+type of hyperbolic geometry; and the magnitude of the constant
+&gamma; in each case simply depends upon the magnitude of the arbitrary
+unit of length in comparison with the natural unit of length
+<span class="pagenum"><a name="page726" id="page726"></a>726</span>
+which each particular instance of either geometry presents.
+The existence of a natural unit of length is a peculiarity common
+both to hyperbolic and elliptic geometries, and differentiates
+them from Euclidean geometry. It is the reason for the failure
+of the theory of similarity in them. If &gamma; is very large, that is,
+if the natural unit is very large compared to the arbitrary unit,
+and if the lengths involved in the figures considered are not large
+compared to the arbitrary unit, then both the elliptic and
+hyperbolic geometries approximate to the Euclidean. For from
+formulae (4) and (5) and also from (12) and (13) we find, after
+retaining only the lowest powers of small quantities, as the
+formulae for any triangle ABC,</p>
+
+<p class="center">a / sin A = b / sin B = c / sin C,</p>
+
+<p class="noind">and</p>
+
+<p class="center">a² = b² + c² &minus; 2bc cos A,</p>
+
+<p class="noind">with two similar equations. Thus the geometries of small
+figures are in both types Euclidean.</p>
+
+<p><i>History.</i>&mdash;&ldquo;In pulcherrimo Geometriae corpore,&rdquo; wrote Sir
+Henry Savile in 1621, &ldquo;duo sunt naevi, duae labes nec quod
+sciam plures, in quibus eluendis et emaculendis cum
+veterum tum recentiorum ... vigilavit industria.&rdquo;
+<span class="sidenote">Theory of parallels before Gauss.</span>
+These two blemishes are the theory of parallels and
+the theory of proportion. The &ldquo;industry of the
+moderns,&rdquo; in both respects, has given rise to important branches
+of mathematics, while at the same time showing that Euclid
+is in these respects more free from blemish than had been
+previously credible. It was from endeavours to improve the
+theory of parallels that non-Euclidean geometry arose; and
+though it has now acquired a far wider scope, its historical
+origin remains instructive and interesting. Euclid&rsquo;s &ldquo;axiom
+of parallels&rdquo; appears as Postulate V. to the first book of his
+<i>Elements</i>, and is stated thus, &ldquo;And that, if a straight line falling
+on two straight lines make the angles, internal and on the same
+side, less than two right angles, the two straight lines, being
+produced indefinitely, meet on the side on which are the
+angles less than two right angles.&rdquo; The original Greek is
+<span class="grk" title="kai ean eis duo eutheias eutheia empiptousa tas entos kai epi ta
+auta merź gōnias duo orthōn elassonas poiź, ekballomenas tas
+duo eutheias ep&rsquo; apeiron sympiptein, eph&rsquo; ha merź eisin hai tōn duo
+orthōn elassones">
+&#954;&#945;&#8054; &#7952;&#8048;&#957; &#949;&#7984;&#962; &#948;&#973;&#959; &#949;&#8016;&#952;&#949;&#943;&#945;&#962; &#949;&#8016;&#952;&#949;&#8150;&#945; &#7952;&#956;&#960;&#943;&#960;&#964;&#959;&#965;&#963;&#945; &#964;&#8048;&#962; &#7952;&#957;&#964;&#8056;&#962; &#954;&#945;&#8054; &#7952;&#960;&#8054; &#964;&#8048;
+&#945;&#8016;&#964;&#8048; &#956;&#941;&#961;&#951; &#947;&#969;&#957;&#943;&#945;&#962; &#948;&#973;&#959; &#8000;&#961;&#952;&#8182;&#957; &#7952;&#955;&#940;&#963;&#963;&#959;&#957;&#945;&#962; &#960;&#959;&#953;&#8135;, &#7952;&#954;&#946;&#945;&#955;&#955;&#959;&#956;&#941;&#957;&#945;&#962; &#964;&#8048;&#962;
+&#948;&#973;&#959; &#949;&#8016;&#952;&#949;&#943;&#945;&#962; &#7952;&#960;&#8125; &#7940;&#960;&#949;&#953;&#961;&#959;&#957; &#963;&#965;&#956;&#960;&#943;&#960;&#964;&#949;&#953;&#957;, &#7952;&#966;&#8125; &#7939; &#956;&#941;&#961;&#951; &#949;&#7984;&#963;&#8054;&#957; &#945;&#7985; &#964;&#8182;&#957; &#948;&#973;&#959;
+&#8000;&#961;&#952;&#8182;&#957; &#7952;&#955;&#940;&#963;&#963;&#959;&#957;&#949;&#962;</span>.</p>
+
+<p>To Euclid&rsquo;s successors this axiom had signally failed to appear
+self-evident, and had failed equally to appear indemonstrable.
+Without the use of the postulate its converse is proved in Euclid&rsquo;s
+28th proposition, and it was hoped that by further efforts the
+postulate itself could be also proved. The first step consisted
+in the discovery of equivalent axioms. Christoph Clavius in
+1574 deduced the axiom from the assumption that a line whose
+points are all equidistant from a straight line is itself straight.
+John Wallis in 1663 showed that the postulate follows from the
+possibility of similar triangles on different scales. Girolamo
+Saccheri (1733) showed that it is sufficient to have a single
+triangle, the sum of whose angles is two right angles. Other
+equivalent forms may be obtained, but none shows any essential
+superiority to Euclid&rsquo;s. Indeed plausibility, which is chiefly
+aimed at, becomes a positive demerit where it conceals a real
+assumption.</p>
+
+<p>A new method, which, though it failed to lead to the desired
+goal, proved in the end immensely fruitful, was invented by
+Saccheri, in a work entitled <i>Euclides ab omni naevo
+vindicatus</i> (Milan, 1733). If the postulate of parallels
+<span class="sidenote">Saccheri.</span>
+is involved in Euclid&rsquo;s other assumptions, contradictions must
+emerge when it is denied while the others are maintained. This
+led Saccheri to attempt a <i>reductio ad absurdum</i>, in which he
+mistakenly believed himself to have succeeded. What is interesting,
+however, is not his fallacious conclusion, but the non-Euclidean
+results which he obtains in the process. Saccheri
+distinguishes three hypotheses (corresponding to what are now
+known as Euclidean or parabolic, elliptic and hyperbolic geometry),
+and proves that some one of the three must be universally
+true. His three hypotheses are thus obtained: equal
+perpendiculars AC, BD are drawn from a straight line AB,
+and CD are joined. It is shown that the angles ACD, BDC are
+equal. The first hypothesis is that these are both right angles;
+the second, that they are both obtuse; and the third, that they
+are both acute. Many of the results afterwards obtained by
+Lobatchewsky and Bolyai are here developed. Saccheri fails
+to be the founder of non-Euclidean geometry only because he
+does not perceive the possible truth of his non-Euclidean hypotheses.</p>
+
+<p>Some advance is made by Johann Heinrich Lambert in his
+<i>Theorie der Parallellinien</i> (written 1766; posthumously published
+1786). Though he still believed in the necessary
+truth of Euclidean geometry, he confessed that, in
+<span class="sidenote">Lambert.</span>
+all his attempted proofs, something remained undemonstrated.
+He deals with the same three hypotheses as Saccheri, showing
+that the second holds on a sphere, while the third would hold on
+a sphere of purely imaginary radius. The second hypothesis
+he succeeds in condemning, since, like all who preceded Bernhard
+Riemann, he is unable to conceive of the straight line as finite
+and closed. But the third hypothesis, which is the same as
+Lobatchewsky&rsquo;s, is not even professedly refuted.<a name="fa6d" id="fa6d" href="#ft6d"><span class="sp">6</span></a></p>
+
+<p>Non-Euclidean geometry proper begins with Karl Friedrich
+Gauss. The advance which he made was rather philosophical
+than mathematical: it was he (probably) who first
+recognized that the postulate of parallels is possibly
+<span class="sidenote">Three periods of non-Euclidean geometry.</span>
+false, and should be empirically tested by measuring
+the angles of large triangles. The history of non-Euclidean
+geometry has been aptly divided by Felix
+Klein into three very distinct periods. The first&mdash;which contains
+only Gauss, Lobatchewsky and Bolyai&mdash;is characterized by its
+synthetic method and by its close relation to Euclid. The
+attempt at indirect proof of the disputed postulate would seem
+to have been the source of these three men&rsquo;s discoveries; but
+when the postulate had been denied, they found that the results,
+so far from showing contradictions, were just as self-consistent
+as Euclid. They inferred that the postulate, if true at all, can
+only be proved by observations and measurements. Only one
+kind of non-Euclidean space is known to them, namely, that
+which is now called hyperbolic. The second period is analytical,
+and is characterized by a close relation to the theory of surfaces.
+It begins with Riemann&rsquo;s inaugural dissertation, which regards
+space as a particular case of a <i>manifold</i>; but the characteristic
+standpoint of the period is chiefly emphasized by Eugenio
+Beltrami. The conception of measure of curvature is extended
+by Riemann from surfaces to spaces, and a new kind of space,
+finite but unbounded (corresponding to the second hypothesis
+of Saccheri and Lambert), is shown to be possible. As opposed
+to the second period, which is purely metrical, the third period
+is essentially projective in its method. It begins with Arthur
+Cayley, who showed that metrical properties are projective
+properties relative to a certain fundamental quadric, and that
+different geometries arise according as this quadric is real,
+imaginary or degenerate. Klein, to whom the development of
+Cayley&rsquo;s work is due, showed further that there are two forms
+of Riemann&rsquo;s space, called by him the elliptic and the spherical.
+Finally, it has been shown by Sophus Lie, that if figures are to be
+freely movable throughout all space in &infin;<span class="sp">6</span> ways, no other
+three-dimensional spaces than the above four are possible.</p>
+
+<p>Gauss published nothing on the theory of parallels, and it
+was not generally known until after his death that he had
+interested himself in that theory from a very early
+date. In 1799 he announces that Euclidean geometry
+<span class="sidenote">Gauss.</span>
+would follow from the assumption that a triangle can be drawn
+greater than any given triangle. Though unwilling to assume
+this, we find him in 1804 still hoping to prove the postulate of
+parallels. In 1830 he announces his conviction that geometry
+is not an a priori science; in the following year he explains that
+non-Euclidean geometry is free from contradictions, and that,
+in this system, the angles of a triangle diminish without limit
+when all the sides are increased. He also gives for the
+<span class="pagenum"><a name="page727" id="page727"></a>727</span>
+circumference of a circle of radius r the formula &pi;k(e<span class="sp">r/k</span> &minus; e<span class="sp">r &minus;/k</span>),
+where k is a constant depending upon the nature of the space. In
+1832, in reply to the receipt of Bolyai&rsquo;s <i>Appendix</i>, he gives an
+elegant proof that the amount by which the sum of the angles of a
+triangle falls short of two right angles is proportional to the area
+of the triangle. From these and a few other remarks it appears
+that Gauss possessed the foundations of hyperbolic geometry,
+which he was probably the first to regard as perhaps true. It
+is not known with certainty whether he influenced Lobatchewsky
+and Bolyai, but the evidence we possess is against such a view.<a name="fa7d" id="fa7d" href="#ft7d"><span class="sp">7</span></a></p>
+
+<p>The first to publish a non-Euclidean geometry was Nicholas
+Lobatchewsky, professor of mathematics in the new university
+of Kazań.<a name="fa8d" id="fa8d" href="#ft8d"><span class="sp">8</span></a> In the place of the disputed postulate
+he puts the following: &ldquo;All straight lines which, in
+<span class="sidenote">Lobatchewsky.</span>
+a plane, radiate from a given point, can, with respect
+to any other straight line in the same plane, be divided into
+two classes, the <i>intersecting</i> and the <i>non-intersecting</i>. The
+<i>boundary line</i> of the one and the other class is called <i>parallel
+to the given line</i>.&rdquo; It follows that there are two parallels to the
+given line through any point, each meeting the line at infinity,
+like a Euclidean parallel. (Hence a line has two distinct points
+at infinity, and not one only as in ordinary geometry.) The
+two parallels to a line through a point make equal acute angles
+with the perpendicular to the line through the point. If p be
+the length of the perpendicular, either of these angles is denoted
+by &Pi;(p). The determination of &Pi;(p) is the chief problem (cf.
+equation (6) above); it appears finally that, with a suitable
+choice of the unit of length,</p>
+
+<p class="center">tan ½ &Pi;(p) = e<span class="sp">&minus;p</span>.</p>
+
+<p>Before obtaining this result it is shown that spherical trigonometry
+is unchanged, and that the normals to a circle or a sphere
+still pass through its centre. When the radius of the circle or
+sphere becomes infinite all these normals become parallel, but the
+circle or sphere does not become a straight line or plane. It
+becomes what Lobatchewsky calls a limit-line or limit-surface.
+The geometry on such a surface is shown to be Euclidean, limit-lines
+replacing Euclidean straight lines. (It is, in fact, a surface
+of zero measure of curvature.) By the help of these propositions
+Lobatchewsky obtains the above value of &Pi;(p), and thence the
+solution of triangles. He points out that his formulae result
+from those of spherical trigonometry by substituting ia, ib, ic,
+for the sides a, b, c.</p>
+
+<p>John Bolyai, a Hungarian, obtained results closely corresponding
+to those of Lobatchewsky. These he published in an appendix
+to a work by his father, entitled <i>Appendix Scientiam
+spatii absolute veram exhibens: a veritate aut falsitate</i>
+<span class="sidenote">Bolyai.</span>
+<i>Axiomatis XI. Euclidei</i> (<i>a priori haud unquam decidenda</i>) <i>independentem:
+adjecta ad casum falsitatis, quadratura circuli
+geometrica</i>.<a name="fa9d" id="fa9d" href="#ft9d"><span class="sp">9</span></a> This work was published in 1831, but its conception
+dates from 1823. It reveals a profounder appreciation of the
+importance of the new ideas, but otherwise differs little from
+Lobatchewsky&rsquo;s. Both men point out that Euclidean geometry
+as a limiting case of their own more general system, that the
+geometry of very small spaces is always approximately Euclidean,
+that no a priori grounds exist for a decision, and that observation
+can only give an approximate answer. Bolyai gives also, as his
+title indicates, a geometrical construction, in hyperbolic space,
+for the quadrature of the circle, and shows that the area of the
+greatest possible triangle, which has all its sides parallel and all
+its angles zero, is &pi;&iota;², where i is what we should now call the
+space-constant.</p>
+
+<p>The works of Lobatchewsky and Bolyai, though known and
+valued by Gauss, remained obscure and ineffective until, in 1866,
+they were translated into French by J. Hoüel. But
+<span class="sidenote">Riemann.</span>
+at this time Riemann&rsquo;s dissertation, <i>Über die Hypothesen,
+welche der Geometrie zu Grunde liegen</i>,<a name="fa10d" id="fa10d" href="#ft10d"><span class="sp">10</span></a> was already about to be
+published. In this work Riemann, without any knowledge of
+his predecessors in the same field, inaugurated a far more profound
+discussion, based on a far more general standpoint; and by
+its publication in 1867 the attention of mathematicians and
+philosophers was at last secured. (The dissertation dates from
+1854, but owing to changes which Riemann wished to make in it,
+it remained unpublished until after his death.)</p>
+
+<p>Riemann&rsquo;s work contains two fundamental conceptions, that
+of a manifold and that of the <i>measure of curvature</i> of a continuous
+manifold possessed of what he calls flatness in the smallest parts.
+By means of these conceptions space is made to appear
+<span class="sidenote">Definition of a manifold.</span>
+at the end of a gradual series of more and more specialized
+conceptions. Conceptions of magnitude, he explains,
+are only possible where we have a general conception
+capable of determination in various ways. The manifold consists
+of all these various determinations, each of which is an element
+of the manifold. The passage from one element to another may
+be discrete or continuous; the manifold is called discrete or
+continuous accordingly. Where it is discrete two portions of
+it can be compared, as to magnitude, by counting; where
+continuous, by measurement. But measurement demands
+superposition, and consequently some magnitude independent
+of its place in the manifold. In passing, in a continuous manifold,
+from one element to another in a determinate way, we pass
+through a series of intermediate terms, which form a one-dimensional
+manifold. If this whole manifold be similarly
+caused to pass over into another, each of its elements passes
+through a one-dimensional manifold, and thus on the whole
+a two-dimensional manifold is generated. In this way we can
+proceed to n dimensions. Conversely, a manifold of n dimensions
+can be analysed into one of one dimension and one of (n &minus; 1)
+dimensions. By repetitions of this process the position of an
+element may be at last determined by n magnitudes. We may
+here stop to observe that the above conception of a manifold
+is akin to that due to Hermann Grassmann in the first edition
+(1847) of his <i>Ausdehnungslehre</i>.<a name="fa11d" id="fa11d" href="#ft11d"><span class="sp">11</span></a></p>
+
+<p>Both concepts have been elaborated and superseded by the
+modern procedure in respect to the axioms of geometry, and by
+the conception of abstract geometry involved therein.
+Riemann proceeds to specialize the manifold by considerations
+<span class="sidenote">Measure of curvature.</span>
+as to measurement. If measurement is to
+be possible, some magnitude, we saw, must be independent of
+position; let us consider manifolds in which lengths of lines are
+such magnitudes, so that every line is measurable by every
+other. The coordinates of a point being x<span class="su">1</span>, x<span class="su">2</span>, ... x<span class="su">n</span>, let us confine
+ourselves to lines along which the ratios dx<span class="su">1</span> : dx<span class="su">2</span> : ... : dx<span class="su">n</span>
+alter continuously. Let us also assume that the element of
+length, ds, is unchanged (to the first order) when all its points
+undergo the same infinitesimal motion. Then if all the increments
+dx be altered in the same ratio, ds is also altered in this ratio.
+Hence ds is a homogeneous function of the first degree of the
+increments dx. Moreover, ds must be unchanged when all the
+dx change sign. The simplest possible case is, therefore, that in
+which ds is the square root of a quadratic function of the dx.
+This case includes space, and is alone considered in what follows.
+It is called the case of flatness in the smallest parts. Its further
+discussion depends upon the measure of curvature, the second
+of Riemann&rsquo;s fundamental conceptions. This conception, derived
+from the theory of surfaces, is applied as follows. Any one of
+the shortest lines which issue from a given point (say the origin)
+is completely determined by the initial ratios of the dx. Two
+such lines, defined by dx and &delta;x say, determine a pencil, or one-dimensional
+series, of shortest lines, any one of which is defined
+<span class="pagenum"><a name="page728" id="page728"></a>728</span>
+by &lambda;dx + &mu;&delta;x, where the parameter &lambda; : &mu; may have any value.
+This pencil generates a two-dimensional series of points, which
+may be regarded as a surface, and for which we may apply
+Gauss&rsquo;s formula for the measure of curvature at any point.
+Thus at every point of our manifold there is a measure of curvature
+corresponding to every such pencil; but all these can be found
+when n·<span class="ov">n &minus; 1</span>/2 of them are known. If figures are to be freely
+movable, it is necessary and sufficient that the measure of
+curvature should be the same for all points and all directions
+at each point. Where this is the case, <span class="correction" title="amended from it">if</span> &alpha; be the measure of
+curvature, the linear element can be put into the form</p>
+
+<p class="center">ds = &radic;(&Sigma;dx²) / (1 + ¼&alpha;&Sigma;x²).</p>
+
+<p class="noind">If &alpha; be positive, space is finite, though still unbounded, and
+every straight line is closed&mdash;a possibility first recognized by
+Riemann. It is pointed out that, since the possible values of
+a form a continuous series, observations cannot prove that our
+space is strictly Euclidean. It is also regarded as possible that,
+in the infinitesimal, the measure of curvature of our space should
+be variable.</p>
+
+<p>There are four points in which this profound and epoch-making
+work is open to criticism or development&mdash;(1) the idea of a manifold
+requires more precise determination; (2) the introduction
+of coordinates is entirely unexplained and the requisite presuppositions
+are unanalysed; (3) the assumption that ds is the
+square root of a quadratic function of dx<span class="su">1</span>, dx<span class="su">2</span>, ... is arbitrary;
+(4) the idea of superposition, or congruence, is not adequately
+analysed. The modern solution of these difficulties is properly
+considered in connexion with the general subject of the axioms
+of geometry.</p>
+
+<p>The publication of Riemann&rsquo;s dissertation was closely followed
+by two works of Hermann von Helmholtz,<a name="fa12d" id="fa12d" href="#ft12d"><span class="sp">12</span></a> again undertaken
+in ignorance of the work of predecessors. In these a
+<span class="sidenote">Helmholtz.</span>
+proof is attempted that ds must be a rational integral
+quadratic function of the increments of the coordinates. This
+proof has since been shown by Lie to stand in need of correction
+(see VII. <i>Axioms of Geometry</i>). Helmholtz&rsquo;s remaining works
+on the subject<a name="fa13d" id="fa13d" href="#ft13d"><span class="sp">13</span></a> are of almost exclusively philosophical interest.
+We shall return to them later.</p>
+
+<p>The only other writer of importance in the second period is
+Eugenio Beltrami, by whom Riemann&rsquo;s work was brought into
+connexion with that of Lobatchewsky and Bolyai.
+As he gave, by an elegant method, a convenient
+<span class="sidenote">Beltrami.</span>
+Euclidean interpretation of hyperbolic plane geometry, his
+results will be stated at some length<a name="fa14d" id="fa14d" href="#ft14d"><span class="sp">14</span></a>. The <i>Saggio</i> shows that
+Lobatchewsky&rsquo;s plane geometry holds in Euclidean geometry
+on surfaces of constant negative curvature, straight lines being
+replaced by geodesics. Such surfaces are capable of a conformal
+representation on a plane, by which geodesics are represented
+by straight lines. Hence if we take, as coordinates on the surface,
+the Cartesian coordinates of corresponding points on the plane,
+the geodesics must have linear equations.</p>
+
+<div class="condensed">
+<p>Hence it follows that</p>
+
+<p class="center">ds² = R²w<span class="sp">&minus;4</span> {(&alpha;² &minus; v²) du² + 2uvdudv + (&alpha;² &minus; u²)dv²}</p>
+
+<p class="noind">where w² = &alpha;² &minus; u² &minus; v², and &minus;1/R² is the measure of curvature
+of our surface (note that k = &gamma; as used above). The angle between
+two geodesics u = const., v = const. is &theta;, where</p>
+
+<p class="center">cos &theta; = uv / &radic; {(&alpha;² &minus; u²) (&alpha;² &minus; v²)}, sin &theta; = aw / &radic; {(a² &minus; u²) (a² &minus; v²)}.</p>
+
+<p class="noind">Thus u = 0 is orthogonal to all geodesies v = const., and vice versa.
+In order that sin &theta; may be real, w² must be positive; thus geodesics
+have no real intersection when the corresponding straight
+lines intersect outside the circle u² + v² = &alpha;². When they intersect on
+this circle, &theta; = 0. Thus Lobatchewsky&rsquo;s parallels are represented
+by straight lines intersecting on the circle. Again, transforming
+to polar coordinates u = r cos &mu;, v = r sin &mu;, and calling &rho; the geodesic
+distance of u, v from the origin, we have, for a geodesic through the
+origin,</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">d&rho; = Radr / (a² &minus; r²), &rho; = ½R log</td> <td>a + r</td>
+<td rowspan="2">, r = a tan h (&rho; / R).</td></tr>
+<tr><td class="denom">a &minus; r</td></tr></table>
+
+<p class="noind">Thus points on the surface corresponding to points in the plane
+on the limiting circle r = a, are all at an infinite distance from the
+origin. Again, considering r constant, the arc of a geodesic circle
+subtending an angle &mu; at the origin is</p>
+
+<p class="center">&sigma; = Rr&mu; / &radic; (a² &minus; r²) = &mu;R sin h (&rho;/R),</p>
+
+<p class="noind">whence the circumference of a circle of radius &rho; is 2&pi;R sin h (&rho;/R).
+Again, if &alpha; be the angle between any two geodesics</p>
+
+<p class="center">V &minus; v = m (U &minus; u), V &minus; v = n (U &minus; u),</p>
+
+<p class="noind">then</p>
+
+<p class="center">tan &alpha; = a (n &minus; m)w / {(1 + mn)a² &minus; (v &minus; mu) (v &minus; nu)}.</p>
+
+<p class="noind">Thus &alpha; is imaginary when u, v is outside the limiting circle, and
+is zero when, and only when, u, v is on the limiting circle. All
+these results agree with those of Lobatchewsky and Bolyai. The
+maximum triangle, whose angles are all zero, is represented in the
+auxiliary plane by a triangle inscribed in the limiting circle. The
+angle of parallelism is also easily obtained. The perpendicular
+to v = 0 at a distance &delta; from the origin is u = a tan h (&delta;/R), and the
+parallel to this through the origin is u = v sin h (&delta;/R). Hence &Pi; (&delta;),
+the angle which this parallel makes with v = 0, is given by</p>
+
+<p class="center">tan &Pi;(&delta;) . sin h (&delta;/R) = 1, or tan ½&Pi;(&delta;) = e<span class="sp">&minus;&delta;/R</span></p>
+
+<p class="noind">which is Lobatchewsky&rsquo;s formula. We also obtain easily for the
+area of a triangle the formula R²(&pi; &minus; A &minus; B &minus; C).</p>
+
+<p>Beltrami&rsquo;s treatment connects two curves which, in the earlier
+treatment, had no connexion. These are limit-lines and curves
+of constant distance from a straight line. Both may be regarded
+as circles, the first having an infinite, the second an imaginary
+radius. The equation to a circle of radius &rho; and centre u<span class="su">0</span>v<span class="su">0</span> is</p>
+
+<p class="center">(a² &minus; uu<span class="su">0</span> &minus; vv<span class="su">0</span>)² = cos h² (&rho;/R) w<span class="su">0</span>²w² = C²w²</p>
+<div class="aut">(say).</div>
+
+<p class="noind">This equation remains real when &rho; is a pure imaginary, and remains
+finite when w<span class="su">0</span> = 0, provided &rho; becomes infinite in such a way that
+w<span class="su">0</span> cos h (&rho;/R) remains finite. In the latter case the equation represents
+a limit-line. In the former case, by giving different values to C,
+we obtain concentric circles with the imaginary centre u<span class="su">0</span>v<span class="su">0</span>. One of
+these, obtained by putting C = 0, is the straight line a² &minus; uu<span class="su">0</span> &minus; vv<span class="su">0</span> = 0.
+Hence the others are each throughout at a constant distance from
+this line. (It may be shown that all motions in a hyperbolic plane
+consist, in a general sense, of rotations; but three types must
+be distinguished according as the centre is real, imaginary or at
+infinity. All points describe, accordingly, one of the three types of
+circles.)</p>
+
+<p>The above Euclidean interpretation fails for three or more dimensions.
+In the <i>Teoria fondamentale</i>, accordingly, where n dimensions
+are considered, Beltrami treats hyperbolic space in a purely analytical
+spirit. The paper shows that Lobatchewsky&rsquo;s space of any number
+of dimensions has, in Riemann&rsquo;s sense, a constant negative measure
+of curvature. Beltrami starts with the formula (analogous to that
+of the <i>Saggio</i>)</p>
+
+<p class="center">ds² = R²x<span class="sp">&minus;2</span> (dx² + dx<span class="su">1</span>² + dx<span class="su">2</span>² + ... + dx<span class="su">n</span>²)</p>
+
+<p class="noind">where</p>
+
+<p class="center">x² + x<span class="su">1</span>² + x<span class="su">2</span>² +  ... + x<span class="su">n</span>² = a².</p>
+
+<p class="noind">He shows that geodesics are represented by linear equations between
+x<span class="su">1</span>, x<span class="su">2</span>, ..., x<span class="su">n</span>, and that the geodesic distance &rho; between two
+points x and x&prime; is given by</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">cos h</td> <td>&rho;</td>
+<td rowspan="2">=</td> <td>a² &minus; x<span class="su">1</span>x&prime;<span class="su">1</span> &minus; x<span class="su">2</span>x&prime;<span class="su">2</span> &minus; ... &minus; x<span class="su">n</span>x&prime;<span class="su">n</span></td></tr>
+<tr><td class="denom">R</td> <td class="denom">{(a² &minus; x<span class="su">1</span>² &minus; x<span class="su">2</span>² &minus; ... &minus; x<span class="su">n</span>²) (a² &minus; x&prime;<span class="su">1</span>² &minus; x&prime;<span class="su">2</span>² &minus; ... &minus; x&prime;<span class="su">n</span>²)}<span class="sp">1/2</span></td></tr></table>
+
+<p class="noind">(a formula practically identical with Cayley&rsquo;s, though obtained by
+a very different method). In order to show that the measure of
+curvature is constant, we make the substitutions</p>
+
+<p class="center">x<span class="su">1</span> = r&lambda;<span class="su">1</span>, x<span class="su">2</span> = r&lambda;<span class="su">2</span> ... x<span class="su">n</span> = r&lambda;<span class="su">n</span>, where &Sigma;&lambda;² = 1.</p>
+
+<p class="noind">Hence</p>
+
+<p class="center">ds² = (Radr / <span class="ov">a² &minus; r²</span>)² + R²r²d&Delta;² / (a² &minus; r²).</p>
+
+<p class="noind">where</p>
+
+<p class="center">d&Delta;² = &Sigma;d&lambda;².</p>
+
+<p class="noind">Also calling &rho; the geodesic distance from the origin, we have</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">cos h (&rho;/R) =</td> <td>a</td>
+<td rowspan="2">, sin h (&rho;/R) =</td> <td>r</td>
+<td rowspan="2">.</td></tr>
+<tr><td class="denom">&radic;(a² &minus; r²)</td> <td class="denom">&radic;(a² &minus; r²)</td></tr></table>
+
+<p class="noind">Hence</p>
+
+<p class="center">ds² = d&rho;² + (R sin h (&rho;/R))² d&Delta;².</p>
+
+<p class="noind">Putting</p>
+
+<p class="center">z<span class="su">1</span> = &rho;&lambda;<span class="su">1</span>, z<span class="su">2</span> = &rho;&lambda;<span class="su">2</span>, ... z<span class="su">n</span> = &rho;&lambda;<span class="su">n</span>,</p>
+
+<p class="noind">we obtain</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">ds² = &Sigma;dz² +</td> <td>1</td>
+<td rowspan="2"><span class="f150">{ (</span></td> <td>R</td>
+<td rowspan="2">sinh</td> <td>&rho;</td>
+<td rowspan="2"><span class="f150">)</span></td> <td>²</td>
+<td rowspan="2">&minus; 1 <span class="f150">}</span> &Sigma; (z<span class="su">i</span>dz<span class="su">k</span> &minus; z<span class="su">k</span>dz<span class="su">i</span>)².</td></tr>
+<tr><td class="denom">&rho;²</td> <td class="denom">&rho;</td>
+<td class="denom">R</td> <td>&nbsp;</td></tr></table>
+
+<p class="noind">Hence when &rho; is small, we have approximately</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">ds² = &Sigma;dz² +</td> <td>1</td>
+<td rowspan="2">&Sigma; (z<span class="su">i</span>dz<span class="su">k</span> &minus; z<span class="su">k</span>dz<span class="su">i</span>)²</td></tr>
+<tr><td class="denom">3R²</td></tr></table>
+<div class="aut">(1).</div>
+
+<p class="noind">Considering a surface element through the origin, we may choose
+our axes so that, for this element,</p>
+
+<p class="center">z<span class="su">3</span> = z<span class="su">4</span> = ... = z<span class="su">n</span> = 0.</p>
+
+<p class="noind">Thus</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">dz<span class="su">1</span>² + dz<span class="su">2</span>² +</td> <td>1</td>
+<td rowspan="2">(z<span class="su">1</span>dz<span class="su">2</span> &minus; z<span class="su">2</span>dz<span class="su">1</span>)²</td></tr>
+<tr><td class="denom">3R²</td></tr></table>
+<div class="aut">(2).</div>
+
+<p class="noind">Now the area of the triangle whose vertices are (0, 0), (z<span class="su">1</span>, z<span class="su">2</span>),
+(dz<span class="su">1</span>, dz<span class="su">2</span>) is ½(z<span class="su">1</span>, dz<span class="su">2</span> &minus; z<span class="su">2</span>dz<span class="su">1</span>). Hence the quotient when the terms of
+the fourth order in (2) are divided by the square of this triangle is
+<span class="pagenum"><a name="page729" id="page729"></a>729</span>
+4/3R²; hence, returning to general axes, the same is the quotient
+when the terms of the fourth order in (1) are divided by the square
+of the triangle whose vertices are (0, 0, ... 0), (z<span class="su">1</span>, z<span class="su">2</span>, z<span class="su">3</span>, ... z<span class="su">n</span>),
+(dz<span class="su">1</span>, dz<span class="su">2</span>, dz<span class="su">3</span> ... dz<span class="su">n</span>). But &minus;¾ of this quotient is defined by Riemann
+as the measure of curvature.<a name="fa15d" id="fa15d" href="#ft15d"><span class="sp">15</span></a> Hence the measure of curvature is
+&minus;1/R², <i>i.e.</i> is constant and negative. The properties of parallels,
+triangles, &amp;c., are as in the <i>Saggio</i>. It is also shown that the analogues
+of limit surfaces have zero curvature; and that spheres of
+radius &rho; have constant positive curvature 1/R² sinh² (&rho;/R), so that
+spherical geometry may be regarded as contained in the pseudo-spherical
+(as Beltrami calls Lobatchewsky&rsquo;s system).</p>
+</div>
+
+<p>The <i>Saggio</i>, as we saw, gives a Euclidean interpretation
+confined to two dimensions. But a consideration of the auxiliary
+plane suggests a different interpretation, which may be
+extended to any number of dimensions. If, instead
+<span class="sidenote">Transition to the projective method.</span>
+of referring to the pseudosphere, we merely <i>define</i>
+distance and angle, in the Euclidean plane, as those
+functions of the coordinates which gave us distance and angle
+on the pseudosphere, we find that the geometry of our plane has
+become Lobatchewsky&rsquo;s. All the points of the limiting circle
+are now at infinity, and points beyond it are imaginary. If we
+give our circle an imaginary radius the geometry on the plane
+becomes elliptic. Replacing the circle by a sphere, we obtain
+an analogous representation for three dimensions. Instead of
+a circle or sphere we may take any conic or quadric. With this
+definition, if the fundamental quadric be &Sigma;<span class="su">xx</span> = 0, and if &Sigma;<span class="su">xx</span>&prime;
+be the polar form of &Sigma;<span class="su">xx</span>, the distance &rho; between x and x&prime; is
+given by the projective formula</p>
+
+<p class="center">cos(&rho;/k) = &Sigma;<span class="su">xx</span>&prime; / {&Sigma;<span class="su">xx</span>·&Sigma;<span class="su">x</span>&prime;<span class="su">x</span>&prime;}<span class="sp">1/2</span>.</p>
+
+<p class="noind">That this formula is projective is rendered evident by observing
+that e<span class="sp">&minus;2i&rho;/k</span> is the anharmonic ratio of the range consisting of
+the two points and the intersections of the line joining them with
+the fundamental quadric. With this we are brought to the third
+or projective period. The method of this period is due to Cayley;
+its application to previous non-Euclidean geometry is due to
+Klein. The projective method contains a generalization of discoveries
+already made by Laguerre<a name="fa16d" id="fa16d" href="#ft16d"><span class="sp">16</span></a> in 1853 as regards Euclidean
+geometry. The arbitrariness of this procedure of deriving
+metrical geometry from the properties of conics is removed by
+Lie&rsquo;s theory of congruence. We then arrive at the stage of
+thought which finds its expression in the modern treatment of
+the axioms of geometry.</p>
+
+<p>The projective method leads to a discrimination, first made
+by Klein,<a name="fa17d" id="fa17d" href="#ft17d"><span class="sp">17</span></a> of two varieties of Riemann&rsquo;s space; Klein calls
+these elliptic and spherical. They are also called the
+polar and antipodal forms of elliptic space. The latter
+<span class="sidenote">The two kinds of elliptic space.</span>
+names will here be used. The difference is strictly
+analogous to that between the diameters and the points
+of a sphere. In the polar form two straight lines in a plane
+always intersect in one and only one point; in the antipodal
+form they intersect always in two points, which are antipodes.
+According to the definition of geometry adopted in section VII.
+(<i>Axioms of Geometry</i>), the antipodal form is not to be termed
+&ldquo;geometry,&rdquo; since any pair of coplanar straight lines intersect
+each other in two points. It may be called a &ldquo;quasi-geometry.&rdquo;
+Similarly in the antipodal form two diameters always determine
+a plane, but two points on a sphere do not determine a great
+circle when they are antipodes, and two great circles always
+intersect in two points. Again, a plane does not form a boundary
+among lines through a point: we can pass from any one such
+line to any other without passing through the plane. But a great
+circle does divide the surface of a sphere. So, in the polar form,
+a complete straight line does not divide a plane, and a plane does
+not divide space, and does not, like a Euclidean plane, have two
+sides.<a name="fa18d" id="fa18d" href="#ft18d"><span class="sp">18</span></a> But, in the antipodal form, a plane is, in these respects,
+like a Euclidean plane.</p>
+
+<p>It is explained in section VII. in what sense the metrical
+geometry of the material world can be considered to be determinate
+and not a matter of arbitrary choice. The scientific
+question as to the best available evidence concerning the nature
+of this geometry is one beset with difficulties of a peculiar kind.
+We are obstructed by the fact that all existing physical science
+assumes the Euclidean hypothesis. This hypothesis has been
+involved in all actual measurements of large distances, and in all
+the laws of astronomy and physics. The principle of simplicity
+would therefore lead us, in general, where an observation conflicted
+with one or more of those laws, to ascribe this anomaly,
+not to the falsity of Euclidean geometry, but to the falsity of the
+laws in question. This applies especially to astronomy. On the
+earth our means of measurement are many and direct, and so
+long as no great accuracy is sought they involve few scientific
+laws. Thus we acquire, from such direct measurements, a
+very high degree of probability that the space-constant, if not
+infinite, is yet large as compared with terrestrial distances. But
+astronomical distances and triangles can only be measured by
+means of the received laws of astronomy and optics, all of which
+have been established by assuming the truth of the Euclidean
+hypothesis. It therefore remains possible (until a detailed proof
+of the contrary is forthcoming) that a large but finite space-constant,
+with different laws of astronomy and optics, would
+have equally explained the phenomena. We cannot, therefore,
+accept the measurements of stellar parallaxes, &amp;c., as conclusive
+evidence that the space-constant is large as compared with stellar
+distances. For the present, on grounds of simplicity, we may
+rightly adopt this view; but it must remain possible that, in
+view of some hitherto undiscovered discrepancy, a slight correction
+of the sort suggested might prove the simplest alternative.
+But conversely, a finite parallax for very distant stars, or a
+negative parallax for any star, could not be accepted as conclusive
+evidence that our geometry is non-Euclidean, unless it were
+shown&mdash;and this seems scarcely possible&mdash;that no modification
+of astronomy or optics could account for the phenomenon.
+Thus although we may admit a probability that the space-constant
+is large in comparison with stellar distances, a conclusive
+proof or disproof seems scarcely possible.</p>
+
+<p>Finally, it is of interest to note that, though it is theoretically
+possible to prove, by scientific methods, that our geometry is
+non-Euclidean, it is wholly impossible to prove by such methods
+that it is accurately Euclidean. For the unavoidable errors of
+observation must always leave a slight margin in our measurements.
+A triangle might be found whose angles were certainly
+greater, or certainly less, than two right angles; but to prove
+them <i>exactly</i> equal to two right angles must always be beyond our
+powers. If, therefore, any man cherishes a hope of proving the
+exact truth of Euclid, such a hope must be based, not upon
+scientific, but upon philosophical considerations.</p>
+
+<div class="condensed">
+<p><span class="sc">Bibliography.</span>&mdash;The bibliography appended to section VII. should
+be consulted in this connexion. Also, in addition to the citations
+already made, the following works may be mentioned.</p>
+
+<p>For Lobatchewsky&rsquo;s writings, cf. <i>Urkunden zur Geschichte der
+nichteuklidischen Geometrie</i>, i., <i>Nikolaj Iwanowitsch Lobatschefsky</i>,
+by F. Engel and P. Stäckel (Leipzig, 1898). For John Bolyai&rsquo;s
+<i>Appendix</i>, cf. <i>Absolute Geometrie nach Johann Bolyai</i>, by J. Frischauf
+(Leipzig, 1872), and also the new edition of his father&rsquo;s large work,
+<i>Tentamen</i> ..., published by the Mathematical Society of Budapest;
+the second volume contains the appendix. Cf. also J. Frischauf,
+<i>Elemente der absoluten Geometrie</i> (Leipzig, 1876); M.L. Gérard, <i>Sur
+la géométrie non-Euclidienne</i> (thesis for doctorate) (Paris, 1892);
+de Tilly, <i>Essai sur les principes fondamentales de la géométrie et de la
+mécanique</i> (Bordeaux, 1879); Sir R.S. Ball, &ldquo;On the Theory of
+Content,&rdquo; <i>Trans. Roy. Irish Acad.</i> vol. xxix. (1889); F. Lindemann,
+&ldquo;Mechanik bei projectiver Maasbestimmung,&rdquo; <i>Math. Annal.</i> vol.
+vii.; W.K. Clifford, &ldquo;Preliminary Sketch of Biquaternions,&rdquo; <i>Proc.
+of Lond. Math. Soc.</i> (1873), and <i>Coll. Works</i>; A. Buchheim, &ldquo;On the
+Theory of Screws in Elliptic Space,&rdquo; <i>Proc. Lond. Math. Soc.</i> vols. xv.,
+xvi., xvii.; H. Cox, &ldquo;On the Application of Quaternions and
+Grassmann&rsquo;s Algebra to different Kinds of Uniform Space,&rdquo; <i>Trans.
+Camb. Phil. Soc.</i> (1882); M. Dehn, &ldquo;Die Legendarischen Sätze über
+die Winkelsumme im Dreieck,&rdquo; Math. Ann. vol. 53 (1900), and
+&ldquo;Über den Rauminhalt,&rdquo; <i>Math. Annal.</i> vol. 55 (1902).</p>
+
+<p>For expositions of the whole subject, cf. F. Klein, <i>Nicht-Euklidische
+Geometrie</i> (Göttingen, 1893); R. Bonola, <i>La Geometria non-Euclidea</i>
+(Bologna, 1906); P. Barbarin, <i>La Géométrie non-Euclidienne</i> (Paris,
+1902); W. Killing, <i>Die nicht-Euklidischen Raumformen in analytischer
+Behandlung</i> (Leipzig, 1885). The last-named work also deals with
+geometry of more than three dimensions; in this connexion cf. also
+G. Veronese, <i>Fondamenti di geometria a pił dimensioni ed a pił specie</i>
+<span class="pagenum"><a name="page730" id="page730"></a>730</span>
+<i>di unitą rettilinee</i> ... (Padua, 1891, German translation, Leipzig,
+1894); G. Fontené, <i>L&rsquo;Hyperespace ą (n-1) dimensions</i> (Paris, 1892);
+and A.N. Whitehead, <i>loc. cit.</i> Cf. also E. Study, &ldquo;Über nicht-Euklidische
+und Liniengeometrie,&rdquo; <i>Jahr. d. Deutsch. Math. Ver.</i>
+vol. xv. (1906); W. Burnside, &ldquo;On the Kinematics of non-Euclidean
+Space,&rdquo; <i>Proc. Lond. Math. Soc.</i> vol. xxvi. (1894). A bibliography
+on the subject up to 1878 has been published by G.B. Halsted,
+<i>Amer. Journ. of Math.</i> vols. i. and ii.; and one up to 1900 by R.
+Bonola, <i>Index operum ad geometriam absolutam spectantium</i> ...
+(1902, and Leipzig, 1903).</p>
+</div>
+<div class="author">(B. A. W. R.; A. N. W.)</div>
+
+<p class="pt2 center sc">VII. Axioms of Geometry</p>
+
+<p>Until the discovery of the non-Euclidean geometries (Lobatchewsky,
+1826 and 1829; J. Bolyai, 1832; B. Riemann, 1854),
+geometry was universally considered as being exclusively
+the science of existent space. (See section
+<span class="sidenote">Theories of space.</span>
+VI. <i>Non-Euclidean Geometry</i>.) In respect to the
+science, as thus conceived, two controversies may be noticed.
+First, there is the controversy respecting the absolute and
+relational theories of space. According to the absolute theory,
+which is the traditional view (held explicitly by Newton), space
+has an existence, in some sense whatever it may be, independent
+of the bodies which it contains. The bodies occupy space, and
+it is not intrinsically unmeaning to say that any definite body
+occupies <i>this</i> part of space, and not <i>that</i> part of space, without
+reference to other bodies occupying space. According to the
+relational theory of space, of which the chief exponent was
+Leibnitz,<a name="fa19d" id="fa19d" href="#ft19d"><span class="sp">19</span></a> space is nothing but a certain assemblage of the relations
+between the various particular bodies in space. The idea of
+space with no bodies in it is absurd. Accordingly there can be
+no meaning in saying that a body is <i>here</i> and not <i>there</i>, apart
+from a reference to the other bodies in the universe. Thus, on
+this theory, absolute motion is intrinsically unmeaning. It is
+admitted on all hands that in practice only relative motion is
+directly measurable. Newton, however, maintains in the
+<i>Principia</i> (scholium to the 8th definition) that it is indirectly
+measurable by means of the effects of &ldquo;centrifugal force&rdquo; as
+it occurs in the phenomena of rotation. This irrelevance of
+absolute motion (if there be such a thing) to science has led to
+the general adoption of the relational theory by modern men
+of science. But no decisive argument for either view has at
+present been elaborated.<a name="fa20d" id="fa20d" href="#ft20d"><span class="sp">20</span></a> Kant&rsquo;s view of space as being a form
+of perception at first sight appears to cut across this controversy.
+But he, saturated as he was with the spirit of the Newtonian
+physics, must (at least in both editions of the <i>Critique</i>) be classed
+with the upholders of the absolute theory. The form of perception
+has a type of existence proper to itself independently
+of the particular bodies which it contains. For example he
+writes:<a name="fa21d" id="fa21d" href="#ft21d"><span class="sp">21</span></a> &ldquo;Space does not represent any quality of objects by
+themselves, or objects in their relation to one another, <i>i.e.</i> space
+does not represent any determination which is inherent in the
+objects themselves, and would remain, even if all subjective
+conditions of intuition were removed.&rdquo;</p>
+
+<p>The second controversy is that between the view that the
+axioms applicable to space are known only from experience,
+and the view that in some sense these axioms are
+given <i>a priori</i>. Both these views, thus broadly stated,
+<span class="sidenote">Axioms.</span>
+are capable of various subtle modifications, and a discussion
+of them would merge into a general treatise on epistemology.
+The cruder forms of the <i>a priori</i> view have been made quite
+untenable by the modern mathematical discoveries. Geometers
+now profess ignorance in many respects of the exact axioms
+which apply to existent space, and it seems unlikely that a
+profound study of the question should thus obliterate <i>a priori</i>
+intuitions.</p>
+
+<p>Another question irrelevant to this article, but with some
+relevance to the above controversy, is that of the derivation
+of our perception of existent space from our various types of
+sensation. This is a question for psychology.<a name="fa22d" id="fa22d" href="#ft22d"><span class="sp">22</span></a></p>
+
+<p><i>Definition of Abstract Geometry.</i>&mdash;Existent space is the subject
+matter of only one of the applications of the modern science of
+abstract geometry, viewed as a branch of pure mathematics.
+Geometry has been defined<a name="fa23d" id="fa23d" href="#ft23d"><span class="sp">23</span></a> as &ldquo;the study of series of two or more
+dimensions.&rdquo; It has also been defined<a name="fa24d" id="fa24d" href="#ft24d"><span class="sp">24</span></a> as &ldquo;the science of cross
+classification.&rdquo; These definitions are founded upon the actual
+practice of mathematicians in respect to their use of the term
+&ldquo;Geometry.&rdquo; Either of them brings out the fact that geometry
+is not a science with a determinate subject matter. It is concerned
+with any subject matter to which the formal axioms may apply.
+Geometry is not peculiar in this respect. All branches of pure
+mathematics deal merely with types of relations. Thus the
+fundamental ideas of geometry (<i>e.g.</i> those of <i>points</i> and of
+<i>straight lines</i>) are not ideas of determinate entities, but of any
+entities for which the axioms are true. And a set of formal
+geometrical axioms cannot in themselves be true or false, since
+they are not determinate propositions, in that they do not refer
+to a determinate subject matter. The axioms are propositional
+functions.<a name="fa25d" id="fa25d" href="#ft25d"><span class="sp">25</span></a> When a set of axioms is given, we can ask (1)
+whether they are consistent, (2) whether their &ldquo;existence
+theorem&rdquo; is proved, (3) whether they are independent. Axioms
+are consistent when the contradictory of any axiom cannot be
+deduced from the remaining axioms. Their existence theorem
+is the proof that they are true when the fundamental ideas are
+considered as denoting some determinate subject matter, so
+that the axioms are developed into determinate propositions.
+It follows from the logical law of contradiction that the proof
+of the existence theorem proves also the consistency of the
+axioms. This is the only method of proof of consistency. The
+axioms of a set are independent of each other when no axiom
+can be deduced from the remaining axioms of the set. The
+independence of a given axiom is proved by establishing the
+consistency of the remaining axioms of the set, together with the
+contradictory of the given axiom. The enumeration of the
+axioms is simply the enumeration of the hypotheses<a name="fa26d" id="fa26d" href="#ft26d"><span class="sp">26</span></a> (with
+respect to the undetermined subject matter) of which some at
+least occur in each of the subsequent propositions.</p>
+
+<p>Any science is called a &ldquo;geometry&rdquo; if it investigates the
+theory of the classification of a set of entities (the points) into
+classes (the straight lines), such that (1) there is one and only
+one class which contains any given pair of the entities, and (2)
+every such class contains more than two members. In the two
+geometries, important from their relevance to existent space,
+axioms which secure an order of the points on any line also
+occur. These geometries will be called &ldquo;Projective Geometry&rdquo;
+and &ldquo;Descriptive Geometry.&rdquo; In projective geometry any
+two straight lines in a plane intersect, and the straight lines
+are closed series which return into themselves, like the circumference
+of a circle. In descriptive geometry two straight lines in
+a plane do not necessarily intersect, and a straight line is an open
+series without beginning or end. Ordinary Euclidean geometry
+is a descriptive geometry; it becomes a projective geometry
+when the so-called &ldquo;points at infinity&rdquo; are added.</p>
+
+<p class="pt2 center"><i>Projective Geometry.</i></p>
+
+<p>Projective geometry may be developed from two undefined
+fundamental ideas, namely, that of a &ldquo;point&rdquo; and that of a
+&ldquo;straight line.&rdquo; These undetermined ideas take different
+specific meanings for the various specific subject matters to
+which projective geometry can be applied. The number of the
+axioms is always to some extent arbitrary, being dependent
+upon the verbal forms of statement which are adopted. They will
+<span class="pagenum"><a name="page731" id="page731"></a>731</span>
+be presented<a name="fa27d" id="fa27d" href="#ft27d"><span class="sp">27</span></a> here as twelve in number, eight being &ldquo;axioms
+of classification,&rdquo; and four being &ldquo;axioms of order.&rdquo;</p>
+
+<p><i>Axioms of Classification.</i>&mdash;The eight axioms of classification
+are as follows:</p>
+
+<p>1. Points form a class of entities with at least two members.</p>
+
+<p>2. Any straight line is a class of points containing at least
+three members.</p>
+
+<p>3. Any two distinct points lie in one and only one straight
+line.</p>
+
+<p>4. There is at least one straight line which does not contain
+all the points.</p>
+
+<p>5. If A, B, C are non-collinear points, and A&prime; is on the straight
+line BC, and B&prime; is on the straight line CA, then the straight lines
+AA&prime; and BB&prime; possess a point in common.</p>
+
+<div class="condensed">
+<p><i>Definition.</i>&mdash;If A, B, C are any three non-collinear points, the
+<i>plane</i> ABC is the class of points lying on the straight lines joining
+A with the various points on the straight line BC.</p>
+</div>
+
+<p>6. There is at least one plane which does not contain all the
+points.</p>
+
+<p>7. There exists a plane &alpha;, and a point A not incident in &alpha;,
+such that any point lies in some straight line which contains
+both A and a point in &alpha;.</p>
+
+<div class="condensed">
+<p><i>Definition.</i>&mdash;Harm. (ABCD) symbolizes the following conjoint
+statements: (1) that the points A, B, C, D are collinear, and (2)
+that a quadrilateral can be found with one pair of opposite sides
+intersecting at A, with the other pair intersecting at C, and with its
+diagonals passing through B and D respectively. Then B and D are
+said to be &ldquo;harmonic conjugates&rdquo; with respect to A and C.</p>
+</div>
+
+<p>8. Harm. (ABCD) implies that B and D are distinct points.</p>
+
+<p>In the above axioms 4 secures at least two dimensions, axiom
+5 is the fundamental axiom of the plane, axiom 6 secures at
+least three dimensions, and axiom 7 secures at most three
+dimensions. From axioms 1-5 it can be proved that any two
+distinct points in a straight line determine that line, that any
+three non-collinear points in a plane determine that plane, that
+the straight line containing any two points in a plane lies wholly
+in that plane, and that any two straight lines in a plane intersect.
+From axioms 1-6 Desargue&rsquo;s well-known theorem on triangles
+in perspective can be proved.</p>
+
+<div class="condensed">
+<p>The enunciation of this theorem is as follows: If ABC and
+A&prime;B&prime;C&prime; are two coplanar triangles such that the lines AA&prime;, BB&prime;,
+CC&prime; are concurrent, then the three points of intersection of BC and
+B&prime;C&prime; of CA and C&prime;A&prime;, and of AB and A&prime;B&prime; are collinear; and
+conversely if the three points of intersection are collinear, the three
+lines are concurrent. The proof which can be applied is the usual
+projective proof by which a third triangle A&Prime;B&Prime;C&Prime; is constructed
+not coplanar with the other two, but in perspective with each
+of them.</p>
+
+<p>It has been proved<a name="fa28d" id="fa28d" href="#ft28d"><span class="sp">28</span></a> that Desargues&rsquo;s theorem cannot be deduced
+from axioms 1-5, that is, if the geometry be confined to two
+dimensions. All the proofs proceed by the method of producing a
+specification of &ldquo;points&rdquo; and &ldquo;straight lines&rdquo; which satisfies
+axioms 1-5, and such that Desargues&rsquo;s theorem does not hold.</p>
+
+<p>It follows from axioms 1-5 that Harm. (ABCD) implies Harm.
+(ADCB) and Harm. (CBAD), and that, if A, B, C be any three
+distinct collinear points, there exists at least one point D such that
+Harm. (ABCD). But it requires Desargues&rsquo;s theorem, and hence
+axiom 6, to prove that Harm. (ABCD) and Harm. (ABCD&prime;) imply
+the identity of D and D&prime;.</p>
+</div>
+
+<p>The necessity for axiom 8 has been proved by G. Fano,<a name="fa29d" id="fa29d" href="#ft29d"><span class="sp">29</span></a> who
+has produced a three dimensional geometry of fifteen points,
+<i>i.e.</i> a method of cross classification of fifteen entities, in which
+each straight line contains three points, and each plane contains
+seven straight lines. In this geometry axiom 8 does not hold.
+Also from axioms 1-6 and 8 it follows that Harm. (ABCD)
+implies Harm. (BCDA).</p>
+
+<div class="condensed">
+<p><i>Definitions.</i>&mdash;When two plane figures can be derived from one
+another by a single projection, they are said to be in <i>perspective</i>.
+When two plane figures can be derived one from the other by a finite
+series of perspective relations between intermediate figures, they
+are said to be <i>projectively</i> related. Any property of a plane figure
+which necessarily also belongs to any projectively related figure, is
+called a <i>projective</i> property.</p>
+
+<p>The following theorem, known from its importance as &ldquo;the
+fundamental theorem of projective geometry,&rdquo; cannot be proved<a name="fa30d" id="fa30d" href="#ft30d"><span class="sp">30</span></a>
+from axioms 1-8. The enunciation is: &ldquo;A projective correspondence
+between the points on two straight lines is completely determined
+when the correspondents of three distinct points on one line
+are determined on the other.&rdquo; This theorem is equivalent<a name="fa31d" id="fa31d" href="#ft31d"><span class="sp">31</span></a>
+(assuming axioms 1-8) to another theorem, known as Pappus&rsquo;s
+Theorem, namely: &ldquo;If l and l&prime; are two distinct coplanar lines, and
+A, B, C are three distinct points on l, and A&prime;, B&prime;, C&prime; are three distinct
+points on l&prime;, then the three points of intersection of AA&prime; and B&prime;C,
+of A&prime;B and CC&prime;, of BB&prime; and C&prime;A, are collinear.&rdquo; This theorem is
+obviously Pascal&rsquo;s well-known theorem respecting a hexagon
+inscribed in a conic, for the special case when the conic has degenerated
+into the two lines l and l&prime;. Another theorem also
+equivalent (assuming axioms 1-8) to the fundamental theorem is
+the following:<a name="fa32d" id="fa32d" href="#ft32d"><span class="sp">32</span></a> If the three collinear pairs of points, A and A&prime;,
+B and B&prime;, C and C&prime;, are such that the three pairs of opposite sides
+of a complete quadrangle pass respectively through them, <i>i.e.</i> one
+pair through A and A&prime; respectively, and so on, and if also the three
+sides of the quadrangle which pass through A, B, and C, are concurrent
+in one of the corners of the quadrangle, then another quadrangle
+can be found with the same relation to the three pairs of points,
+except that its three sides which pass through A, B, and C, are not
+concurrent.</p>
+
+<p>Thus, if we choose to take any one of these three theorems as an
+axiom, all the theorems of projective geometry which do not require
+ordinal or metrical ideas for their enunciation can be proved. Also
+a conic can be defined as the locus of the points found by the usual
+construction, based upon Pascal&rsquo;s theorem, for points on the conic
+through five given points. But it is unnecessary to assume here
+any one of the suggested axioms; for the fundamental theorem can
+be deduced from the axioms of order together with axioms 1-8.</p>
+</div>
+
+<p><i>Axioms of Order.</i>&mdash;It is possible to define (cf. Pieri, <i>loc. cit.</i>)
+the property upon which the order of points on a straight line
+depends. But to secure that this property does in fact range
+the points in a serial order, some axioms are required. A straight
+line is to be a closed series; thus, when the points are in order,
+it requires two points on the line to divide it into two distinct
+complementary segments, which do not overlap, and together
+form the whole line. Accordingly the problem of the definition
+of order reduces itself to the definition of these two segments
+formed by any two points on the line; and the axioms are
+stated relatively to these segments.</p>
+
+<div class="condensed">
+<p><i>Definition.</i>&mdash;If A, B, C are three collinear points, the points on the
+<i>segment</i> ABC are defined to be those points such as X, for which
+there exist two points Y and Y&prime; with the property that Harm.
+(AYCY&prime;) and Harm. (BYXY&prime;) both hold. The <i>supplementary
+segment</i> ABC is defined to be the rest of the points on the line.
+This definition is elucidated by noticing that with our ordinary
+geometrical ideas, if B and X are any two points between A and C,
+then the two pairs of points, A and C, B and X, define an involution
+with real double points, namely, the Y and Y&prime; of the above definition.
+The property of belonging to a segment ABC is projective, since
+the harmonic relation is projective.</p>
+</div>
+
+<p>The first three axioms of order (cf. Pieri, <i>loc. cit.</i>) are:</p>
+
+<p>9. If A, B, C are three distinct collinear points, the supplementary
+segment ABC is contained within the segment BCA.</p>
+
+<p>10. If A, B, C are three distinct collinear points, the common
+part of the segments BCA and CAB is contained in the supplementary
+segment ABC.</p>
+
+<p>11. If A, B, C are three distinct collinear points, and D lies
+In the segment ABC, then the segment ADC is contained
+within the segment ABC.</p>
+
+<p>From these axioms all the usual properties of a closed order
+follow. It will be noticed that, if A, B, C are any three collinear
+points, C is necessarily traversed in passing from A to B by one
+route along the line, and is not traversed in passing from A to B
+along the other route. Thus there is no meaning, as referred
+to closed straight lines, in the simple statement that C lies
+between A and B. But there may be a relation of separation
+between two pairs of collinear points, such as A and C, and
+B and D. The couple B and D is said to separate A and C, if
+<span class="pagenum"><a name="page732" id="page732"></a>732</span>
+the four points are collinear and D lies in the segment complementary
+to the segment ABC. The property of the separation
+of pairs of points by pairs of points is projective. Also it can be
+proved that Harm. (ABCD) implies that B and D separate
+A and C.</p>
+
+<div class="condensed">
+<p><i>Definitions.</i>&mdash;A series of entities arranged in a serial order, open
+or closed, is said to be <i>compact</i>, if the series contains no immediately
+consecutive entities, so that in traversing the series from any one
+entity to any other entity it is necessary to pass through entities
+distinct from either. It was the merit of R. Dedekind and of
+G. Cantor explicitly to formulate another fundamental property of
+series. The Dedekind property<a name="fa33d" id="fa33d" href="#ft33d"><span class="sp">33</span></a> as applied to an open series can
+be defined thus: An open series possesses the Dedekind property,
+if, however, it be divided into two mutually exclusive classes u and
+v, which (1) contain between them the whole series, and (2) are
+such that every member of u precedes in the serial order every
+member of v, there is always a member of the series, belonging to one
+of the two, u or v, which precedes every member of v (other than
+itself if it belong to v), and also succeeds every member of u (other
+than itself if it belong to u). Accordingly in an open series with the
+Dedekind property there is always a member of the series marking
+the junction of two classes such as u and v. An open series is <i>continuous</i>
+if it is compact and possesses the Dedekind property. A
+closed series can always be transformed into an open series by taking
+any arbitrary member as the first term and by taking one of the two
+ways round as the ascending order of the series. Thus the definitions
+of compactness and of the Dedekind property can be at once transferred
+to a closed series.</p>
+</div>
+
+<p>12. The last axiom of order is that there exists at least one
+straight line for which the point order possesses the Dedekind
+property.</p>
+
+<p>It follows from axioms 1-12 by projection that the Dedekind
+property is true for all lines. Again the <i>harmonic system</i> ABC,
+where A, B, C are collinear points, is defined<a name="fa34d" id="fa34d" href="#ft34d"><span class="sp">34</span></a> thus: take the
+harmonic conjugates A&prime;, B&prime;, C&prime; of each point with respect to
+the other two, again take the harmonic conjugates of each of
+the six points A, B, C, A&prime;, B&prime;, C&prime; with respect to each pair of the
+remaining five, and proceed in this way by an unending series
+of steps. The set of points thus obtained is called the harmonic
+system ABC. It can be proved that a harmonic system is
+compact, and that every segment of the line containing it
+possesses members of it. Furthermore, it is easy to prove that
+the fundamental theorem holds for harmonic systems, in the
+sense that, if A, B, C are three points on a line l, and A&prime;, B&prime;, C&prime;
+are three points on a line l&prime;, and if by any two distinct series
+of projections A, B, C are projected into A&prime;, B&prime;, C&prime;, then any point
+of the harmonic system ABC corresponds to the same point of
+the harmonic system A&prime;B&prime;C&prime; according to both the projective
+relations which are thus established between l and l&prime;. It now
+follows immediately that the fundamental theorem must hold for
+all the points on the lines l and l&prime;, since (as has been pointed out)
+harmonic systems are &ldquo;everywhere dense&rdquo; on their containing
+lines. Thus the fundamental theorem follows from the axioms
+of order.</p>
+
+<p>A system of numerical coordinates can now be introduced,
+possessing the property that linear equations represent planes
+and straight lines. The outline of the argument by which this
+remarkable problem (in that &ldquo;distance&rdquo; is as yet undefined) is
+solved, will now be given. It is first proved that the points on
+any line can in a certain way be definitely associated with all
+the positive and negative real numbers, so as to form with them
+a one-one correspondence. The arbitrary elements in the
+establishment of this relation are the points on the line associated
+with 0, 1 and &infin;.</p>
+
+<p>This association<a name="fa35d" id="fa35d" href="#ft35d"><span class="sp">35</span></a> is most easily effected by considering a
+class of projective relations of the line with itself, called by
+F. Schur (<i>loc. cit.</i>) <i>prospectivities</i>.</p>
+
+<table class="flt" style="float: right; width: 260px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:209px; height:150px" src="images/img732a.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 69.</span></td></tr>
+<tr><td class="figright1"><img style="width:230px; height:158px" src="images/img732b.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 70.</span></td></tr>
+<tr><td class="figright1"><img style="width:216px; height:122px" src="images/img732c.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 71.</span></td></tr>
+<tr><td class="figright1"><img style="width:202px; height:156px" src="images/img732d.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 72.</span></td></tr></table>
+
+<div class="condensed">
+<p>Let l (fig. 69) be the given line, m and n any two lines intersecting
+at U on l, S and S&prime; two points on n. Then a projective relation
+between l and itself is formed by projecting l from S on to m, and
+then by projecting m from S&prime; back on to l. All such projective
+relations, however m, n, S and S&prime; be varied, are called &ldquo;prospectivities,&rdquo;
+and U is the double point of the prospectivity. If a point
+O on l is related to A by a prospectivity, then all prospectivities,
+which (1) have the same double point
+U, and (2) relate O to A, give the same
+correspondent (Q, in figure) to any
+point P on the line l; in fact they are
+all the same prospectivity, however
+m, n, S, and S&prime; may have been varied
+subject to these conditions. Such
+a prospectivity will be denoted by
+(OAU²).</p>
+
+<p>The sum of two prospectivities,
+written (OAU²) + (OBU²), is defined
+to be that transformation of the line
+l into itself which is obtained by first applying the prospectivity
+(OAU²) and then applying the prospectivity (OBU²). Such a
+transformation, when the two summands have the same double
+point, is itself a prospectivity with that double point.</p>
+
+<p>With this definition of addition it can be proved that prospectivities
+with the same double point satisfy all the axioms of magnitude.
+Accordingly they can be associated in a one-one correspondence
+with the positive and negative real numbers. Let E
+(fig. 70) be any point on l, distinct from O and U. Then the
+prospectivity (OEU²) is associated with unity, the prospectivity
+(OOU²) is associated with zero,
+and (OUU²) with &infin;. The prospectivities
+of the type (OPU²),
+where P is any point on the segment
+OEU, correspond to the positive
+numbers; also if P&prime; is the
+harmonic conjugate of P with
+respect to O and U, the prospectivity
+(OP&prime;U²) is associated with
+the corresponding negative number.
+(The subjoined figure explains this
+relation of the positive and negative
+prospectivities.) Then any
+point P on l is associated with the same number as is the prospectivity
+(OPU²).</p>
+
+<p>It can be proved that the order of the numbers in algebraic order
+of magnitude agrees with the order on the line of the associated
+points. Let the numbers, assigned according to the preceding
+specification, be said to be associated with the points according to
+the &ldquo;numeration-system (OEU).&rdquo; The introduction of a coordinate
+system for a plane is now managed
+as follows: Take any triangle OUV
+in the plane, and on the lines OU
+and OV establish the numeration
+systems (OE<span class="su">1</span>U) and (OE<span class="su">2</span>V), where
+E<span class="su">1</span> and E<span class="su">2</span> are arbitrarily chosen.
+Then (cf. fig. 71) if M and N are
+associated with the numbers x and
+y according to these systems, the
+coordinates of P are x and y. It then
+follows that the equation of a straight
+line is of the form ax + by + c = 0. Both coordinates of any point on
+the line UV are infinite. This can be avoided by introducing
+homogeneous coordinates X, Y, Z, where x = X/Z, and y = Y/Z, and
+Z = 0 is the equation of UV.</p>
+
+<p>The procedure for three dimensions is similar. Let OUVW
+(fig. 72) be any tetrahedron, and associate points on OU, OV, OW
+with numbers according to the numeration
+systems (OE<span class="su">1</span>U), (OE<span class="su">2</span>V), and
+(OE<span class="su">3</span>W). Let the planes VWP, WUP,
+UVP cut OU, OV, OW in L, M, N respectively;
+and let x, y, z be the numbers
+associated with L, M, N respectively.
+Then P is the point (x, y, z). Also
+homogeneous coordinates can be introduced
+as before, thus avoiding the
+infinities on the plane UVW.</p>
+
+<p>The cross ratio of a range of four
+collinear points can now be defined
+as a number characteristic of that range. Let the coordinates of any
+point P<span class="su">r</span> of the range P<span class="su">1</span> P<span class="su">2</span> P<span class="su">3</span> P<span class="su">4</span> be</p>
+
+<table class="math0" summary="math">
+<tr><td>&lambda;<span class="su">r</span>a + &mu;<span class="su">r</span> + a&prime;</td>
+<td rowspan="2">, &emsp;</td> <td>&lambda;<span class="su">r</span>b + &mu;<span class="su">r</span>b&prime;</td>
+<td rowspan="2">, &emsp;</td> <td>&lambda;<span class="su">r</span>c + &mu;<span class="su">r</span>c&prime;</td>
+<td rowspan="2">, &emsp; (r = 1, 2, 3, 4)</td></tr>
+<tr><td class="denom">&lambda;<span class="su">r</span> + &mu;<span class="su">r</span></td> <td class="denom">&lambda;<span class="su">r</span> + &mu;<span class="su">r</span></td>
+<td class="denom">&lambda;<span class="su">r</span> + &mu;<span class="su">r</span></td></tr></table>
+
+<p class="noind">and let (&lambda;<span class="su">r</span>&mu;<span class="su">s</span>) be written for &lambda;<span class="su">r</span>&mu;<span class="su">s</span> -&lambda;<span class="su">s</span>&mu;<span class="su">r</span>. Then the cross ratio
+{P<span class="su">1</span> P<span class="su">2</span> P<span class="su">3</span> P<span class="su">4</span>} is defined to be the number
+(&lambda;<span class="su">1</span>&mu;<span class="su">2</span>)(&lambda;<span class="su">3</span>&mu;<span class="su">4</span>) / (&lambda;<span class="su">1</span>&mu;<span class="su">4</span>)(&lambda;<span class="su">3</span>&mu;<span class="su">2</span>).
+The equality of the cross ratios of the ranges (P<span class="su">1</span> P<span class="su">2</span> P<span class="su">3</span> P<span class="su">4</span>) and
+(Q<span class="su">1</span> Q<span class="su">2</span> Q<span class="su">3</span> Q<span class="su">4</span>) is proved to be the necessary and sufficient condition
+for their mutual projectivity. The cross ratios of all harmonic
+ranges are then easily seen to be all equal to -1, by comparing with
+the range (OE<span class="su">1</span>UE&prime;<span class="su">1</span>) on the axis of x.</p>
+
+<p>Thus all the ordinary propositions of geometry in which distance
+and angular measure do not enter otherwise than in cross ratios
+can now be enunciated and proved. Accordingly the greater part of
+the analytical theory of conics and quadrics belongs to geometry
+<span class="pagenum"><a name="page733" id="page733"></a>733</span>
+at this stage The theory of distance will be considered after the
+principles of descriptive geometry have been developed.</p>
+</div>
+
+<p class="pt2 center"><i>Descriptive Geometry.</i></p>
+
+<p>Descriptive geometry is essentially the science of multiple
+order for open series. The first satisfactory system of axioms
+was given by M. Pasch.<a name="fa36d" id="fa36d" href="#ft36d"><span class="sp">36</span></a> An improved version is due to G.
+Peano.<a name="fa37d" id="fa37d" href="#ft37d"><span class="sp">37</span></a> Both these authors treat the idea of the class of points
+constituting the segment lying <i>between</i> two points as an undefined
+fundamental idea. Thus in fact there are in this system two
+fundamental ideas, namely, of points and of segments. It is
+then easy enough to define the prolongations of the segments,
+so as to form the complete straight lines. D. Hilbert&rsquo;s<a name="fa38d" id="fa38d" href="#ft38d"><span class="sp">38</span></a> formulation
+of the axioms is in this respect practically based on the same
+fundamental ideas. His work is justly famous for some of the
+mathematical investigations contained in it, but his exposition of
+the axioms is distinctly inferior to that of Peano. Descriptive
+geometry can also be considered<a name="fa39d" id="fa39d" href="#ft39d"><span class="sp">39</span></a> as the science of a class of
+relations, each relation being a two-termed serial relation, as
+considered in the logic of relations, ranging the points between
+which it holds into a linear open order. Thus the relations are
+the straight lines, and the terms between which they hold are
+the points. But a combination of these two points of view
+yields<a name="fa40d" id="fa40d" href="#ft40d"><span class="sp">40</span></a> the simplest statement of all. Descriptive geometry is
+then conceived as the investigation of an undefined fundamental
+relation between three terms (points); and when the relation
+holds between three points A, B, C, the points are said to be &ldquo;in
+the [linear] order ABC.&rdquo;</p>
+
+<p>O. Veblen&rsquo;s axioms and definitions, slightly modified, are as
+follows:&mdash;</p>
+
+<p>1. If the points A, B, C are in the order ABC, they are in the
+order CBA.</p>
+
+<p>2. If the points A, B, C are in the order ABC, they are not
+in the order BCA.</p>
+
+<p>3. If the points A, B, C are in the order ABC, A is distinct
+from C.</p>
+
+<p>4. If A and B are any two distinct points, there exists a point
+C such that A, B, C are in the order ABC.</p>
+
+<div class="condensed">
+<p><i>Definition.</i>&mdash;The <i>line</i> AB (A &#8800; B) consists of A and B, and of all
+points X in one of the possible orders, ABX, AXB, XAB. The
+points X in the order AXB constitute the <i>segment</i> AB.</p>
+</div>
+
+<p>5. If points C and D (C &#8800; D) lie on the line AB, then A lies on
+the line CD.</p>
+
+<p>6. There exist three distinct points A, B, C not in any of the
+orders ABC, BCA, CAB.</p>
+
+<table class="flt" style="float: right; width: 260px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:211px; height:136px" src="images/img733a.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 73.</span></td></tr></table>
+
+<p>7. If three distinct points A, B, C (fig. 73) do not lie on the
+same line, and D and E are two distinct points in the orders
+BCD and CEA, then a point F exists
+in the order AFB, and such that
+D, E, F are collinear.</p>
+
+<div class="condensed">
+<p><i>Definition.</i>&mdash;If A, B, C are three
+non-collinear points, the <i>plane</i> ABC
+is the class of points which lie on any
+one of the lines joining any two of the
+points belonging to the <i>boundary</i> of
+the triangle ABC, the boundary being
+formed by the segments BC, CA and
+AB. The <i>interior</i> of the triangle ABC is formed by the points in
+segments such as PQ, where P and Q are points respectively on
+two of the segments BC, CA, AB.</p>
+</div>
+
+<p>8. There exists a plane ABC, which does not contain all the
+points.</p>
+
+<div class="condensed">
+<p><i>Definition.</i>&mdash;If A, B, C, D are four non-coplanar points, the space
+ABCD is the class of points which lie on any of the lines containing
+two points on the surface of the tetrahedron ABCD, the <i>surface</i>
+being formed by the interiors of the triangles ABC, BCD, DCA,
+DAB.</p>
+</div>
+
+<p>9. There exists a space ABCD which contains all the points.</p>
+
+<p>10. The Dedekind property holds for the order of the points
+on any straight line.</p>
+
+<p>It follows from axioms 1-9 that the points on any straight line
+are arranged in an open serial order. Also all the ordinary
+theorems respecting a point dividing a straight line into two
+parts, a straight line dividing a plane into two parts, and a plane
+dividing space into two parts, follow.</p>
+
+<div class="condensed">
+<p>Again, in any plane &alpha; consider a line l and a point A (fig. 74).</p>
+
+<table class="flt" style="float: left; width: 250px;" summary="Illustration">
+<tr><td class="figleft1"><img style="width:198px; height:155px" src="images/img733b.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 74.</span></td></tr></table>
+
+<p>Let any point B divide l into two half-lines l<span class="su">1</span> and l<span class="su">2</span>. Then it can
+be proved that the set of half-lines, emanating from A and intersecting
+l<span class="su">1</span> (such as m), are bounded by two half-lines, of which ABC
+is one. Let r be the other. Then it can be proved that r does not
+intersect l<span class="su">1</span>. Similarly for the half-line,
+such as n, intersecting l<span class="su">2</span>. Let s be its
+bounding half-line. Then two cases are
+possible. (1) The half-lines r and s are
+collinear, and together form one complete
+line. In this case, there is one and
+only one line (viz. r + s) through A and
+lying in &alpha; which does not intersect l.
+This is the Euclidean case, and the
+assumption that this case holds is the
+<i>Euclidean parallel axiom</i>. But (2) the
+half-lines r and s may not be collinear.
+In this case there will be an infinite
+number of lines, such as k for instance, containing A and lying in &alpha;,
+which do not intersect l. Then the lines through A in &alpha; are divided
+into two classes by reference to l, namely, the <i>secant</i> lines which
+intersect l, and the <i>non-secant</i> lines which do not intersect l. The
+two boundary non-secant lines, of which r and s are respectively
+halves, may be called the two parallels to l through A.</p>
+
+<p>The perception of the possibility of case 2 constituted the starting-point
+from which Lobatchewsky constructed the first explicit
+coherent theory of non-Euclidean geometry, and thus created a
+revolution in the philosophy of the subject. For many centuries
+the speculations of mathematicians on the foundations of geometry
+were almost confined to hopeless attempts to prove the &ldquo;parallel
+axiom&rdquo; without the introduction of some equivalent axiom.<a name="fa41d" id="fa41d" href="#ft41d"><span class="sp">41</span></a></p>
+</div>
+
+<p><i>Associated Projective and Descriptive Spaces.</i>&mdash;A region of a
+projective space, such that one, and only one, of the two supplementary
+segments between any pair of points within it lies
+entirely within it, satisfies the above axioms (1-10) of descriptive
+geometry, where the points of the region are the descriptive
+points, and the portions of straight lines within the region are
+the descriptive lines. If the excluded part of the original projective
+space is a single plane, the Euclidean parallel axiom also
+holds, otherwise it does not hold for the descriptive space of the
+limited region. Again, conversely, starting from an original
+descriptive space an associated projective space can be constructed
+by means of the concept of <i>ideal points</i>.<a name="fa42d" id="fa42d" href="#ft42d"><span class="sp">42</span></a> These are also
+called <i>projective points</i>, where it is understood that the simple
+points are the points of the original descriptive space. An
+<i>ideal point</i> is the class of straight lines which is composed of two
+coplanar lines a and b, together with the lines of intersection of
+all pairs of intersecting planes which respectively contain a and b,
+together with the lines of intersection with the plane ab of all
+planes containing any one of the lines (other than a or b) already
+specified as belonging to the ideal point. It is evident that, if
+the two original lines a and b intersect, the corresponding ideal
+point is nothing else than the whole class of lines which are
+concurrent at the point ab. But the essence of the definition is
+that an ideal point has an existence when the lines a and b do
+not intersect, so long as they are coplanar. An ideal point is
+termed <i>proper</i>, if the lines composing it intersect; otherwise it
+is <i>improper</i>.</p>
+
+<p>A theorem essential to the whole theory is the following: if
+any two of the three lines a, b, c are coplanar, but the three lines
+are not all coplanar, and similarly for the lines a, b, d, then c
+and d are coplanar. It follows that any two lines belonging to an
+ideal point can be used as the pair of guiding lines in the definition.
+An ideal point is said to be <i>coherent</i> with a plane, if any of the
+lines composing it lie in the plane. An <i>ideal line</i> is the class of
+ideal points each of which is coherent with two given planes.
+<span class="pagenum"><a name="page734" id="page734"></a>734</span>
+If the planes intersect, the ideal line is termed <i>proper</i>, otherwise
+it is <i>improper</i>. It can be proved that any two planes, with which
+any two of the ideal points are both coherent, will serve as the
+guiding planes used in the definition. The ideal planes are
+defined as in projective geometry, and all the other definitions
+(for segments, order, &amp;c.) of projective geometry are applied
+to the ideal elements. If an ideal plane contains some proper
+ideal points, it is called <i>proper</i>, otherwise it is <i>improper</i>. Every
+ideal plane contains some improper ideal points.</p>
+
+<p>It can now be proved that all the axioms of projective geometry
+hold of the ideal elements as thus obtained; and also that the
+order of the ideal points as obtained by the projective method
+agrees with the order of the proper ideal points as obtained from
+that of the associated points of the descriptive geometry. Thus
+a projective space has been constructed out of the ideal elements,
+and the proper ideal elements correspond element by element with
+the associated descriptive elements. Thus the proper ideal
+elements form a region in the projective space within which the
+descriptive axioms hold. Accordingly, by substituting ideal
+elements, a descriptive space can always be considered as a
+region within a projective space. This is the justification for the
+ordinary use of the &ldquo;points at infinity&rdquo; in the ordinary Euclidean
+geometry; the reasoning has been transferred from the original
+descriptive space to the associated projective space of ideal
+elements; and with the Euclidean parallel axiom the improper
+ideal elements reduce to the ideal points on a single improper ideal
+plane, namely, the plane at infinity.<a name="fa43d" id="fa43d" href="#ft43d"><span class="sp">43</span></a></p>
+
+<p><i>Congruence and Measurement.</i>&mdash;The property of physical space
+which is expressed by the term &ldquo;measurability&rdquo; has now to be
+considered. This property has often been considered as essential
+to the very idea of space. For example, Kant writes,<a name="fa44d" id="fa44d" href="#ft44d"><span class="sp">44</span></a> &ldquo;Space
+is represented as an infinite given <i>quantity</i>.&rdquo; This quantitative
+aspect of space arises from the measurability of distances, of
+angles, of surfaces and of volumes. These four types of quantity
+depend upon the two first among them as fundamental. The
+measurability of space is essentially connected with the idea of
+<i>congruence</i>, of which the simplest examples are to be found in
+the proofs of equality by the method of superposition, as used
+in elementary plane geometry. The mere concepts of &ldquo;part&rdquo;
+and of &ldquo;whole&rdquo; must of necessity be inadequate as the foundation
+of measurement, since we require the comparison as to
+quantity of regions of space which have no portions in common.
+The idea of congruence, as exemplified by the method of superposition
+in geometrical reasoning, appears to be founded upon
+that of the &ldquo;rigid body,&rdquo; which moves from one position to
+another with its internal spatial relations unchanged. But unless
+there is a previous concept of the metrical relations between the
+parts of the body, there can be no basis from which to deduce
+that they are unchanged.</p>
+
+<p>It would therefore appear as if the idea of the congruence, or
+metrical equality, of two portions of space (as empirically suggested
+by the motion of rigid bodies) must be considered as a
+fundamental idea incapable of definition in terms of those
+geometrical concepts which have already been enumerated.
+This was in effect the point of view of Pasch.<a name="fa45d" id="fa45d" href="#ft45d"><span class="sp">45</span></a> It has, however,
+been proved by Sophus Lie<a name="fa46d" id="fa46d" href="#ft46d"><span class="sp">46</span></a> that congruence is capable of
+definition without recourse to a new fundamental idea. This
+he does by means of his theory of finite continuous groups (see
+<span class="sc"><a href="#artlinks">Groups, Theory of</a></span>), of which the definition is possible in terms
+of our established geometrical ideas, remembering that coordinates
+have already been introduced. The displacement
+of a rigid body is simply a mode of defining to the senses a one-one
+transformation of all space into itself. For at any point of
+space a particle may be conceived to be placed, and to be rigidly
+connected with the rigid body; and thus there is a definite
+correspondence of any point of space with the new point occupied
+by the associated particle after displacement. Again two successive
+displacements of a rigid body from position A to position
+B, and from position B to position C, are the same in effect as one
+displacement from A to C. But this is the characteristic &ldquo;group&rdquo;
+property. Thus the transformations of space into itself defined
+by displacements of rigid bodies form a group.</p>
+
+<p>Call this group of transformations a congruence-group. Now
+according to Lie a congruence-group is defined by the following
+characteristics:&mdash;</p>
+
+<p>1. A congruence-group is a finite continuous group of one-one
+transformations, containing the identical transformation.</p>
+
+<p>2. It is a sub-group of the general projective group, <i>i.e.</i> of
+the group of which any transformation converts planes into
+planes, and straight lines into straight lines.</p>
+
+<p>3. An infinitesimal transformation can always be found satisfying
+the condition that, at least throughout a certain enclosed
+region, any definite line and any definite point on the line are
+latent, <i>i.e.</i> correspond to themselves.</p>
+
+<p>4. No infinitesimal transformation of the group exists, such
+that, at least in the region for which (3) holds, a straight line,
+a point on it, and a plane through it, shall all be latent.</p>
+
+<p>The property enunciated by conditions (3) and (4), taken
+together, is named by Lie &ldquo;Free mobility in the infinitesimal.&rdquo;
+Lie proves the following theorems for a projective space:&mdash;</p>
+
+<div class="condensed">
+<p>1. If the above four conditions are only satisfied by a group
+throughout part of projective space, this part either (&alpha;) must be the
+region enclosed by a real closed quadric, or (&beta;) must be the whole of
+the projective space with the exception of a single plane. In case
+(&alpha;) the corresponding congruence group is the continuous group for
+which the enclosing quadric is latent; and in case (&beta;) an imaginary
+conic (with a real equation) lying in the latent plane is also latent,
+and the congruence group is the continuous group for which the
+plane and conic are latent.</p>
+
+<p>2. If the above four conditions are satisfied by a group throughout
+the whole of projective space, the congruence group is the continuous
+group for which some imaginary quadric (with a real equation) is
+latent.</p>
+
+<p>By a proper choice of non-homogeneous co-ordinates the equation
+of any quadrics of the types considered, either in theorem 1 (&alpha;), or in
+theorem 2, can be written in the form 1 + c(x² + y² + z²) = 0, where c is
+negative for a real closed quadric, and positive for an imaginary
+quadric. Then the general infinitesimal transformation is defined
+by the three equations:</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcl">dx/dt = u &minus; &omega;<span class="su">3</span>y + &omega;<span class="su">2</span>z + cx (ux + vy + wz),</td> <td class="tccm" rowspan="3">(A)</td></tr>
+
+<tr><td class="tcl">dy/dt = v &minus; &omega;<span class="su">1</span>z + &omega;<span class="su">3</span>x + cy (ux + vy + wz),</td></tr>
+
+<tr><td class="tcl">dz/dt = w &minus; &omega;<span class="su">2</span>x + &omega;<span class="su">1</span>y + cz (ux + vy + wz).</td></tr>
+</table>
+
+<p class="noind">In the ease considered in theorem 1 (&beta;), with the proper choice of
+co-ordinates the three equations defining the general infinitesimal
+transformation are:</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcl">dx/dt = u &minus; &omega;<span class="su">3</span>y + &omega;<span class="su">2</span>z,</td> <td class="tccm" rowspan="3">(B)</td></tr>
+
+<tr><td class="tcl">dy/dt = v &minus; &omega;<span class="su">1</span>z + &omega;<span class="su">3</span>x,</td></tr>
+
+<tr><td class="tcl">dz/dt = w &minus; &omega;<span class="su">2</span>x + &omega;<span class="su">1</span>y.</td></tr>
+</table>
+
+<p class="noind">In this case the latent plane is the plane for which at least one of
+x, y, z are infinite, that is, the plane 0.x + 0.y + 0.z + a = 0; and the
+latent conic is the conic in which the cone x² + y² + z² = 0 intersects
+the latent plane.</p>
+</div>
+
+<p>It follows from theorems 1 and 2 that there is not one unique
+congruence-group, but an indefinite number of them. There is
+one congruence-group corresponding to each closed real quadric,
+one to each imaginary quadric with a real equation, and one to
+each imaginary conic in a real plane and with a real equation.
+The quadric thus associated with each congruence-group is
+called the <i>absolute</i> for that group, and in the degenerate case
+of 1 (&beta;) the absolute is the latent plane together with the latent
+imaginary conic. If the absolute is real, the congruence-group
+is <i>hyperbolic</i>; if imaginary, it is <i>elliptic</i>; if the absolute is a
+plane and imaginary conic, the group is parabolic. Metrical
+geometry is simply the theory of the properties of some particular
+congruence-group selected for study.</p>
+
+<div class="condensed">
+<p>The definition of distance is connected with the corresponding
+congruence-group by two considerations in respect to a range of five
+points (A<span class="su">1</span>, A<span class="su">2</span>, P<span class="su">1</span>, P<span class="su">2</span>, P<span class="su">3</span>), of which A<span class="su">1</span> and A<span class="su">2</span> are on the absolute.</p>
+
+<p>Let {A<span class="su">1</span>P<span class="su">1</span>A<span class="su">2</span>P<span class="su">2</span>} stand for the cross ratio (as defined above) of the
+range (A<span class="su">1</span>P<span class="su">1</span>A<span class="su">2</span>P<span class="su">2</span>), with a similar notation for the other ranges.
+Then</p>
+
+<p class="noind">(1)</p>
+
+<p class="center">log {A<span class="su">1</span>P<span class="su">1</span>A<span class="su">2</span>P<span class="su">2</span>} + log {A<span class="su">1</span>P<span class="su">2</span>A<span class="su">2</span>P<span class="su">3</span>} = log {A<span class="su">1</span>P<span class="su">1</span>A<span class="su">2</span>P<span class="su">3</span>},</p>
+
+<p class="noind">and</p>
+
+<p class="noind">(2), if the points A<span class="su">1</span>, A<span class="su">2</span>, P<span class="su">1</span>, P<span class="su">2</span> are transformed into A&prime;<span class="su">1</span>, A&prime;<span class="su">2</span>, P&prime;<span class="su">1</span>, P&prime;<span class="su">2</span>
+by any transformation of the congruence-group, (&alpha;) {A<span class="su">1</span>P<span class="su">1</span>A<span class="su">2</span>P<span class="su">2</span> =
+{A&prime;<span class="su">1</span>P&prime;<span class="su">1</span>A&prime;<span class="su">2</span>P&prime;<span class="su">2</span>}, since the transformation is projective, and (&beta;) A&prime;<span class="su">1</span>, A&prime;<span class="su">2</span>
+are on the absolute since A<span class="su">1</span> and A<span class="su">2</span> are on it. Thus if we define
+<span class="pagenum"><a name="page735" id="page735"></a>735</span>
+the distance P<span class="su">1</span>P<span class="su">2</span> to be ½k log {A<span class="su">1</span>P<span class="su">1</span>A<span class="su">2</span>P<span class="su">2</span>}, where A<span class="su">1</span> and A<span class="su">2</span> are the
+points in which the line P<span class="su">1</span>P<span class="su">2</span> cuts the absolute, and k is some constant,
+the two characteristic properties of distance, namely, (1) the
+addition of consecutive lengths on a straight line, and (2) the invariability
+of distances during a transformation of the congruence-group,
+are satisfied. This is the well-known Cayley-Klein projective
+definition<a name="fa47d" id="fa47d" href="#ft47d"><span class="sp">47</span></a> of distance, which was elaborated in view of the addition
+property alone, previously to Lie&rsquo;s discovery of the theory of congruence-groups.
+For a hyperbolic group when P<span class="su">1</span> and P<span class="su">2</span> are in the
+region enclosed by the absolute, log {A<span class="su">1</span>P<span class="su">1</span>A<span class="su">2</span>P<span class="su">2</span>} is real, and therefore
+k must be real. For an elliptic group A<span class="su">1</span> and A<span class="su">2</span> are conjugate
+imaginaries, and log {A<span class="su">1</span>P<span class="su">1</span>A<span class="su">2</span>P<span class="su">2</span>} is a pure imaginary, and k is chosen
+to be &kappa;/&iota;, where &kappa; is real and &iota; = &radic; &minus;.</p>
+
+<p>Similarly the angle between two planes, p<span class="su">1</span> and p<span class="su">2</span>, is defined to be
+(1/2&iota;) log (t<span class="su">1</span>p<span class="su">1</span>t<span class="su">2</span>p<span class="su">2</span>), where t<span class="su">1</span> and t<span class="su">2</span> are tangent planes to the absolute
+through the line p<span class="su">1</span>p<span class="su">2</span>. The planes t<span class="su">1</span> and t<span class="su">2</span> are imaginary for an
+elliptic group, and also for an hyperbolic group when the planes p<span class="su">1</span>
+and p<span class="su">2</span> intersect at points within the region enclosed by the absolute.
+The development of the consequences of these metrical definitions
+is the <span class="correction" title="amended from subjct">subject</span> of non-Euclidean geometry.</p>
+
+<p>The definitions for the parabolic case can be arrived at as limits
+of those obtained in either of the other two cases by making k
+ultimately to vanish. It is also obvious that, if P<span class="su">1</span> and P<span class="su">2</span> be the
+points (x<span class="su">1</span>, y<span class="su">1</span>, z<span class="su">1</span>) and (x<span class="su">2</span>, y<span class="su">2</span>, z<span class="su">2</span>), it follows from equations (B) above
+that {(x<span class="su">1</span> &minus; x<span class="su">2</span>)² + (y<span class="su">1</span> &minus; y<span class="su">2</span>)² + (z<span class="su">1</span> &minus; z<span class="su">2</span>)²}<span class="sp">1/2</span> is unaltered by a congruence
+transformation and also satisfies the addition property for collinear
+distances. Also the previous definition of an angle can be adapted
+to this case, by making t<span class="su">1</span> and t<span class="su">2</span> to be the tangent planes through
+the line p<span class="su">1</span>p<span class="su">2</span> to the imaginary conic. Similarly if p<span class="su">1</span> and p<span class="su">2</span> are intersecting
+lines, the same definition of an angle holds, where t<span class="su">1</span> and t<span class="su">2</span>
+are now the lines from the point p<span class="su">1</span>p<span class="su">2</span> to the two points where the
+plane p<span class="su">1</span>p<span class="su">2</span> cuts the imaginary conic. These points are in fact the
+&ldquo;circular points at infinity&rdquo; on the plane. The development of
+the consequences of these definitions for the parabolic case gives the
+ordinary Euclidean metrical geometry.</p>
+</div>
+
+<p>Thus the only metrical geometry for the whole of projective
+space is of the elliptic type. But the actual measure-relations
+(though not their general properties) differ according to the
+elliptic congruence-group selected for study. In a descriptive
+space a congruence-group should possess the four characteristics
+of such a group throughout the whole of the space. Then form
+the associated ideal projective space. The associated congruence-group
+for this ideal space must satisfy the four conditions
+throughout the region of the proper ideal points. Thus the
+boundary of this region is the absolute. Accordingly there can
+be no metrical geometry for the whole of a descriptive space
+unless its boundary (in the associated ideal space) is a closed
+quadric or a plane. If the boundary is a closed quadric, there
+is one possible congruence-group of the hyperbolic type. If
+the boundary is a plane (the plane at infinity), the possible
+congruence-groups are parabolic; and there is a congruence-group
+corresponding to each imaginary conic in this plane,
+together with a Euclidean metrical geometry corresponding to
+each such group. Owing to these alternative possibilities, it
+would appear to be more accurate to say that systems of quantities
+can be found in a space, rather than that space is a quantity.</p>
+
+<p>Lie has also deduced<a name="fa48d" id="fa48d" href="#ft48d"><span class="sp">48</span></a> the same results with respect to congruence-groups
+from another set of defining properties, which
+explicitly assume the existence of a quantitative relation (the
+distance) between any two points, which is invariant for any
+transformation of the congruence-group.<a name="fa49d" id="fa49d" href="#ft49d"><span class="sp">49</span></a></p>
+
+<p>The above results, in respect to congruence and metrical
+geometry, considered in relation to existent space, have led to the
+doctrine<a name="fa50d" id="fa50d" href="#ft50d"><span class="sp">50</span></a> that it is intrinsically unmeaning to ask which system
+of metrical geometry is true of the physical world. Any one of
+these systems can be applied, and in an indefinite number of ways.
+The only question before us is one of convenience in respect to
+simplicity of statement of the physical laws. This point of view
+seems to neglect the consideration that science is to be relevant
+to the definite perceiving minds of men; and that (neglecting
+the ambiguity introduced by the invariable slight inexactness
+of observation which is not relevant to this special doctrine)
+we have, in fact, presented to our senses a definite set of transformations
+forming a congruence-group, resulting in a set of
+measure relations which are in no respect arbitrary. Accordingly
+our scientific laws are to be stated relevantly to that particular
+congruence-group. Thus the investigation of the type (elliptic,
+hyperbolic or parabolic) of this special congruence-group is a
+perfectly definite problem, to be decided by experiment. The
+consideration of experiments adapted to this object requires some
+development of non-Euclidean geometry (see section VI.,
+<i>Non-Euclidean Geometry</i>). But if the doctrine means that,
+assuming some sort of objective reality for the material universe,
+beings can be imagined, to whom <i>either</i> all congruence-groups
+are equally important, <i>or</i> some other congruence-group is specially
+important, the doctrine appears to be an immediate deduction
+from the mathematical facts. Assuming a definite congruence-group,
+the investigation of surfaces (or three-dimensional loci
+in space of four dimensions) with geodesic geometries of the form
+of metrical geometries of other types of congruence-groups forms
+an important chapter of non-Euclidean geometry. Arising
+from this investigation there is a widely-spread fallacy, which
+has found its way into many philosophic writings, namely, that
+the possibility of the geometry of existent three-dimensional
+space being other than Euclidean depends on the physical
+existence of Euclidean space of four or more dimensions. The
+foregoing exposition shows the baselessness of this idea.</p>
+
+<div class="condensed">
+<p><span class="sc">Bibliography</span>.&mdash;For an account of the investigations on the
+axioms of geometry during the Greek period, see M. Cantor, <i>Vorlesungen
+über die Geschichte der Mathematik</i>, Bd. i. and iii.; T.L.
+Heath, <i>The Thirteen Books of Euclid&rsquo;s Elements, a New Translation
+from the Greek, with Introductory Essays and Commentary, Historical,
+Critical, and Explanatory</i> (Cambridge, 1908)&mdash;this work is the standard
+source of information; W.B. Frankland, <i>Euclid, Book I., with a
+Commentary</i> (Cambridge, 1905)&mdash;the commentary contains copious
+extracts from the ancient commentators. The next period of really
+substantive importance is that of the 18th century. The leading
+authors are: G. Saccheri, S.J., <i>Euclides ab omni naevo vindicatus</i>
+(Milan, 1733). Saccheri was an Italian Jesuit who unconsciously
+discovered non-Euclidean geometry in the course of his efforts to
+prove its impossibility. J.H. Lambert, <i>Theorie der Parallellinien</i>
+(1766); A.M. Legendre, <i>Éléments de géométrie</i> (1794). An adequate
+account of the above authors is given by P. Stäckel and F. Engel,
+<i>Die Theorie der Parallellinien von Euklid bis auf Gauss</i> (Leipzig,
+1895). The next period of time (roughly from 1800 to 1870) contains
+two streams of thought, both of which are essential to the modern
+analysis of the subject. The first stream is that which produced the
+discovery and investigation of non-Euclidean geometries, the second
+stream is that which has produced the geometry of position, comprising
+both projective and descriptive geometry not very accurately
+discriminated. The leading authors on non-Euclidean geometry
+are K.F. Gauss, in private letters to Schumacher, cf. Stäckel and
+Engel, <i>loc. cit.</i>; N. Lobatchewsky, rector of the university of Kazan,
+to whom the honour of the effective discovery of non-Euclidean
+geometry must be assigned. His first publication was at Kazan
+in 1826. His various memoirs have been re-edited by Engel;
+cf. <i>Urkunden zur Geschichte der nichteuklidischen Geometrie</i> by
+Stäckel and Engel, vol. i. &ldquo;Lobatchewsky.&rdquo; J. Bolyai discovered
+non-Euclidean geometry apparently in independence of Lobatchewsky.
+His memoir was published in 1831 as an appendix to a
+work by his father W. Bolyai, <i>Tentamen juventutem....</i> This
+memoir has been separately edited by J. Frischauf, <i>Absolute Geometrie
+nach J. Bolyai</i> (Leipzig, 1872); B. Riemann, <i>Über die Hypothesen,
+welche der Geometrie zu Grunde liegen</i> (1854); cf. <i>Gesamte Werke</i>, a
+translation in The Collected Papers of W.K. Clifford. This is a
+fundamental memoir on the subject and must rank with the work of
+Lobatchewsky. Riemann discovered elliptic metrical geometry,
+and Lobatchewsky hyperbolic geometry. A full account of Riemann&rsquo;s
+ideas, with the subsequent developments due to Clifford,
+F. Klein and W. Killing, will be found in <i>The Boston Colloquium for
+1903</i> (New York, 1905), article &ldquo;Forms of Non-Euclidean Space,&rdquo;
+by F.S. Woods. A. Cayley, <i>loc. cit.</i> (1859), and F. Klein, &ldquo;Über die
+sogenannte nichteuklidische Geometrie,&rdquo; <i>Math. Annal.</i> vols. iv.
+and vi. (1871 and 1872), between them elaborated the projective
+theory of distance; H. Helmholtz, &ldquo;Über die thatsächlichen
+Grundlagen der Geometrie&rdquo; (1866), and &ldquo;Über die Thatsachen, die
+der Geometrie zu Grunde liegen&rdquo; (1868), both in his <i>Wissenschaftliche
+Abhandlungen</i>, vol. ii., and S. Lie, <i>loc. cit.</i> (1890 and 1893), between
+them elaborated the group theory of congruence.</p>
+
+<p>The numberless works which have been written to suggest equivalent
+alternatives to Euclid&rsquo;s parallel axioms may be neglected as
+being of trivial importance, though many of them are marvels of
+geometric ingenuity.</p>
+
+<p>The second stream of thought confined itself within the circle of
+ideas of Euclidean geometry. Its origin was mainly due to a
+<span class="pagenum"><a name="page736" id="page736"></a>736</span>
+succession of great French mathematicians, for example, G. Monge,
+<i>Géométrie descriptive</i> (1800); J.V. Poncelet, <i>Traité des proprietés
+projectives des figures</i> (1822); M. Chasles, <i>Aperēu historique sur
+l&rsquo;origine et le développement des méthodes en géométrie</i> (Bruxelles, 1837),
+and <i>Traité de géométrie supérieure</i> (Paris, 1852); and many others.
+But the works which have been, and are still, of decisive influence on
+thought as a store-house of ideas relevant to the foundations of
+geometry are K.G.C. von Staudt&rsquo;s two works, <i>Geometrie der Lage</i>
+(Nürnberg, 1847); and <i>Beiträge zur Geometrie der Lage</i> (Nürnberg,
+1856, 3rd ed. 1860).</p>
+
+<p>The final period is characterized by the successful production of
+exact systems of axioms, and by the final solution of problems
+which have occupied mathematicians for two thousand years. The
+successful analysis of the ideas involved in serial continuity is due to
+R. Dedekind, <i>Stetigkeit und irrationale Zahlen</i> (1872), and to G.
+Cantor, <i>Grundlagen einer allgemeinen Mannigfaltigkeitslehre</i> (Leipzig,
+1883), and <i>Acta math.</i> vol. 2.</p>
+
+<p>Complete systems of axioms have been stated by M. Pasch, <i>loc.
+cit.</i>; G. Peano, <i>loc. cit.</i>; M. Pieri, loc. cit.; B. Russell, <i>Principles of
+Mathematics</i>; O. Veblen, <i>loc. cit.</i>; and by G. Veronese in his treatise,
+<i>Fondamenti di geometria</i> (Padua, 1891; German transl. by A. Schepp,
+<i>Grundzüge der Geometrie</i>, Leipzig, 1894). Most of the leading memoirs
+on special questions involved have been cited in the text; in addition
+there may be mentioned M. Pieri, &ldquo;Nuovi principii di geometria
+projettiva complessa,&rdquo; <i>Trans. Accad. R. d. Sci.</i> (Turin, 1905);
+E.H. Moore, &ldquo;On the Projective Axioms of Geometry,&rdquo; <i>Trans.
+Amer. Math. Soc.</i>, 1902; O. Veblen and W.H. Bussey, &ldquo;Finite
+Projective Geometries,&rdquo; <i>Trans. Amer. Math. Soc.</i>, 1905; A.B.
+Kempe, &ldquo;On the Relation between the Logical Theory of Classes
+and the Geometrical Theory of Points,&rdquo; <i>Proc. Lond. Math. Soc.</i>,
+1890; J. Royce, &ldquo;The Relation of the Principles of Logic to the
+Foundations of Geometry,&rdquo; <i>Trans. of Amer. Math. Soc.</i>, 1905;
+A. Schoenflies, &ldquo;Über die Möglichkeit einer projectiven Geometrie
+bei transfiniter (nichtarchimedischer) Massbestimmung,&rdquo; Deutsch.
+<i>M.-V. Jahresb.</i>, 1906.</p>
+
+<p>For general expositions of the bearings of the above investigations,
+cf. Hon. Bertrand Russell, <i>loc. cit.</i>; L. Couturat, <i>Les Principes
+des mathématiques</i> (Paris, 1905); H. Poincaré, <i>loc. cit.</i>; Russell
+and Whitehead, <i>Principia mathematica</i> (Cambridge, Univ. Press).
+The philosophers whose views on space and geometric truth deserve
+especial study are Descartes, Leibnitz, Hume, Kant and J.S.
+Mill.</p>
+</div>
+<div class="author">(A. N. W.)</div>
+
+<hr class="foot" /> <div class="note">
+
+<p><a name="ft1d" id="ft1d" href="#fa1d"><span class="fn">1</span></a> For Egyptian geometry see <span class="sc"><a href="#artlinks">Egypt</a></span>, § <i>Science and Mathematics</i>.</p>
+
+<p><a name="ft2d" id="ft2d" href="#fa2d"><span class="fn">2</span></a> Cf. A.N. Whitehead, <i>Universal Algebra</i>, Bk. vi. (Cambridge,
+1898).</p>
+
+<p><a name="ft3d" id="ft3d" href="#fa3d"><span class="fn">3</span></a> Cf. A.N. Whitehead, <i>loc. cit.</i></p>
+
+<p><a name="ft4d" id="ft4d" href="#fa4d"><span class="fn">4</span></a> Cf. A.N. Whitehead, &ldquo;The Geodesic Geometry of Surfaces in
+non-Euclidean Space,&rdquo; <i>Proc. Lond. Math. Soc.</i> vol. xxix.</p>
+
+<p><a name="ft5d" id="ft5d" href="#fa5d"><span class="fn">5</span></a> Cf. Klein, &ldquo;Zur nicht-Euklidischen Geometrie,&rdquo; <i>Math. Annal.</i>
+vol. xxxvii.</p>
+
+<p><a name="ft6d" id="ft6d" href="#fa6d"><span class="fn">6</span></a> On the theory of parallels before Lobatchewsky, see Stäckel und
+Engel, <i>Theorie der Parallellinien von Euklid bis auf Gauss</i> (Leipzig,
+1895). The foregoing remarks are based upon the materials collected
+in this work.</p>
+
+<p><a name="ft7d" id="ft7d" href="#fa7d"><span class="fn">7</span></a> See Stäckel und Engel, <i>op. cit.</i>, and &ldquo;Gauss, die beiden Bolyai,
+und die nicht-Euklidische Geometrie,&rdquo; <i>Math. Annalen</i>, Bd. xlix.;
+also Engel&rsquo;s translation of Lobatchewsky (Leipzig, 1898), pp. 378 ff.</p>
+
+<p><a name="ft8d" id="ft8d" href="#fa8d"><span class="fn">8</span></a> Lobatchewsky&rsquo;s works on the subject are the following:&mdash;&ldquo;On
+the Foundations of Geometry,&rdquo; <i>Kazań Messenger</i>, 1829-1830;
+&ldquo;New Foundations of Geometry, with a complete Theory of
+Parallels,&rdquo; <i>Proceedings of the University of Kazań</i>, 1835 (both in
+Russian, but translated into German by Engel, Leipzig, 1898);
+&ldquo;Géométrie imaginaire,&rdquo; Crelle&rsquo;s Journal, 1837; <i>Theorie der
+Parallellinien</i> (Berlin, 1840; 2nd ed., 1887; translated by Halsted,
+Austin, Texas, 1891). His results appear to have been set forth in a
+paper (now lost) which he read at Kazań in 1826.</p>
+
+<p><a name="ft9d" id="ft9d" href="#fa9d"><span class="fn">9</span></a> Translated by Halsted (Austin, Texas, 4th ed., 1896.)</p>
+
+<p><a name="ft10d" id="ft10d" href="#fa10d"><span class="fn">10</span></a> <i>Abhandlungen d. Königl. Ges. d. Wiss. zu Göttingen</i>, Bd. xiii.;
+<i>Ges. math. Werke</i>, pp. 254-269; translated by Clifford, <i>Collected
+Mathematical Papers</i>.</p>
+
+<p><a name="ft11d" id="ft11d" href="#fa11d"><span class="fn">11</span></a> Cf. <i>Gesamm. math. und phys. Werke</i>, vol. i. (Leipzig, 1894).</p>
+
+<p><a name="ft12d" id="ft12d" href="#fa12d"><span class="fn">12</span></a> <i>Wiss. Abh.</i> vol. ii. pp. 610, 618 (1866, 1868).</p>
+
+<p><a name="ft13d" id="ft13d" href="#fa13d"><span class="fn">13</span></a> <i>Mind</i>, O.S., vols. i. and iii.; <i>Vorträge und Reden</i>, vol. ii. pp. 1,
+256.</p>
+
+<p><a name="ft14d" id="ft14d" href="#fa14d"><span class="fn">14</span></a> His papers are &ldquo;Saggio di interpretazione della geometria non-Euclidea,&rdquo;
+<i>Giornale di matematiche</i>, vol. vi. (1868); &ldquo;Teoria fondamentale
+degli spazii di curvatura costante,&rdquo; <i>Annali di matematica</i>,
+vol. ii. (1868-1869). Both were translated into French by J. Hoüel,
+<i>Annales scientifiques de l&rsquo;École Normale supérieure</i>, vol. vi. (1869).</p>
+
+<p><a name="ft15d" id="ft15d" href="#fa15d"><span class="fn">15</span></a> Beltrami shows also that this definition agrees with that of Gauss.</p>
+
+<p><a name="ft16d" id="ft16d" href="#fa16d"><span class="fn">16</span></a> &ldquo;Sur la théorie des foyers,&rdquo; <i>Nouv. Ann.</i> vol. xii.</p>
+
+<p><a name="ft17d" id="ft17d" href="#fa17d"><span class="fn">17</span></a> <i>Math. Annalen</i>, iv. vi., 1871-1872.</p>
+
+<p><a name="ft18d" id="ft18d" href="#fa18d"><span class="fn">18</span></a> For an investigation of these and similar properties, see Whitehead,
+<i>Universal Algebra</i> (Cambridge, 1898), bk. vi. ch. ii. The polar
+form was independently discovered by Simon Newcomb in 1877.</p>
+
+<p><a name="ft19d" id="ft19d" href="#fa19d"><span class="fn">19</span></a> For an analysis of Leibnitz&rsquo;s ideas on space, cf. B. Russell, <i>The
+Philosophy of Leibnitz</i>, chs. viii.-x.</p>
+
+<p><a name="ft20d" id="ft20d" href="#fa20d"><span class="fn">20</span></a> Cf. Hon. Bertrand Russell, &ldquo;Is Position in Time and Space
+Absolute or Relative?&rdquo; <i>Mind</i>, n.s. vol. 10 (1901), and A.N. Whitehead,
+&ldquo;Mathematical Concepts of the Material World,&rdquo; <i>Phil. Trans.</i>
+(1906), p. 205.</p>
+
+<p><a name="ft21d" id="ft21d" href="#fa21d"><span class="fn">21</span></a> Cf. <i>Critique of Pure Reason</i>, 1st section: &ldquo;Of Space,&rdquo; conclusion
+A, Max Müller&rsquo;s translation.</p>
+
+<p><a name="ft22d" id="ft22d" href="#fa22d"><span class="fn">22</span></a> Cf. Ernst Mach, <i>Erkenntniss und Irrtum</i> (Leipzig); the relevant
+chapters are translated by T.J. McCormack, <i>Space and Geometry</i>
+(London, 1906); also A. Meinong, <i>Über die Stellung der Gegenstandstheorie
+im System der Wissenschaften</i> (Leipzig, 1907).</p>
+
+<p><a name="ft23d" id="ft23d" href="#fa23d"><span class="fn">23</span></a> Cf. Russell, <i>Principles of Mathematics</i>, § 352 (Cambridge, 1903).</p>
+
+<p><a name="ft24d" id="ft24d" href="#fa24d"><span class="fn">24</span></a> Cf. A.N. Whitehead, <i>The Axioms of Projective Geometry</i>, § 3
+(Cambridge, 1906).</p>
+
+<p><a name="ft25d" id="ft25d" href="#fa25d"><span class="fn">25</span></a> Cf. Russell, <i>Princ. of Math.</i>, ch. i.</p>
+
+<p><a name="ft26d" id="ft26d" href="#fa26d"><span class="fn">26</span></a> Cf. Russell, <i>loc. cit.</i>, and G. Frege, &ldquo;Über die Grundlagen der
+Géométrie,&rdquo; <i>Jahresber. der Deutsch. Math. Ver.</i> (1906).</p>
+
+<p><a name="ft27d" id="ft27d" href="#fa27d"><span class="fn">27</span></a> This formulation&mdash;though not in respect to number&mdash;is in all
+essentials that of M. Pieri, cf. &ldquo;I principii della Geometria di Posizione,&rdquo;
+<i>Accad. R. di Torino</i> (1898); also cf. Whitehead, <i>loc. cit.</i></p>
+
+<p><a name="ft28d" id="ft28d" href="#fa28d"><span class="fn">28</span></a> Cf. G. Peano, &ldquo;Sui fondamenti della Geometria,&rdquo; p. 73, <i>Rivista
+di matematica</i>, vol. iv. (1894), and D. Hilbert, <i>Grundlagen der Geometrie</i>
+(Leipzig, 1899); and R.F. Moulton, &ldquo;A Simple non-Desarguesian
+Plane Geometry,&rdquo; <i>Trans. Amer. Math. Soc.</i>, vol. iii. (1902).</p>
+
+<p><a name="ft29d" id="ft29d" href="#fa29d"><span class="fn">29</span></a> Cf. &ldquo;Sui postulati fondamentali della geometria projettiva,&rdquo;
+<i>Giorn. di matematica</i>, vol. xxx. (1891); also of Pieri, loc. cit., and
+Whitehead, <i>loc. cit.</i></p>
+
+<p><a name="ft30d" id="ft30d" href="#fa30d"><span class="fn">30</span></a> Cf. Hilbert, <i>loc. cit.</i>; for a fuller exposition of Hilbert&rsquo;s proof
+cf. K.T. Vahlen, <i>Abstrakte Geometrie</i> (Leipzig, 1905), also Whitehead,
+<i>loc. cit.</i></p>
+
+<p><a name="ft31d" id="ft31d" href="#fa31d"><span class="fn">31</span></a> Cf. H. Wiener, <i>Jahresber. der Deutsch. Math. Ver.</i> vol. i. (1890);
+and F. Schur, &ldquo;Über den Fundamentalsatz der projectiven Geometrie,&rdquo;
+<i>Math. Ann.</i> vol. li. (1899).</p>
+
+<p><a name="ft32d" id="ft32d" href="#fa32d"><span class="fn">32</span></a> Cf. Hilbert, <i>loc. cit.</i>, and Whitehead, <i>loc. cit.</i></p>
+
+<p><a name="ft33d" id="ft33d" href="#fa33d"><span class="fn">33</span></a> Cf. Dedekind, <i>Stetigkeit und irrationale Zahlen</i> (1872).</p>
+
+<p><a name="ft34d" id="ft34d" href="#fa34d"><span class="fn">34</span></a> Cf. v. Staudt, <i>Geometrie der Lage</i> (1847).</p>
+
+<p><a name="ft35d" id="ft35d" href="#fa35d"><span class="fn">35</span></a> Cf. Pasch, <i>Vorlesungen über neuere Geometrie</i> (Leipzig, 1882), a
+classic work; also Fiedler, <i>Die darstellende Geometrie</i> (1st ed., 1871,
+3rd ed., 1888); Clebsch, <i>Vorlesungen über Geometrie</i>, vol. iii.;
+Hilbert, <i>loc. cit.</i>; F. Schur, <i>Math. Ann. Bd.</i> lv. (1902); Vahlen,
+<i>loc. cit.</i>; Whitehead, <i>loc. cit.</i></p>
+
+<p><a name="ft36d" id="ft36d" href="#fa36d"><span class="fn">36</span></a> Cf. <i>loc. cit.</i></p>
+
+<p><a name="ft37d" id="ft37d" href="#fa37d"><span class="fn">37</span></a> Cf. <i>I Principii di geometria</i> (Turin, 1889) and &ldquo;Sui fondamenti
+della geometria,&rdquo; <i>Rivista di mat.</i> vol. iv. (1894).</p>
+
+<p><a name="ft38d" id="ft38d" href="#fa38d"><span class="fn">38</span></a> Cf. <i>loc. cit.</i></p>
+
+<p><a name="ft39d" id="ft39d" href="#fa39d"><span class="fn">39</span></a> Cf. Vailati, <i>Rivista di mat.</i> vol. iv. and Russell, <i>loc. cit.</i> § 376.</p>
+
+<p><a name="ft40d" id="ft40d" href="#fa40d"><span class="fn">40</span></a> Cf. O. Veblen, &ldquo;On the Projective Axioms of Geometry,&rdquo;
+<i>Trans. Amer. Math. Soc.</i> vol. iii. (1902).</p>
+
+<p><a name="ft41d" id="ft41d" href="#fa41d"><span class="fn">41</span></a> Cf. P. Stäckel and F. Engel, <i>Die Theorie der Parallellinien von
+Euklid bis auf Gauss</i> (Leipzig, 1895).</p>
+
+<p><a name="ft42d" id="ft42d" href="#fa42d"><span class="fn">42</span></a> Cf. Pasch, loc. cit., and R. Bonola, &ldquo;Sulla introduzione degli
+enti improprii in geometria projettive,&rdquo; <i>Giorn. di mat.</i> vol. xxxviii.
+(1900); and Whitehead, <i>Axioms of Descriptive Geometry</i> (Cambridge,
+1907).</p>
+
+<p><a name="ft43d" id="ft43d" href="#fa43d"><span class="fn">43</span></a> The original idea (confined to this particular case) of ideal
+points is due to von Staudt (<i>loc. cit.</i>).</p>
+
+<p><a name="ft44d" id="ft44d" href="#fa44d"><span class="fn">44</span></a> Cf. <i>Critique</i>, &ldquo;Trans. Aesth.&rdquo; Sect. I.</p>
+
+<p><a name="ft45d" id="ft45d" href="#fa45d"><span class="fn">45</span></a> Cf. <i>loc. cit.</i></p>
+
+<p><a name="ft46d" id="ft46d" href="#fa46d"><span class="fn">46</span></a> Cf. <i>Über die Grundlagen der Geometrie</i> (Leipzig, Ber., 1890);
+and <i>Theorie der Transformationsgruppen</i> (Leipzig, 1893), vol. iii.</p>
+
+<p><a name="ft47d" id="ft47d" href="#fa47d"><span class="fn">47</span></a> Cf. A. Cayley, &ldquo;A Sixth Memoir on Quantics,&rdquo; <i>Trans. Roy. Soc.</i>,
+1859, and <i>Coll. Papers</i>, vol. ii.; and F. Klein, <i>Math. Ann.</i> vol. iv.,
+1871.</p>
+
+<p><a name="ft48d" id="ft48d" href="#fa48d"><span class="fn">48</span></a> Cf. <i>loc. cit.</i></p>
+
+<p><a name="ft49d" id="ft49d" href="#fa49d"><span class="fn">49</span></a> For similar deductions from a third set of axioms, suggested in
+essence by Peano, Riv. mat. vol. iv. loc. cit. cf. Whitehead, <i>Desc.
+Geom.</i> <i>loc. cit.</i></p>
+
+<p><a name="ft50d" id="ft50d" href="#fa50d"><span class="fn">50</span></a> Cf. H. Poincaré, <i>La Science et l&rsquo;hypothčse</i>, ch. iii.</p>
+</div>
+
+<hr class="art" />
+
+
+
+
+
+
+
+
+
+<pre>
+
+
+
+
+
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@@ -0,0 +1,24182 @@
+The Project Gutenberg EBook of Encyclopaedia Britannica, 11th Edition,
+Volume 11, Slice 6, by Various
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+Title: Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6
+       "Geodesy" to "Geometry"
+
+Author: Various
+
+Release Date: September 17, 2011 [EBook #37461]
+
+Language: English
+
+Character set encoding: ASCII
+
+*** START OF THIS PROJECT GUTENBERG EBOOK ENCYCLOPAEDIA BRITANNICA ***
+
+
+
+
+Produced by Marius Masi, Don Kretz and the Online
+Distributed Proofreading Team at http://www.pgdp.net
+
+
+
+
+
+
+
+
+
+Transcriber's notes:
+
+(1) Numbers following letters (without space) like C2 were originally
+      printed in subscript. Letter subscripts are preceded by an
+      underscore, like C_n.
+
+(2) Characters following a carat (^) were printed in superscript.
+
+(3) Side-notes were relocated to function as titles of their respective
+      paragraphs.
+
+(4) Macrons and breves above letters and dots below letters were not
+      inserted.
+
+(5) [root] stands for the root symbol; [alpha], [beta], etc. for greek
+      letters.
+
+(6) The following typographical errors have been corrected:
+
+    ARTICLE GEOFFREY: "... his history in chiefly one of quarrels, with
+      the see of Canterbury, with the chancellor William Longchamp, with
+      his half-brothers Richard and John, and especially with his canons
+      at York." 'William' amended from 'Willian'.
+
+    ARTICLE GEOLOGY: "... and at the same time greater appreciation has
+      been shown of the signification and strength of the geological
+      proofs of the high antiquity of our planet." 'strength' amended
+      from 'stength'.
+
+    ARTICLE GEOLOGY: "... it can be demonstrated that sometimes an inch
+      or two of sediment might, on certain horizons, represent the
+      deposit of an enormously longer period than a hundred or a thousand
+      times the same amount of sediment on other horizons." 'might'
+      amended from 'much'.
+
+    ARTICLE GEOLOGY: "But no such extensive disturbance of the normal
+      conditions of the distribution of life can take place without
+      carrying with it many secondary effects, and setting in motion a
+      wide cycle of change and of reaction in the animal and vegetable
+      kingdoms." 'kingdoms' amended from 'kindgoms'.
+
+    ARTICLE GEOMETRY: "The bases and altitudes of equal solid
+      parallelepipeds are reciprocally proportional; and if the bases and
+      altitudes be reciprocally proportional, the solid parallelepipeds
+      are equal." 'are' amended from 'and'.
+
+    ARTICLE GEOMETRY: "An alternative method of testing a relation is
+      illustrated in the following example:--If A, B, C, D, E,
+      F be six collinear points, then" 'following example:--' amended
+      from 'example: following'.
+
+    ARTICLE GEOMETRY: "3. In an hyperbolic involution any two conjugate
+      points are harmonic conjugates with regard to the two foci." 'an'
+      amended from 'a'.
+
+    ARTICLE GEOMETRY: "If two lines, given by their projections,
+      intersect, the intersection of their planes and the intersection of
+      their elevations must lie in a line perpendicular to the axis,
+      because they must be the projections of the point common to the two
+      lines." 'planes' amended from 'plans'.
+
+    ARTICLE GEOMETRY: "Where this is the case, if [alpha] be the measure
+      of curvature, the linear element can be put into the form" 'if'
+      amended from 'it'.
+
+    ARTICLE GEOMETRY: "The development of the consequences of these
+      metrical definitions is the subject of non-Euclidean geometry."
+      'subject' amended from 'subjct'.
+
+
+
+
+          ENCYCLOPAEDIA BRITANNICA
+
+  A DICTIONARY OF ARTS, SCIENCES, LITERATURE
+           AND GENERAL INFORMATION
+
+              ELEVENTH EDITION
+
+
+            VOLUME XI, SLICE VI
+
+            GEODESY to GEOMETRY
+
+
+
+
+ARTICLES IN THIS SLICE:
+
+
+  GEODESY                          GEOFFROY, ETIENNE FRANCOIS
+  GEOFFREY (Martel)                GEOFFROY, JULIEN LOUIS
+  GEOFFREY (Plantagenet)           GEOFFROY SAINT-HILAIRE, ETIENNE
+  GEOFFREY (duke of Brittany)      GEOFFROY SAINT-HILAIRE, ISIDORE
+  GEOFFREY (archbishop of York)    GEOGRAPHY
+  GEOFFREY DE MONTBRAY             GEOID
+  GEOFFREY OF MONMOUTH             GEOK-TEPE
+  GEOFFREY OF PARIS                GEOLOGY
+  GEOFFREY THE BAKER               GEOMETRICAL CONTINUITY
+  GEOFFRIN, MARIE THERESE RODET    GEOMETRY
+
+
+
+
+GEODESY (from the Gr. [Greek: ge], the earth, and [Greek: daiein], to
+divide), the science of surveying (q.v.) extended to large tracts of
+country, having in view not only the production of a system of maps of
+very great accuracy, but the determination of the curvature of the
+surface of the earth, and eventually of the figure and dimensions of the
+earth. This last, indeed, may be the sole object in view, as was the
+case in the operations conducted in Peru and in Lapland by the
+celebrated French astronomers P. Bouguer, C.M. de la Condamine, P.L.M.
+de Maupertuis, A.C. Clairault and others; and the measurement of the
+meridian arc of France by P.F.A. Mechain and J.B.J. Delambre had for
+its end the determination of the true length of the "metre" which was to
+be the legal standard of length of France (see EARTH, FIGURE OF THE).
+
+The basis of every extensive survey is an accurate triangulation, and
+the operations of geodesy consist in the measurement, by theodolites, of
+the angles of the triangles; the measurement of one or more sides of
+these triangles on the ground; the determination by astronomical
+observations of the azimuth of the whole network of triangles; the
+determination of the actual position of the same on the surface of the
+earth by observations, first for latitude at some of the stations, and
+secondly for longitude; the determination of altitude for all stations.
+
+For the computation, the points of the actual surface of the earth are
+imagined as projected along their plumb lines on the mathematical
+figure, which is given by the stationary sea-level, and the extension of
+the sea through the continents by a system of imaginary canals. For many
+purposes the mathematical surface is assumed to be a plane; in other
+cases a sphere of radius 6371 kilometres (20,900,000 ft.). In the case
+of extensive operations the surface must be considered as a compressed
+ellipsoid of rotation, whose minor axis coincides with the earth's axis,
+and whose compression, flattening, or ellipticity is about 1/298.
+
+
+_Measurement of Base Lines._
+
+  To determine by actual measurement on the ground the length of a side
+  of one of the triangles ("base line"), wherefrom to infer the lengths
+  of all the other sides in the triangulation, is not the least
+  difficult operation of a trigonometrical survey. When the problem is
+  stated thus--To determine the number of times that a certain standard
+  or unit of length is contained between two finely marked points on the
+  surface of the earth at a distance of some miles asunder, so that the
+  error of the result may be pronounced to lie between certain very
+  narrow limits,--then the question demands very serious consideration.
+  The representation of the unit of length by means of the distance
+  between two fine lines on the surface of a bar of metal at a certain
+  temperature is never itself free from uncertainty and probable error,
+  owing to the difficulty of knowing at any moment the precise
+  temperature of the bar; and the transference of this unit, or a
+  multiple of it, to a measuring bar will be affected not only with
+  errors of observation, but with errors arising from uncertainty of
+  temperature of both bars. If the measuring bar be not
+  self-compensating for temperature, its expansion must be determined by
+  very careful experiments. The thermometers required for this purpose
+  must be very carefully studied, and their errors of division and index
+  error determined.
+
+  In order to avoid the difficulty in exactly determining the
+  temperature of a bar by the mercury thermometer, F.W. Bessel
+  introduced in 1834 near Konigsberg a compound bar which constituted a
+  metallic thermometer.[1] A zinc bar is laid on an iron bar two toises
+  long, both bars being perfectly planed and in free contact, the zinc
+  bar being slightly shorter and the two bars rigidly united at one end.
+  As the temperature varies, the difference of the lengths of the bars,
+  as perceived by the other end, also varies, and affords a quantitative
+  correction for temperature variations, which is applied to reduce the
+  length to standard temperature. During the measurement of the base
+  line the bars were not allowed to come into contact, the interval
+  being measured by the insertion of glass wedges. The results of the
+  comparisons of four measuring rods with one another and with the
+  standards were elaborately computed by the method of least-squares.
+  The probable error of the measured length of 935 toises (about 6000
+  ft.) has been estimated as 1/863500 or 1.2 [mu] ([mu] denoting a
+  millionth). With this apparatus fourteen base lines were measured in
+  Prussia and some neighbouring states; in these cases a somewhat higher
+  degree of accuracy was obtained.
+
+  The principal triangulation of Great Britain and Ireland has seven
+  base lines: five have been measured by steel chains, and two, more
+  exactly, by the compensation bars of General T.F. Colby, an apparatus
+  introduced in 1827-1828 at Lough Foyle in Ireland. Ten base lines were
+  measured in India in 1831-1869 by the same apparatus. This is a system
+  of six compound-bars self-correcting for temperature. The bars may be
+  thus described: Two bars, one of brass and the other of iron, are laid
+  in parallelism side by side, firmly united at their centres, from
+  which they may freely expand or contract; at the standard temperature
+  they are of the same length. Let AB be one bar, A'B' the other; draw
+  lines through the corresponding extremities AA' (to P) and BB' (to Q),
+  and make A'P = B'Q, AA' being equal to BB'. If the ratio A'P/AP equals
+  the ratio of the coefficients of expansion of the bars A'B' and AB,
+  then, obviously, the distance PQ is constant (or nearly so). In the
+  actual instrument P and Q are finely engraved dots 10 ft. apart. In
+  practice the bars, when aligned, are not in contact, an interval of 6
+  in. being allowed between each bar and its neighbour. This distance is
+  accurately measured by an ingenious micrometrical arrangement
+  constructed on exactly the same principle as the bars themselves.
+
+  The last base line measured in India had a length of 8913 ft. In
+  consequence of some suspicion as to the accuracy of the compensation
+  apparatus, the measurement was repeated four times, the operations
+  being conducted so as to determine the actual values of the probable
+  errors of the apparatus. The direction of the line (which is at Cape
+  Comorin) is north and south. In two of the measurements the brass
+  component was to the west, in the others to the east; the differences
+  between the individual measurements and the mean of the four were
+  +0.0017, -0.0049, -0.0015, +0.0045 ft. These differences are very
+  small; an elaborate investigation of all sources of error shows that
+  the probable error of a base line in India is on the average [+-]2.8
+  [mu]. These compensation bars were also used by Sir Thomas Maclear in
+  the measurement of the base line in his extension of Lacaille's arc at
+  the Cape. The account of this operation will be found in a volume
+  entitled _Verification and Extension of Lacaille's Arc of Meridian at
+  the Cape of Good Hope_, by Sir Thomas Maclear, published in 1866. A
+  rediscussion has been given by Sir David Gill in his _Report on the
+  Geodetic Survey of South Africa, &c., 1896_.
+
+  A very simple base apparatus was employed by W. Struve in his
+  triangulations in Russia from 1817 to 1855. This consisted of four
+  wrought-iron bars, each two toises (rather more than 13 ft.) long; one
+  end of each bar is terminated in a small steel cylinder presenting a
+  slightly convex surface for contact, the other end carries a contact
+  lever rigidly connected with the bar. The shorter arm of the lever
+  terminates below in a polished hemisphere, the upper and longer arm
+  traversing a vertical divided arc. In measuring, the plane end of one
+  bar is brought into contact with the short arm of the contact lever
+  (pushed forward by a weak spring) of the next bar. Each bar has two
+  thermometers, and a level for determining the inclination of the bar
+  in measuring. The manner of transferring the end of a bar to the
+  ground is simply this: under the end of the bar a stake is driven very
+  firmly into the ground, carrying on its upper surface a disk, capable
+  of movement in the direction of the measured line by means of
+  slow-motion screws. A fine mark on this disk is brought vertically
+  under the end of the bar by means of a theodolite which is planted at
+  a distance of 25 ft. from the stake in a direction perpendicular to
+  the base. Struve investigated for each base the probable errors of the
+  measurement arising from each of these seven causes: Alignment,
+  inclination, comparisons with standards, readings of index, personal
+  errors, uncertainties of temperature, and the probable errors of
+  adopted rates of expansion. He found that [+-]0.8 [mu] was the mean of
+  the probable errors of the seven bases measured by him. The
+  Austro-Hungarian apparatus is similar; the distance of the rods is
+  measured by a slider, which rests on one of the ends of each rod.
+  Twenty-two base lines were measured in 1840-1899.
+
+  General Carlos Ibanez employed in 1858-1879, for the measurement of
+  nine base lines in Spain, two apparatus similar to the apparatus
+  previously employed by Porro in Italy; one is complicated, the other
+  simplified. The first, an apparatus of the brothers Brunner of Paris,
+  was a thermometric combination of two bars, one of platinum and one of
+  brass, in length 4 metres, furnished with three levels and four
+  thermometers. Suppose A, B, C three micrometer microscopes very firmly
+  supported at intervals of 4 metres with their axes vertical, and
+  aligned in the plane of the base line by means of a transit
+  instrument, their micrometer screws being in the line of measurement.
+  The measuring bar is brought under say A and B, and those micrometers
+  read; the bar is then shifted and brought under B and C. By repetition
+  of this process, the reading of a micrometer indicating the end of
+  each position of the bar, the measurement is made.
+
+  Quite similar apparatus (among others) has been employed by the French
+  and Germans. Since, however, it only permitted a distance of about 300
+  m. to be measured daily, Ibanez introduced a simplification; the
+  measuring rod being made simply of steel, and provided with inlaid
+  mercury thermometers. This apparatus was used in Switzerland for the
+  measurement of three base lines. The accuracy is shown by the
+  estimated probable errors: [+-]0.2 [mu] to [+-]0.8 [mu]. The distance
+  measured daily amounts at least to 800 m.
+
+  A greater daily distance can be measured with the same accuracy by
+  means of Bessel's apparatus; this permits the ready measurement of
+  2000 m. daily. For this, however, it is important to notice that a
+  large staff and favourable ground are necessary. An important
+  improvement was introduced by Edward Jaderin of Stockholm, who
+  measures with stretched wires of about 24 metres long; these wires are
+  about 1.65 mm. in diameter, and when in use are stretched by an
+  accurate spring balance with a tension of 10 kg.[2] The nature of the
+  ground has a very trifling effect on this method. The difficulty of
+  temperature determinations is removed by employing wires made of
+  invar, an alloy of steel (64%) and nickel (36%) which has practically
+  no linear expansion for small thermal changes at ordinary
+  temperatures; this alloy was discovered in 1896 by Benoit and
+  Guillaume of the International Bureau of Weights and Measures at
+  Breteuil. Apparently the future of base-line measurements rests with
+  the invar wires of the Jaderin apparatus; next comes Porro's apparatus
+  with invar bars 4 to 5 metres long.
+
+  Results have been obtained in the United States, of great importance
+  in view of their accuracy, rapidity of determination and economy. For
+  the measurement of the arc of meridian in longitude 98 deg. E., in
+  1900, nine base lines of a total length of 69.2 km. were measured in
+  six months. The total cost of one base was $1231. At the beginning and
+  at the end of the field-season a distance of exactly 100 m. was
+  measured with R.S. Woodward's "5-m. ice-bar" (invented in 1891); by
+  means of the remeasurement of this length the standardization of the
+  apparatus was done under the same conditions as existed in the case of
+  the base measurements. For the measurements there were employed two
+  steel tapes of 100 m. long, provided with supports at distances of 25
+  m., two of 50 m., and the duplex apparatus of Eimbeck, consisting of
+  four 5-m. rods. Each base was divided into sections of about 1000 m.;
+  one of these, the "test kilometre," was measured with all the five
+  apparatus, the others only with two apparatus, mostly tapes. The
+  probable error was about [+-]0.8 [mu], and the day's work a distance
+  of about 2000 m. Each of the four rods of the duplex apparatus
+  consists of two bars of brass and steel. Mercury thermometers are
+  inserted in both bars; these serve for the measurement of the length
+  of the base lines by each of the bars, as they are brought into their
+  consecutive positions, the contact being made by an elastic-sliding
+  contact. The length of the base lines may be calculated for each bar
+  only, and also by the supposition that both bars have the same
+  temperature. The apparatus thus affords three sets of results, which
+  mutually control themselves, and the contact adjustments permit rapid
+  work. The same device has been applied to the older
+  bimetallic-compensating apparatus of Bache-Wurdemann (six bases,
+  1847-1857) and of Schott. There was also employed a single rod
+  bimetallic apparatus on F. Porro's principle, constructed by the
+  brothers Repsold for some base lines. Excellent results have been more
+  recently obtained with invar tapes.
+
+  The following results show the lengths of the same German base lines
+  as measured by different apparatus:
+
+                                                 metres.
+    Base at Berlin   1864 Apparatus of Bessel   2336.3920
+        "     "      1880     "        Brunner      .3924
+    Base at Strehlen 1854     "        Bessel   2762.5824
+        "     "      1879     "        Brunner      .5852
+    Old base at Bonn 1847     "        Bessel   2133.9095
+        "     "      1892     "          "          .9097
+    New base at Bonn 1892     "          "      2512.9612
+        "     "      1892     "        Brunner      .9696
+
+  It is necessary that the altitude above the level of the sea of every
+  part of a base line be ascertained by spirit levelling, in order that
+  the measured length may be reduced to what it would have been had the
+  measurement been made on the surface of the sea, produced in
+  imagination. Thus if l be the length of a measuring bar, h its height
+  at any given position in the measurement, r the radius of the earth,
+  then the length radially projected on to the level of the sea is l(1 -
+  h/r). In the Salisbury Plain base line the reduction to the level of
+  the sea is -0.6294 ft.
+
+  The total number of base lines measured in Europe up to the present
+  time is about one hundred and ten, nineteen of which do not exceed in
+  length 2500 metres, or about 1-1/2 miles, and three--one in France, the
+  others in Bavaria--exceed 19,000 metres. The question has been
+  frequently discussed whether or not the advantage of a long base is
+  sufficiently great to warrant the expenditure of time that it
+  requires, or whether as much precision is not obtainable in the end by
+  careful triangulation from a short base. But the answer cannot be
+  given generally; it must depend on the circumstances of each
+  particular case. With Jaderin's apparatus, provided with invar wires,
+  bases of 20 to 30 km. long are obtained without difficulty.
+
+  [Illustration: FIG. 1.]
+
+  In working away from a base line ab, stations c, d, e, f are carefully
+  selected so as to obtain from well-shaped triangles gradually
+  increasing sides. Before, however, finally leaving the base line, it
+  is usual to verify it by triangulation thus: during the measurement
+  two or more points, as p, q (fig. 1), are marked in the base in
+  positions such that the lengths of the different segments of the line
+  are known; then, taking suitable external stations, as h, k, the
+  angles of the triangles bhp, phq, hqk, kqa are measured. From these
+  angles can be computed the ratios of the segments, which must agree,
+  if all operations are correctly performed, with the ratios resulting
+  from the measures. Leaving the base line, the sides increase up to
+  10, 30 or 50 miles occasionally, but seldom reaching 100 miles. The
+  triangulation points may either be natural objects presenting
+  themselves in suitable positions, such as church towers; or they may
+  be objects specially constructed in stone or wood on mountain tops or
+  other prominent ground. In every case it is necessary that the precise
+  centre of the station be marked by some permanent mark. In India no
+  expense is spared in making permanent the principal trigonometrical
+  stations--costly towers in masonry being erected. It is essential that
+  every trigonometrical station shall present a fine object for
+  observation from surrounding stations.
+
+
+  _Horizontal Angles._
+
+  In placing the theodolite over a station to be observed from, the
+  first point to be attended to is that it shall rest upon a perfectly
+  solid foundation. The method of obtaining this desideratum must depend
+  entirely on the nature of the ground; the instrument must if possible
+  be supported on rock, or if that be impossible a solid foundation must
+  be obtained by digging. When the theodolite is required to be raised
+  above the surface of the ground in order to command particular points,
+  it is necessary to build two scaffolds,--the outer one to carry the
+  observatory, the inner one to carry the instrument,--and these two
+  edifices must have no point of contact. Many cases of high scaffolding
+  have occurred on the English Ordnance Survey, as for instance at
+  Thaxted church, where the tower, 80 ft. high, is surmounted by a spire
+  of 90 ft. The scaffold for the observatory was carried from the base
+  to the top of the spire; that for the instrument was raised from a
+  point of the spire 140 ft. above the ground, having its bearing upon
+  timbers passing through the spire at that height. Thus the instrument,
+  at a height of 178 ft. above the ground, was insulated, and not
+  affected by the action of the wind on the observatory.
+
+  At every station it is necessary to examine and correct the
+  adjustments of the theodolite, which are these: the line of
+  collimation of the telescope must be perpendicular to its axis of
+  rotation; this axis perpendicular to the vertical axis of the
+  instrument; and the latter perpendicular to the plane of the horizon.
+  The micrometer microscopes must also measure correct quantities on the
+  divided circle or circles. The method of observing is this. Let A, B,
+  C ... be the stations to be observed taken in order of azimuth; the
+  telescope is first directed to A and the cross-hairs of the telescope
+  made to bisect the object presented by A, then the microscopes or
+  verniers of the horizontal circle (also of the vertical circle if
+  necessary) are read and recorded. The telescope is then turned to B,
+  which is observed in the same manner; then C and the other stations.
+  Coming round by continuous motion to A, it is again observed, and the
+  agreement of this second reading with the first is some test of the
+  stability of the instrument. In taking this round of angles--or "arc,"
+  as it is called on the Ordnance Survey--it is desirable that the
+  interval of time between the first and second observations of A should
+  be as small as may be consistent with due care. Before taking the next
+  arc the horizontal circle is moved through 20 deg. or 30 deg.; thus a
+  different set of divisions of the circle is used in each arc, which
+  tends to eliminate the errors of division.
+
+  It is very desirable that all arcs at a station should contain one
+  point in common, to which all angular measurements are thus
+  referred,--the observations on each arc commencing and ending with
+  this point, which is on the Ordnance Survey called the "referring
+  object." It is usual for this purpose to select, from among the points
+  which have to be observed, that one which affords the best object for
+  precise observation. For mountain tops a "referring object" is
+  constructed of two rectangular plates of metal in the same vertical
+  plane, their edges parallel and placed at such a distance apart that
+  the light of the sky seen through appears as a vertical line about 10"
+  in width. The best distance for this object is from 1 to 2 miles.
+
+  This method seems at first sight very advantageous; but if, however,
+  it be desired to attain the highest accuracy, it is better, as shown
+  by General Schreiber of Berlin in 1878, to measure only single angles,
+  and as many of these as possible between the directions to be
+  determined. Division-errors are thus more perfectly eliminated, and
+  errors due to the variation in the stability, &c., of the instruments
+  are diminished. This method is rapidly gaining precedence.
+
+  The theodolites used in geodesy vary in pattern and in size--the
+  horizontal circles ranging from 10 in. to 36 in. in diameter. In
+  Ramsden's 36-in. theodolite the telescope has a focal length of 36 in.
+  and an aperture of 2.5 in., the ordinarily used magnifying power being
+  54; this last, however, can of course be changed at the requirements
+  of the observer or of the weather. The probable error of a single
+  observation of a fine object with this theodolite is about 0".2. Fig.
+  2 represents an altazimuth theodolite of an improved pattern used on
+  the Ordnance Survey. The horizontal circle of 14-in. diameter is read
+  by three micrometer microscopes; the vertical circle has a diameter of
+  12 in., and is read by two microscopes. In the great trigonometrical
+  survey of India the theodolites used in the more important parts of
+  the work have been of 2 and 3 ft. diameter--the circle read by five
+  equidistant microscopes. Every angle is measured twice in each
+  position of the zero of the horizontal circle, of which there are
+  generally ten; the entire number of measures of an angle is never
+  less than 20. An examination of 1407 angles showed that the probable
+  error of an observed angle is on the average [+-] 0".28.
+
+  For the observations of very distant stations it is usual to employ a
+  heliotrope (from the Gr. [Greek: helios], sun; [Greek: tropos], a
+  turn), invented by Gauss at Gottingen in 1821. In its simplest form
+  this is a plane mirror, 4, 6, or 8 in. in diameter, capable of
+  rotation round a horizontal and a vertical axis. This mirror is placed
+  at the station to be observed, and in fine weather it is kept so
+  directed that the rays of the sun reflected by it strike the distant
+  observing telescope. To the observer the heliotrope presents the
+  appearance of a star of the first or second magnitude, and is
+  generally a pleasant object for observing.
+
+  Observations at night, with the aid of light-signals, have been
+  repeatedly made, and with good results, particularly in France by
+  General Francois Perrier, and more recently in the United States by
+  the Coast and Geodetic Survey; the signal employed being an acetylene
+  bicycle-lamp, with a lens 5 in. in diameter. Particularly noteworthy
+  are the trigonometrical connexions of Spain and Algeria, which were
+  carried out in 1879 by Generals Ibanez and Perrier (over a distance of
+  270 km.), of Sicily and Malta in 1900, and of the islands of Elba and
+  Sardinia in 1902 by Dr Guarducci (over distances up to 230 km.); in
+  these cases artificial light was employed: in the first case electric
+  light and in the two others acetylene lamps.
+
+  [Illustration: FIG. 2.--Altazimuth Theodolite.]
+
+
+  _Astronomical Observations._
+
+  The direction of the meridian is determined either by a theodolite or
+  a portable transit instrument. In the former case the operation
+  consists in observing the angle between a terrestrial
+  object--generally a mark specially erected and capable of illumination
+  at night--and a close circumpolar star at its greatest eastern or
+  western azimuth, or, at any rate, when very near that position. If the
+  observation be made t minutes of time before or after the time of
+  greatest azimuth, the azimuth then will differ from its maximum value
+  by (450t)^2 sin 1" sin 2[delta]/ sin z, in seconds of angle, omitting
+  smaller terms, [delta] being the star's declination and z its zenith
+  distance. The collimation and level errors are very carefully
+  determined before and after these observations, and it is usual to
+  arrange the observations by the reversal of the telescope so that
+  collimation error shall disappear. If b, c be the level and
+  collimation errors, the correction to the circle reading is b cot z
+  [+-] c cosec z, b being positive when the west end of the axis is
+  high. It is clear that any uncertainty as to the real state of the
+  level will produce a corresponding uncertainty in the resulting value
+  of the azimuth,--an uncertainty which increases with the latitude and
+  is very large in high latitudes. This may be partly remedied by
+  observing in connexion with the star its reflection in mercury. In
+  determining the value of "one division" of a level tube, it is
+  necessary to bear in mind that in some the value varies considerably
+  with the temperature. By experiments on the level of Ramsden's 3-foot
+  theodolite, it was found that though at the ordinary temperature of 66
+  deg. the value of a division was about one second, yet at 32 deg. it
+  was about five seconds.
+
+  In a very excellent portable transit used on the Ordnance Survey, the
+  uprights carrying the telescope are constructed of mahogany, each
+  upright being built of several pieces glued and screwed together; the
+  base, which is a solid and heavy plate of iron, carries a reversing
+  apparatus for lifting the telescope out of its bearings, reversing it
+  and letting it down again. Thus is avoided the change of temperature
+  which the telescope would incur by being lifted by the hands of the
+  observer. Another form of transit is the German diagonal form, in
+  which the rays of light after passing through the object-glass are
+  turned by a total reflection prism through one of the transverse arms
+  of the telescope, at the extremity of which arm is the eye-piece. The
+  unused half of the ordinary telescope being cut away is replaced by a
+  counterpoise. In this instrument there is the advantage that the
+  observer without moving the position of his eye commands the whole
+  meridian, and that the level may remain on the pivots whatever be the
+  elevation of the telescope. But there is the disadvantage that the
+  flexure of the transverse axis causes a variable collimation error
+  depending on the zenith distance of the star to which it is directed;
+  and moreover it has been found that in some cases the personal error
+  of an observer is not the same in the two positions of the telescope.
+
+  To determine the direction of the meridian, it is well to erect two
+  marks at nearly equal angular distances on either side of the north
+  meridian line, so that the pole star crosses the vertical of each mark
+  a short time before and after attaining its greatest eastern and
+  western azimuths.
+
+  If now the instrument, perfectly levelled, is adjusted to have its
+  centre wire on one of the marks, then when elevated to the star, the
+  star will traverse the wire, and its exact position in the field at
+  any moment can be measured by the micrometer wire. Alternate
+  observations of the star and the terrestrial mark, combined with
+  careful level readings and reversals of the instrument, will enable
+  one, even with only one mark, to determine the direction of the
+  meridian in the course of an hour with a probable error of less than a
+  second. The second mark enables one to complete the station more
+  rapidly and gives a check upon the work. As an instance, at Findlay
+  Seat, in latitude 57 deg. 35', the resulting azimuths of the two marks
+  were 177 deg. 45' 37".29 [+-] 0".20 and 182 deg. 17' 15".61 [+-] 0".13,
+  while the angle between the two marks directly measured by a
+  theodolite was found to be 4 deg. 31' 37".43 [+-] 0".23.
+
+  [Illustration: FIG. 3.]
+
+  We now come to the consideration of the determination of time with the
+  transit instrument. Let fig. 3 represent the sphere stereographically
+  projected on the plane of the horizon,--ns being the meridian, we the
+  prime vertical, Z, P the zenith and the pole. Let p be the point in
+  which the production of the axis of the instrument meets the celestial
+  sphere, S the position of a star when observed on a wire whose
+  distance from the collimation centre is c. Let a be the azimuthal
+  deviation, namely, the angle wZp, b the level error so that Zp = 90
+  deg. - b. Let also the hour angle corresponding to p be 90 deg. - n,
+  and the declination of the same = m, the star's declination being
+  [delta], and the latitude [phi]. Then to find the hour angle ZPS =
+  [tau] of the star when observed, in the triangles pPS, pPZ we have,
+  since pPS = 90 + [tau] - n,
+
+         -Sin c = sin m sin [delta] + cos m cos [delta] sin (n - [tau]),
+          Sin m = sin b sin [phi]   - cos b cos [phi] sin a,
+    Cos m sin n = sin b cos [phi]   + cos b sin [phi] sin a.
+
+  And these equations solve the problem, however large be the errors of
+  the instrument. Supposing, as usual, a, b, m, n to be small, we have
+  at once [tau] = n + c sec [delta] + m tan [delta], which is the
+  correction to the observed time of transit. Or, eliminating m and n by
+  means of the second and third equations, and putting z for the zenith
+  distance of the star, t for the observed time of transit, the
+  corrected time is t + (a sin z + b cos z + c) / cos [delta]. Another
+  very convenient form for stars near the zenith is [tau] = b sec [phi]
+  + c sec [delta] + m (tan [delta] - tan [phi]).
+
+  Suppose that in commencing to observe at a station the error of the
+  chronometer is not known; then having secured for the instrument a
+  very solid foundation, removed as far as possible level and
+  collimation errors, and placed it by estimation nearly in the
+  meridian, let two stars differing considerably in declination be
+  observed--the instrument not being reversed between them. From these
+  two stars, neither of which should be a close circumpolar star, a good
+  approximation to the chronometer error can be obtained; thus let
+  [epsilon]1, [epsilon]2, be the apparent clock errors given by these
+  stars if [delta]1, [delta]2 be their declinations the real error is
+
+    [epsilon] = [epsilon]1 + ([epsilon]1 - [epsilon]2)
+      (tan [phi] - tan [delta]1) / (tan [delta]1 - tan [delta]2).
+
+  Of course this is still only approximate, but it will enable the
+  observer (who by the help of a table of natural tangents can compute
+  [epsilon] in a few minutes) to find the meridian by placing at the
+  proper time, which he now knows approximately, the centre wire of his
+  instrument on the first star that passes--not near the zenith.
+
+  The transit instrument is always reversed at least once in the course
+  of an evening's observing, the level being frequently read and
+  recorded. It is necessary in most instruments to add a correction for
+  the difference in size of the pivots.
+
+  The transit instrument is also used in the prime vertical for the
+  determination of latitudes. In the preceding figure let q be the point
+  in which the northern extremity of the axis of the instrument produced
+  meets the celestial sphere. Let nZq be the azimuthal deviation = a,
+  and b being the level error, Zq = 90 deg. - b; let also nPq = [tau]
+  and Pq = [psi]. Let S' be the position of a star when observed on a
+  wire whose distance from the collimation centre is c, positive when to
+  the south, and let h be the observed hour angle of the star, viz.
+  ZPS'. Then the triangles qPS', gPZ give
+
+    -Sin c = sin [delta] cos [psi] - cos [delta] sin [psi] cos (h + [tau]),
+     Cos [psi] = sin b sin [phi] + cos b cos [phi] cos a,
+     Sin [psi] sin [tau] = cos b sin a.
+
+  Now when a and b are very small, we see from the last two equations
+  that [psi] = [phi] - b, a = [tau] sin [psi], and if we calculate
+  [phi]' by the formula cot [phi]' = cot [delta] cos h, the first
+  equation leads us to this result--
+
+    [phi] = [phi]' + (a sin z + b cos z + c)/cos z,
+
+  the correction for instrumental error being very similar to that
+  applied to the observed time of transit in the case of meridian
+  observations. When a is not very small and z is small, the formulae
+  required are more complicated.
+
+  [Illustration: FIG. 4.--Zenith Telescope constructed for the
+  International Stations at Mizusawa, Carloforte, Gaithersburg and
+  Ukiah, by Hermann Wanschaff, Berlin.]
+
+  The method of determining latitude by transits in the prime vertical
+  has the disadvantage of being a somewhat slow process, and of
+  requiring a very precise knowledge of the time, a disadvantage from
+  which the zenith telescope is free. In principle this instrument is
+  based on the proposition that when the meridian zenith distances of
+  two stars at their upper culminations--one being to the north and the
+  other to the south of the zenith--are equal, the latitude is the mean
+  of their declinations; or, if the zenith distance of a star
+  culminating to the south of the zenith be Z, its declination being
+  [delta], and that of another culminating to the north with zenith
+  distance Z' and declination [delta]', then clearly the latitude is
+  1/2([delta] + [delta]') + 1/2(Z - Z'). Now the zenith telescope does
+  away with the divided circle, and substitutes the measurement
+  micrometrically of the quantity Z' - Z.
+
+  In fig. 4 is shown a zenith telescope by H. Wanschaff of Berlin, which
+  is the type used (according to the Central Bureau at Potsdam) since
+  about 1890 for the determination of the variations of latitude due to
+  different, but as yet imperfectly understood, influences. The
+  instrument is supported on a strong tripod, fitted with levelling
+  screws; to this tripod is fixed the azimuth circle and a long vertical
+  steel axis. Fitting on this axis is a hollow axis which carries on its
+  upper end a short transverse horizontal axis with a level. This latter
+  carries the telescope, which, supported at the centre of its length,
+  is free to rotate in a vertical plane. The telescope is thus mounted
+  eccentrically with respect to the vertical axis around which it
+  revolves. Two extremely sensitive levels are attached to the
+  telescope, which latter carries a micrometer in its eye-piece, with a
+  screw of long range for measuring differences of zenith distance. Two
+  levels are employed for controlling and increasing the accuracy. For
+  this instrument stars are selected in pairs, passing north and south
+  of the zenith, culminating within a few minutes of time and within
+  about twenty minutes (angular) of zenith distance of each other. When
+  a pair of stars is to be observed, the telescope is set to the mean of
+  the zenith distances and in the plane of the meridian. The first star
+  on passing the central meridional wire is bisected by the micrometer;
+  then the telescope is rotated very carefully through 180 deg. round
+  the vertical axis, and the second star on passing through the field is
+  bisected by the micrometer on the centre wire. The micrometer has thus
+  measured the difference of the zenith distances, and the calculation
+  to get the latitude is most simple. Of course it is necessary to read
+  the level, and the observations are not necessarily confined to the
+  centre wire. In fact if n, s be the north and south readings of the
+  level for the south star, n', s' the same for the north star, l the
+  value of one division of the level, m the value of one division of the
+  micrometer, r, r' the refraction corrections, [mu], [mu]' the
+  micrometer readings of the south and north star, the micrometer being
+  supposed to read from the zenith, then, supposing the observation made
+  on the centre wire,--
+
+    [phi] = 1/2([delta] + [delta]') + 1/2([mu] - [mu]')m + 1/4(n + n' - s - s')l +
+      1/2(r - r').
+
+  It is of course of the highest importance that the value m of the
+  screw be well determined. This is done most effectually by observing
+  the vertical movement of a close circumpolar star when at its greatest
+  azimuth.
+
+  In a single night with this instrument a very accurate result, say
+  with a probable error of about 0".2, could be obtained for latitude
+  from, say, twenty pair of stars; but when the latitude is required to
+  be obtained with the highest possible precision, two nights at least
+  are necessary. The weak point of the zenith telescope lies in the
+  circumstance that its requirements prevent the selection of stars
+  whose positions are well fixed; very frequently it is necessary to
+  have the declinations of the stars selected for this instrument
+  specially observed at fixed observatories. The zenith telescope is
+  made in various sizes from 30 to 54 in. in focal length; a 30-in.
+  telescope is sufficient for the highest purposes and is very portable.
+  The net observation probable-error for one pair of stars is only
+  [+-]0".1.
+
+  The zenith telescope is a particularly pleasant instrument to work
+  with, and an observer has been known (a sergeant of Royal Engineers,
+  on one occasion) to take every star in his list during eleven hours on
+  a stretch, namely, from 6 o'clock P.M. until 5 A.M., and this on a
+  very cold November night on one of the highest points of the
+  Grampians. Observers accustomed to geodetic operations attain
+  considerable powers of endurance. Shortly after the commencement of
+  the observations on one of the hills in the Isle of Skye a storm
+  carried away the wooden houses of the men and left the observatory
+  roofless. Three observatory roofs were subsequently demolished, and
+  for some time the observatory was used without a roof, being filled
+  with snow every night and emptied every morning. Quite different,
+  however, was the experience of the same party when on the top of Ben
+  Nevis, 4406 ft. high. For about a fortnight the state of the
+  atmosphere was unusually calm, so much so, that a lighted candle could
+  often be carried between the tents of the men and the observatory,
+  whilst at the foot of the hill the weather was wild and stormy.
+
+  The determination of the difference of longitude between two stations
+  A and B resolves itself into the determination of the local time at
+  each of the stations, and the comparison by signals of the clocks at A
+  and B. Whenever telegraphic lines are available these comparisons are
+  made by telegraphy. A small and delicately-made apparatus introduced
+  into the mechanism of an astronomical clock or chronometer breaks or
+  closes by the action of the clock an electric circuit every second. In
+  order to record the minutes as well as seconds, one second in each
+  minute, namely that numbered 0 or 60, is omitted. The seconds are
+  recorded on a chronograph, which consists of a cylinder revolving
+  uniformly at the rate of one revolution per minute covered with white
+  paper, on which a pen having a slow movement in the direction of the
+  axis of the cylinder describes a continuous spiral. This pen is
+  deflected through the agency of an electromagnet every second, and
+  thus the seconds of the clock are recorded on the chronograph by
+  offsets from the spiral curve. An observer having his hand on a
+  contact key in the same circuit can record in the same manner his
+  observed times of transits of stars. The method of determination of
+  difference of longitude is, therefore, virtually as follows. After the
+  necessary observations for instrumental corrections, which are
+  recorded only at the station of observation, the clock at A is put in
+  connexion with the circuit so as to write on both chronographs,
+  namely, that at A and that at B. Then the clock at B is made to write
+  on both chronographs. It is clear that by this double operation one
+  can eliminate the effect of the small interval of time consumed in the
+  transmission of signals, for the difference of longitude obtained from
+  the one chronograph will be in excess by as much as that obtained from
+  the other will be in defect. The determination of the personal errors
+  of the observers in this delicate operation is a matter of the
+  greatest importance, as therein lies probably the chief source of
+  residual error.
+
+  These errors can nevertheless be almost entirely avoided by using the
+  impersonal micrometer of Dr Repsold (Hamburg, 1889). In this device
+  there is a movable micrometer wire which is brought by hand into
+  coincidence with the star and moved along with it; at fixed points
+  there are electrical contacts, which replace the fixed wires.
+  Experiments at the Geodetic Institute and Central Bureau at Potsdam in
+  1891 gave the following personal equations in the case of four
+  observers:--
+
+          Older Procedure.   New Procedure.
+
+    A-B      -0^s.108          -0^s.004
+    A-G      -0^s.314          -0^s.035
+    A-S      -0^s.184          -0^s.027
+    B-G      -0^s.225          +0^s.013
+    B-S      -0^s.086          -0^s.023
+    G-S      +0^s.109          -0^s.006
+
+  These results show that in the later method the personal equation is
+  small and not so variable; and consequently the repetition of
+  longitude determinations with exchanged observers and apparatus
+  entirely eliminates the constant errors, the probable error of such
+  determinations on ten nights being scarcely [+-]0^s.01.
+
+
+  _Calculation of Triangulation._
+
+  The surface of Great Britain and Ireland is uniformly covered by
+  triangulation, of which the sides are of various lengths from 10 to
+  111 miles. The largest triangle has one angle at Snowdon in Wales,
+  another on Slieve Donard in Ireland, and a third at Scaw Fell in
+  Cumberland; each side is over a hundred miles and the spherical excess
+  is 64". The more ordinary method of triangulation is, however, that of
+  chains of triangles, in the direction of the meridian and
+  perpendicular thereto. The principal triangulations of France, Spain,
+  Austria and India are so arranged. Oblique chains of triangles are
+  formed in Italy, Sweden and Norway, also in Germany and Russia, and in
+  the United States. Chains are composed sometimes merely of consecutive
+  plain triangles; sometimes, and more frequently in India, of
+  combinations of triangles forming consecutive polygonal figures. In
+  this method of triangulating, the sides of the triangles are generally
+  from 20 to 30 miles in length--seldom exceeding 40.
+
+  The inevitable errors of observation, which are inseparable from all
+  angular as well as other measurements, introduce a great difficulty
+  into the calculation of the sides of a triangulation. Starting from a
+  given base in order to get a required distance, it may generally be
+  obtained in several different ways--that is, by using different sets
+  of triangles. The results will certainly differ one from another, and
+  probably no two will agree. The experience of the computer will then
+  come to his aid, and enable him to say which is the most trustworthy
+  result; but no experience or ability will carry him through a large
+  network of triangles with anything like assurance. The only way to
+  obtain trustworthy results is to employ the method of least squares.
+  We cannot here give any illustration of this method as applied to
+  general triangulation, for it is most laborious, even for the simplest
+  cases.
+
+  Three stations, projected on the surface of the sea, give a spherical
+  or spheroidal triangle according to the adoption of the sphere or the
+  ellipsoid as the form of the surface. A spheroidal triangle differs
+  from a spherical triangle, not only in that the curvatures of the
+  sides are different one from another, but more especially in this
+  that, while in the spherical triangle the normals to the surface at
+  the angular points meet at the centre of the sphere, in the spheroidal
+  triangle the normals at the angles A, B, C meet the axis of revolution
+  of the spheroid in three different points, which we may designate
+  [alpha], [beta], [gamma] respectively. Now the angle A of the triangle
+  as measured by a theodolite is the inclination of the planes BA[alpha]
+  and CA[alpha], and the angle at B is that contained by the planes
+  AB[beta] and CB[beta]. But the planes AB[alpha] and AB[beta]
+  containing the line AB in common cut the surface in two distinct plane
+  curves. In order, therefore, that a spheroidal triangle may be exactly
+  defined, it is necessary that the nature of the lines joining the
+  three vertices be stated. In a mathematical point of view the most
+  natural definition is that the sides be geodetic or shortest lines.
+  C.C.G. Andrae, of Copenhagen, has also shown that other lines give a
+  less convenient computation.
+
+  K.F. Gauss, in his treatise, _Disquisitiones generales circa
+  superficies curvas_, entered fully into the subject of geodetic (or
+  geodesic) triangles, and investigated expressions for the angles of a
+  geodetic triangle whose sides are given, not certainly finite
+  expressions, but approximations inclusive of small quantities of the
+  fourth order, the side of the triangle or its ratio to the radius of
+  the nearly spherical surface being a small quantity of the first
+  order. The terms of the fourth order, as given by Gauss for any
+  surface in general, are very complicated even when the surface is a
+  spheroid. If we retain small quantities of the second order only, and
+  put [A], [B], [C] for the angles of the geodetic triangle, while A, B,
+  C are those of a plane triangle having sides equal respectively to
+  those of the geodetic triangle, then, [sigma] being the area of the
+  plane triangle and [a], [b], [c] the measures of curvature at the
+  angular points,
+
+    [A] = A + [sigma](2[a] + [b] + [c])/12,
+    [B] = B + [sigma]([a] + 2[b] + [c])/12,
+    [C] = C + [sigma]([a] + [b] + 2[c])/12.
+
+  For the sphere [a] = [b] = [r], and making this simplification, we
+  obtain the theorem previously given by A.M. Legendre. With the terms
+  of the fourth order, we have (after Andrae):
+
+              [epsilon]   [sigma]   /m^2 - a^2    [a] - k \
+    [A] - A = --------- + -------k ( ---------k + -------  ),
+                  3          3      \    20          4k   /
+
+              [epsilon]   [sigma]   /m^2 - b^2    [b] - k \
+    [B] - B = --------- + -------k ( ---------k + -------- ),
+                  3          3      \    20          4k   /
+
+              [epsilon]   [sigma]   /m^2 - c^2    [c] - k \
+    [C] - C = --------- + -------k ( ---------k + -------- ),
+                  3          3      \    20          4k   /
+
+  in which [epsilon] = [sigma] k {1 + (m^2k / 8)}, 3m^2 = a^2 + b^2 +
+  c^2, 3k = [a] + [b] + [c]. For the ellipsoid of rotation the measure
+  of curvature is equal to 1 / [rho]n, [rho] and n being the radii of
+  curvature of the meridian and perpendicular.
+
+  It is rarely that the terms of the fourth order are required. As a
+  rule spheroidal triangles are calculated as spherical (after
+  Legendre), i.e. like plane triangles with a decrease of each angle of
+  about [epsilon] / 3; [epsilon] must, however, be calculated for each
+  triangle separately with its mean measure of curvature k.
+
+  The geodetic line being the shortest that can be drawn on any surface
+  between two given points, we may be conducted to its most important
+  characteristics by the following considerations: let p, q be adjacent
+  points on a curved surface; through s the middle point of the chord pq
+  imagine a plane drawn perpendicular to pq, and let S be any point in
+  the intersection of this plane with the surface; then pS + Sq is
+  evidently least when sS is a minimum, which is when sS is a normal to
+  the surface; hence it follows that of all plane curves on the surface
+  joining p, q, when those points are indefinitely near to one another,
+  that is the shortest which is made by the normal plane. That is to
+  say, the osculating plane at any point of a geodetic line contains the
+  normal to the surface at that point. Imagine now three points in
+  space, A, B, C, such that AB = BC = c; let the direction cosines of AB
+  be l, m, n, those of BC l', m', n', then x, y, z being the
+  co-ordinates of B, those of A and C will be respectively--
+
+    x - cl : y - cm : z - cn
+    x + cl': y + cm': z + cn'.
+
+  Hence the co-ordinates of the middle point M of AC are x + 1/2c(l' - l),
+  y + 1/2c(m' - m), z + 1/2c(n' - n), and the direction cosines of BM are
+  therefore proportional to l' - l : m' - m : n' - n. If the angle made
+  by BC with AB be indefinitely small, the direction cosines of BM are
+  as [delta]l : [delta]m : [delta]n. Now if AB, BC be two contiguous
+  elements of a geodetic, then BM must be a normal to the surface, and
+  since [delta]l, [delta]m, [delta]n are in this case represented by
+  [delta](dx/ds), [delta](dy/ds), [delta](dz/ds), and if the equation of
+  the surface be u = 0, we have
+
+    d^2x   / du   d^2y   / du   d^2z   / du
+    ----  /  -- = ----  /  -- = ----  /  --,
+    ds^2 /   dx   ds^2 /   dy   ds^2 /   dz
+
+  which, however, are equivalent to only one equation. In the case of
+  the spheroid this equation becomes
+
+      d^2x     d^2y
+    y ---- - x ---- = 0,
+      ds^2     ds^2
+
+  which integrated gives ydx - xdy = Cds. This again may be put in the
+  form r sin a = C, where a is the azimuth of the geodetic at any
+  point--the angle between its direction and that of the meridian--and r
+  the distance of the point from the axis of revolution.
+
+  From this it may be shown that the azimuth at A of the geodetic
+  joining AB is not the same as the astronomical azimuth at A of B or
+  that determined by the vertical plane A[alpha]B. Generally speaking,
+  the geodetic lies between the two plane section curves joining A and B
+  which are formed by the two vertical planes, supposing these points
+  not far apart. If, however, A and B are nearly in the same latitude,
+  the geodetic may cross (between A and B) that plane curve which lies
+  nearest the adjacent pole of the spheroid. The condition of crossing
+  is this. Suppose that for a moment we drop the consideration of the
+  earth's non-sphericity, and draw a perpendicular from the pole C on
+  AB, meeting it in S between A and B. Then A being that point which is
+  nearest the pole, the geodetic will cross the plane curve if AS be
+  between 1/4AB and 3/8 AB. If AS lie between this last value and 1/2AB,
+  the geodetic will lie wholly to the north of both plane curves, that
+  is, supposing both points to be in the northern hemisphere.
+
+  The difference of the azimuths of the vertical section AB and of the
+  geodetic AB, i.e. the astronomical and geodetic azimuths, is very
+  small for all observable distances, being approximately:--
+
+  Geod. azimuth = Astr. azimuth -(1/12) [e^2/(1 - e^2)] (s^2/[rho]n)
+  (cos^2[phi] sin 2[alpha] + (s/4a)|sin 2[phi] sin [alpha]), in which: e
+  and a are the numerical eccentricity and semi-major axis respectively
+  of the meridian ellipse, [phi] and [alpha] are the latitude and
+  azimuth at A, s = AB, and [rho] and n are the radii of curvature of
+  the meridian and perpendicular at A. For s = 100 kilometres, only the
+  first term is of moment; its value is 0".028 cos^2 [phi] sin 2[alpha],
+  and it lies well within the errors of observation. If we imagine the
+  geodetic AB, it will generally trisect the angles between the vertical
+  sections at A and B, so that the geodetic at A is near the vertical
+  section AB, and at B near the section BA.[3] The greatest distance of
+  the vertical sections one from another is e^2s^3 cos^2 [phi]0 sin
+  2[alpha]0/16a^2, in which [phi]0 and [alpha]0 are the mean latitude
+  and azimuth respectively of the middle point of AB. For the value s =
+  64 kilometres, the maximum distance is 3 mm.
+
+  An idea of the course of a longer geodetic line may be gathered from
+  the following example. Let the line be that joining Cadiz and St
+  Petersburg, whose approximate positions are--
+
+        Cadiz.               St Petersburg.
+    Lat. 36 deg. 22' N.      59 deg. 56' N.
+    Long. 6 deg. 18' W.      30 deg. 17' E.
+
+  If G be the point on the geodetic corresponding to F on that one of
+  the plane curves which contains the normal at Cadiz (by
+  "corresponding" we mean that F and G are on a meridian) then G is to
+  the north of F; at a quarter of the whole distance from Cadiz GF is
+  458 ft., at half the distance it is 637 ft., and at three-quarters it
+  is 473 ft. The azimuth of the geodetic at Cadiz differs 20" from that
+  of the vertical plane, which is the astronomical azimuth.
+
+  The azimuth of a geodetic line cannot be observed, so that the line
+  does not enter of necessity into practical geodesy, although many
+  formulae connected with its use are of great simplicity and elegance.
+  The geodetic line has always held a more important place in the
+  science of geodesy among the mathematicians of France, Germany and
+  Russia than has been assigned to it in the operations of the English
+  and Indian triangulations. Although the observed angles of a
+  triangulation are not geodetic angles, yet in the calculation of the
+  distance and reciprocal bearings of two points which are far apart,
+  and are connected by a long chain of triangles, we may fall upon the
+  geodetic line in this manner:--
+
+  If A, Z be the points, then to start the calculation from A, we obtain
+  by some preliminary calculation the approximate azimuth of Z, or the
+  angle made by the direction of Z with the side AB or AC of the first
+  triangle. Let P1 be the point where this line intersects BC; then, to
+  find P2, where the line cuts the next triangle side CD, we make the
+  angle BP1P2 such that BP1P2 + BP1A = 180 deg. This fixes P2, and P3 is
+  fixed by a repetition of the same process; so for P4, P5 .... Now it
+  is clear that the points P1, P2, P3 so computed are those which would
+  be actually fixed by an observer with a theodolite, proceeding in the
+  following manner. Having set the instrument up at A, and turned the
+  telescope in the direction of the computed bearing, an assistant
+  places a mark P1 on the line BC, adjusting it till bisected by the
+  cross-hairs of the telescope at A. The theodolite is then placed over
+  P1, and the telescope turned to A; the horizontal circle is then moved
+  through 180 deg. The assistant then places a mark P2 on the line CD,
+  so as to be bisected by the telescope, which is then moved to P2, and
+  in the same manner P3 is fixed. Now it is clear that the series of
+  points P1, P2, P3 approaches to the geodetic line, for the plane of
+  any two consecutive elements P_(n-1) P_n, P_n P_(n+1) contains the
+  normal at P_n.
+
+  If the objection be raised that not the geodetic azimuths but the
+  astronomical azimuths are observed, it is necessary to consider that
+  the observed vertical sections do not correspond to points on the
+  sea-level but to elevated points. Since the normals of the ellipsoid
+  of rotation do not in general intersect, there consequently arises an
+  influence of the height on the azimuth. In the case of the measurement
+  of the azimuth from A to B, the instrument is set to a point A' over
+  the surface of the ellipsoid (the sea-level), and it is then adjusted
+  to a point B', also over the surface, say at a height h'. The vertical
+  plane containing A' and B' also contains A but not B: it must
+  therefore be rotated through a small azimuth in order to contain B.
+  The correction amounts approximately to -e^2h' cos^2[phi] sin
+  2[alpha]/2a; in the case of h' = 1000 m., its value is 0".108
+  cos^2[phi] sin 2[alpha].
+
+  This correction is therefore of greater importance in the case of
+  observed azimuths and horizontal angles than in the previously
+  considered case of the astronomical and the geodetic azimuths. The
+  observed azimuths and horizontal angles must therefore also be
+  corrected in the case, where it is required to dispense with geodetic
+  lines.
+
+  When the angles of a triangulation have been adjusted by the method of
+  least squares, and the sides are calculated, the next process is to
+  calculate the latitudes and longitudes of all the stations starting
+  from one given point. The calculated latitudes, longitudes and
+  azimuths, which are designated geodetic latitudes, longitudes and
+  azimuths, are not to be confounded with the observed latitudes,
+  longitudes and azimuths, for these last are subject to somewhat large
+  errors. Supposing the latitudes of a number of stations in the
+  triangulation to be observed, practically the mean of these determines
+  the position in latitude of the network, taken as a whole. So the
+  orientation or general azimuth of the whole is inferred from all the
+  azimuth observations. The triangulation is then supposed to be
+  projected on a spheroid of given elements, representing as nearly as
+  one knows the real figure of the earth. Then, taking the latitude of
+  one point and the direction of the meridian there as given--obtained,
+  namely, from the astronomical observations there--one can compute the
+  latitudes of all the other points with any degree of precision that
+  may be considered desirable. It is necessary to employ for this
+  purpose formulae which will give results true even for the longest
+  distances to the second place of decimals of seconds, otherwise there
+  will arise an accumulation of errors from imperfect calculation which
+  should always be avoided. For very long distances, eight places of
+  decimals should be employed in logarithmic calculations; if seven
+  places only are available very great care will be required to keep the
+  last place true. Now let [phi], [phi]' be the latitudes of two
+  stations A and B; [alpha], [alpha]^* their mutual azimuths counted
+  from north by east continuously from 0 deg. to 360 deg.; [omega] their
+  difference of longitude measured from west to east; and s the distance
+  AB.
+
+  First compute a latitude [phi]1 by means of the formula [phi]1 = [phi]
+  + (s cos [alpha]) / [rho], where [rho] is the radius of curvature of
+  the meridian at the latitude [phi]; this will require but four places
+  of logarithms. Then, in the first two of the following, five places
+  are sufficient--
+
+                  s^2                                  s^2
+    [epsilon] = ------- sin [alpha] cos a,   [eta] = ------- sin^2[alpha] tan[phi]1,
+                2[rho]n                              2[rho]n
+
+                      s
+    [phi]' - [phi] = ---- cos ([alpha] - 2/3[epsilon]) - [eta],
+                     rho0
+
+              s sin (alpha - 1/3[epsilon])
+    [omega] = ----------------------------,
+               n cos ([phi]' + 1/3[eta])
+
+    [alpha]^* - [alpha] = [omega] sin ([phi]' + 2/3[eta]) - [epsilon] + 180 deg.
+
+  Here n is the normal or radius of curvature perpendicular to the
+  meridian; both n and [rho] correspond to latitude [phi]1, and [rho]0
+  to latitude 1/2([phi] + [phi]'). For calculations of latitude and
+  longitude, tables of the logarithmic values of [rho] sin 1", n sin 1",
+  and 2 n [rho] sin 1" are necessary. The following table contains these
+  logarithms for every ten minutes of latitude from 52 deg. to 53 deg.
+  computed with the elements a = 20926060 and a : b = 295 : 294 :--
+
+    +------+------------------+--------------+--------------------+
+    |      |          1       |        1     |           1        |
+    | Lat. | Log.------------.| Log.--------.| Log.--------------.|
+    |      |     [rho] sin 1" |     n sin 1" |     2[rho]n sin 1" |
+    +------+------------------+--------------+--------------------+
+    |deg. '|                  |              |                    |
+    |52  0 |    7.9939434     |  7.9928231   |      0.37131       |
+    |   10 |         9309     |       8190   |           29       |
+    |   20 |         9185     |       8148   |           28       |
+    |   30 |         9060     |       8107   |           26       |
+    |   40 |         8936     |       8065   |           24       |
+    |   50 |         8812     |       8024   |           23       |
+    |53  0 |         8688     |       7982   |           22       |
+    +------+------------------+--------------+--------------------+
+
+  The logarithm in the last column is that required also for the
+  calculation of spherical excesses, the spherical excess of a triangle
+  being expressed by a b sin (C/2[rho]n) sin 1".
+
+  It is frequently necessary to obtain the co-ordinates of one point
+  with reference to another point; that is, let a perpendicular arc be
+  drawn from B to the meridian of A meeting it in P, then, [alpha] being
+  the azimuth of B at A, the co-ordinates of B with reference to A are
+
+    AP = s cos ([alpha] - 2/3[epsilon]), BP = s sin ([alpha] -
+      1/3[epsilon]),
+
+  where [epsilon] is the spherical excess of APB, viz. s^2 sin [alpha]
+  cos [alpha] multiplied by the quantity whose logarithm is in the
+  fourth column of the above table.
+
+  If it be necessary to determine the geographical latitude and
+  longitude as well as the azimuths to a greater degree of accuracy than
+  is given by the above formulae, we make use of the following formula:
+  given the latitude [phi] of A, and the azimuth [alpha] and the
+  distance s of B, to determine the latitude [phi]' and longitude
+  [omega] of B, and the back azimuth [alpha]'. Here it is understood
+  that [alpha]' is symmetrical to [alpha], so that [alpha]^* + [alpha]'
+  = 360 deg.
+
+  Let
+
+    [theta] = s [Delta] / a, where [Delta] = (1 - e^2 sin^2 [phi])^1/2
+
+  and
+
+           e^2 [theta]^2
+    [xi] = -------------  cos^2 [phi] sin 2[alpha],
+           (4 (1 - e^2)
+
+            e^2 [theta]^3
+    [xi]' = ------------- cos^2 [phi] cos^2 [alpha];
+            (6 (1 - e^2)
+
+  [xi], [xi]' are always very minute quantities even for the longest
+  distances; then, putting [kappa] = 90 deg. - [phi],
+
+       [alpha]' + [xi] - [omega]   sin 1/2([kappa] - [theta] - [xi]')     [alpha]
+    tan------------------------- = ---------------------------------- cot -------
+                   2               sin 1/2([kappa] + [theta] + [xi]')        2
+
+       [alpha]' + [xi] + [omega]   cos 1/2([kappa] - [theta] - [xi]')     [alpha]
+    tan------------------------- = ---------------------------------- cot -------
+                   2               cos 1/2([kappa] + [theta] + [xi]')        2
+
+                        s sin 1/2([alpha]' + [xi] - [alpha])    /    [theta]^2      [alpha]' - [alpha]\
+    [phi]' - [phi] = ----------------------------------------- ( 1 + ---------cos^2 ------------------ );
+                     [rho]0 sin 1/2([alpha]' + [xi] + [alpha])  \       12                   2        /
+
+  here [rho]0 is the radius of curvature of the meridian for the mean
+  latitude 1/2([phi] + [phi]'). These formulae are approximate only, but
+  they are sufficiently precise even for very long distances.
+
+  For lines of any length the formulae of F.W. Bessel (_Astr. Nach._,
+  1823, iv. 241) are suitable.
+
+  If the two points A and B be defined by their geographical
+  co-ordinates, we can accurately calculate the corresponding
+  astronomical azimuths, i.e. those of the vertical section, and then
+  proceed, in the case of not too great distances, to determine the
+  length and the azimuth of the shortest lines. For _any_ distances
+  recourse must again be made to Bessel's formula.[4]
+
+  Let [alpha], [alpha]' be the mutual azimuths of two points A, B on a
+  spheroid, k the chord line joining them, [mu], [mu]' the angles made
+  by the chord with the normals at A and B, [phi], [phi]', [omega] their
+  latitudes and difference of longitude, and (x^2 + y^2)/a^2 + z^2 b^2 =
+  1 the equation of the surface; then if the plane xz passes through A
+  the co-ordinates of A and B will be
+
+    x = (a/[Delta]) cos [phi], x' = (a/[Delta]') cos [phi]' cos [omega],
+
+    y = 0                      y' = (a/[Delta]') cos [phi]' sin [omega],
+
+    z = (a/[Delta]) (1 - e^2) sin [phi], z' = (a/[Delta]') (1 - e^2) sin [phi]',
+
+  where [Delta] = (1 - e^2 sin^2 [phi])^1/2, [Delta]' = (1 - e^2 sin^2
+  [phi]')^1/2, and e is the eccentricity. Let f, g, h be the direction
+  cosines of the normal to that plane which contains the normal at A and
+  the point B, and whose inclinations to the meridian plane of A is =
+  [alpha]; let also l, m, n and l', m', n' be the direction cosines of
+  the normal at A, and of the tangent to the surface at A which lies in
+  the plane passing through B, then since the first line is
+  perpendicular to each of the other two and to the chord k, whose
+  direction cosines are proportional to x' - x, y' - y, z' - z, we have
+  these three equations
+
+    f(x' - x) + gy' + h(z' - z) = 0
+
+                fl + gm + hn    = 0
+
+                fl' + gm' + hn' = 0.
+
+  Eliminate f, g, h from these equations, and substitute
+
+    l = cos [phi]   l' = - sin [phi] cos [alpha]
+
+    m = 0           m' = sin [alpha]
+
+    n = sin [phi]   n' = cos [phi] cos [alpha],
+
+  and we get
+
+    (x' - x) sin [phi] + y' cot [alpha] - (z' - z) cos [phi] = 0.
+
+  The substitution of the values of x, z, x', y', z' in this equation
+  will give immediately the value of cot [alpha]; and if we put [zeta],
+  [zeta]' for the corresponding azimuths on a sphere, or on the
+  supposition e = 0, the following relations exist
+
+                                      cos [phi] Q
+    cot [alpha] - cot [zeta] = e^2 ------------------
+                                   cos [phi]' [Delta]
+
+                                        cos [phi]' Q
+    cot [alpha]' - cot [zeta]' = e^2 ------------------
+                                     cos [phi] [Delta]'
+
+    [Delta]' sin [phi] - [Delta] sin [phi]' = Q sin [omega].
+
+  If from B we let fall a perpendicular on the meridian plane of A, and
+  from A let fall a perpendicular on the meridian plane of B, then the
+  following equations become geometrically evident:
+
+    k sin [mu] sin [alpha] = (a/[Delta]') cos [phi]' sin [omega]
+
+    k sin [mu]' sin [alpha]' = (a/[Delta]) cos [phi] sin [omega].
+
+  Now in any surface u = 0 we have
+
+    k^2 = (x' - x)^2 + (y' - y)^2 + (z' - z)^2
+                  _                                      _
+                 |         du            du            du |  /   / du^2   du^2   du^2 \ 1/2
+    -cos [mu] =  |(x' - x) -- + (y' - y) -- + (z' - z) -- | / k (  ---- + ---- + ----  )
+                 |_        dx            dy            dz_|/     \ dx^2   dy^2   dz^2 /
+                  _                                         _
+                 |         du             du             du  |  /   / du^2    du^2    du^2  \ 1/2
+    -cos [mu]' = |(x' - x) --- + (y' - y) --- + (z' - z) --- | / k (  ----- + ----- + -----  ).
+                 |_        dx'            dy'            dz'_|/     \ dx'^2   dy'^2   dz'^2 /
+
+  In the present case, if we put
+
+        xx'   zz'
+    1 - --- - --- = U,
+        a^2   b^2
+
+  then
+
+    k^2             /z' - z \ ^2
+    --- = 2U - e^2 ( ------  )
+    a^2             \   b   /
+
+    cos [mu] = (a/k) [Delta]U; cos [mu]' = (a/k) [Delta]'U.
+
+  Let u be such an angle that
+
+    (1 - e^2)^1/2 sin [phi] = [Delta] sin u
+
+               cos [phi] = [Delta] cos u,
+
+  then on expressing x, x', z, z' in terms of u and u',
+
+    U = 1 - cos u cos u' cos [omega] - sin u sin u';
+
+  also, if v be the third side of a spherical triangle, of which two
+  sides are 1/2[pi] - u and 1/2[pi] - u' and the included angle [omega],
+  using a subsidiary angle [psi] such that
+
+    sin [psi] sin 1/2v = e sin 1/2(u' - u) cos 1/2(u' + u),
+
+  we obtain finally the following equations:--
+
+                      k = 2a cos [psi] sin 1/2v
+
+                  cos [mu] = [Delta] sec [psi] sin 1/2v
+
+                 cos [mu]' = [Delta]' sec [psi] sin 1/2v
+
+      sin [mu] sin [alpha] = (a/k) cos u' sin [omega]
+
+    sin [mu]' sin [alpha]' = (a/k) cos u sin [omega].
+
+  These determine rigorously the distance, and the mutual zenith
+  distances and azimuths, of any two points on a spheroid whose
+  latitudes and difference of longitude are given.
+
+  By a series of reductions from the equations containing [zeta],
+  [zeta]' it may be shown that
+
+    [alpha] + [alpha]' = [zeta] + [zeta]' + 1/4e^4[omega]([phi]' - [phi])^2
+      cos^4 [phi]0 sin [phi]0 + ...,
+
+  where [phi]0 is the mean of [phi] and [phi]', and the higher powers of
+  e are neglected. A short computation will show that the small quantity
+  on the right-hand side of this equation cannot amount even to the
+  thousandth part of a second for k < 0.1a, which is, practically
+  speaking, zero; consequently the sum of the azimuths [alpha] +
+  [alpha]' on the spheroid is equal to the sum of the spherical
+  azimuths, whence follows this very important theorem (known as Dalby's
+  theorem). If [phi], [phi]' be the latitudes of two points on the
+  surface of a spheroid, [omega] their difference of longitude, [alpha],
+  [alpha]' their reciprocal azimuths,
+
+    tan 1/2[omega] = cot 1/2([alpha] + [alpha]') {cos 1/2([phi]' - [phi])/
+      sin 1/2([phi]' + [phi])}.
+
+  The computation of the geodetic from the astronomical azimuths has
+  been given above. From k we can now compute the length s of the
+  vertical section, and from this the shortest length. The difference of
+  length of the geodetic line and either of the plane curves is
+
+    e^4 s^5 cos^4 [phi]0 sin^2 2[alpha]0/360 a^4.
+
+  At least this is an approximate expression. Supposing s = 0.1a, this
+  quantity would be less than one-hundredth of a millimetre. The line s
+  is now to be calculated as a circular arc with a mean radius r along
+  AB. If [phi]0 = 1/2([phi] + [phi]'), [alpha]0 = 1/2(180 deg. + [alpha]
+  - [alpha]'), [Delta]0 = (1 - e^2 sin^2 [phi]0)^1/2, then 1/r =
+  [Delta]0/a [1 + e^2/(1 - e^2) (cos^2 [phi]0 cos^2 [alpha]0)], and
+  approximately sin (s/2r) = k/2r. These formulae give, in the case of k
+  = 0.1a, values certain to eight logarithmic decimal places. An
+  excellent series of formulae for the solution of the problem, to
+  determine the azimuths, chord and distance along the surface from the
+  geographical co-ordinates, was given in 1882 by Ch. M. Schols
+  (_Archives Neerlandaises_, vol. xvii.).
+
+
+  _Irregularities of the Earth's Surface._
+
+  In considering the effect of unequal distribution of matter in the
+  earth's crust on the form of the surface, we may simplify the matter
+  by disregarding the considerations of rotation and eccentricity. In
+  the first place, supposing the earth a sphere covered with a film of
+  water, let the density [rho] be a function of the distance from the
+  centre so that surfaces of equal density are concentric spheres. Let
+  now a disturbance of the arrangement of matter take place, so that the
+  density is no longer to be expressed by [rho], a function of r only,
+  but is expressed by [rho] + [rho]', where [rho]' is a function of
+  three co-ordinates [theta], [phi], r. Then [rho]' is the density of
+  what may be designated disturbing matter; it is positive in some
+  places and negative in others, and the whole quantity of matter whose
+  density is [rho]' is zero. The previously spherical surface of the sea
+  of radius a now takes a new form. Let P be a point on the disturbed
+  surface, P' the corresponding point vertically below it on the
+  undisturbed surface, PP' = N. The knowledge of N over the whole
+  surface gives us the form of the disturbed or actual surface of the
+  sea; it is an equipotential surface, and if V be the potential at P of
+  the disturbing matter [rho]', M the mass of the earth (the
+  attraction-constant is assumed equal to unity)
+
+      M             M     M
+    ----- + V = C = -- - --- N + V.
+    a + N           a    a^2
+
+  As far as we know, N is always a very small quantity, and we have with
+  sufficient approximation N = 3V/4[pi][delta]a, where [delta] is the
+  mean density of the earth. Thus we have the disturbance in elevation
+  of the sea-level expressed in terms of the potential of the disturbing
+  matter. If at any point P the value of N remain constant when we pass
+  to any adjacent point, then the actual surface is there parallel to
+  the ideal spherical surface; as a rule, however, the normal at P is
+  inclined to that at P', and astronomical observations have shown that
+  this inclination, the deflection or deviation, amounting ordinarily to
+  one or two seconds, may in some cases exceed 10", or, as at the foot
+  of the Himalayas, even 60". By the expression "mathematical figure of
+  the earth" we mean the surface of the sea produced in imagination so
+  as to percolate the continents. We see then that the effect of the
+  uneven distribution of matter in the crust of the earth is to produce
+  small elevations and depressions on the mathematical surface which
+  would be otherwise spheroidal. No geodesist can proceed far in his
+  work without encountering the irregularities of the mathematical
+  surface, and it is necessary that he should know how they affect his
+  astronomical observations. The whole of this subject is dealt with in
+  his usual elegant manner by Bessel in the _Astronomische Nachrichten_,
+  Nos. 329, 330, 331, in a paper entitled "Ueber den Einfluss der
+  Unregelmassigkeiten der Figur der Erde auf geodatische Arbeiten, &c."
+  But without entering into further details it is not difficult to see
+  how local attraction at any station affects the determinations of
+  latitude, longitude and azimuth there.
+
+  Let there be at the station an attraction to the north-east throwing
+  the zenith to the south-west, so that it takes in the celestial sphere
+  a position Z', its undisturbed position being Z. Let the rectangular
+  components of the displacement ZZ' be [xi] measured southwards and
+  [eta] measured westwards. Now the great circle joining Z' with the
+  pole of the heavens P makes there an angle with the meridian PZ =
+  [eta] cosec PZ' = [eta] sec [phi], where [phi] is the latitude of the
+  station. Also this great circle meets the horizon in a point whose
+  distance from the great circle PZ is [eta] sec [phi] sin [phi] = [eta]
+  tan [phi]. That is, a meridian mark, fixed by observations of the pole
+  star, will be placed that amount to the east of north. Hence the
+  observed latitude requires the correction [xi]; the observed longitude
+  a correction [eta] sec [phi]; and any observed azimuth a correction
+  [eta] tan [phi]. Here it is supposed that azimuths are measured from
+  north by east, and longitudes eastwards. The horizontal angles are
+  also influenced by the deflections of the plumb-line, in fact, just as
+  if the direction of the vertical axis of the theodolite varied by the
+  same amount. This influence, however, is slight, so long as the sights
+  point almost horizontally at the objects, which is always the case in
+  the observation of distant points.
+
+  The expression given for N enables one to form an approximate estimate
+  of the effect of a compact mountain in raising the sea-level. Take,
+  for instance, Ben Nevis, which contains about a couple of cubic miles;
+  a simple calculation shows that the elevation produced would only
+  amount to about 3 in. In the case of a mountain mass like the
+  Himalayas, stretching over some 1500 miles of country with a breadth
+  of 300 and an average height of 3 miles, although it is difficult or
+  impossible to find an expression for V, yet we may ascertain that an
+  elevation amounting to several hundred feet may exist near their base.
+  The geodetical operations, however, rather negative this idea, for it
+  was shown by Colonel Clarke (_Phil. Mag._, 1878) that the form of the
+  sea-level along the Indian arc departs but slightly from that of the
+  mean figure of the earth. If this be so, the action of the Himalayas
+  must be counteracted by subterranean tenuity.
+
+  Suppose now that A, B, C, ... are the stations of a network of
+  triangulation projected on or lying on a spheroid of semiaxis major
+  and eccentricity a, e, this spheroid having its axis parallel to the
+  axis of rotation of the earth, and its surface coinciding with the
+  mathematical surface of the earth at A. Then basing the calculations
+  on the observed elements at A, the calculated latitudes, longitudes
+  and directions of the meridian at the other points will be the true
+  latitudes, &c., of the points as projected on the spheroid. On
+  comparing these geodetic elements with the corresponding astronomical
+  determinations, there will appear a system of differences which
+  represent the inclinations, at the various points, of the actual
+  irregular surface to the surface of the spheroid of reference. These
+  differences will suggest two things,--first, that we may improve the
+  agreement of the two surfaces, by not restricting the spheroid of
+  reference by the condition of making its surface coincide with the
+  mathematical surface of the earth at A; and secondly, by altering the
+  form and dimensions of the spheroid. With respect to the first
+  circumstance, we may allow the spheroid two degrees of freedom, that
+  is, the normals of the surfaces at A may be allowed to separate a
+  small quantity, compounded of a meridional difference and a difference
+  perpendicular to the same. Let the spheroid be so placed that its
+  normal at A lies to the north of the normal to the earth's surface by
+  the small quantity [xi] and to the east by the quantity [eta]. Then in
+  starting the calculation of geodetic latitudes, longitudes and
+  azimuths from A, we must take, not the observed elements [phi],
+  [alpha], but for [phi], [phi] + [xi], and for [alpha], [alpha] + [eta]
+  tan [phi], and zero longitude must be replaced by [eta] sec [phi]. At
+  the same time suppose the elements of the spheroid to be altered from
+  a, e to a + da, e + de. Confining our attention at first to the two
+  points A, B, let ([phi]'), ([alpha]'), ([omega]) be the numerical
+  elements at B as obtained in the first calculation, viz. before the
+  shifting and alteration of the spheroid; they will now take the form
+
+    ([phi]') + f[xi] + g[eta] + hda + kde,
+
+    ([alpha]') + f'[xi] + g'[eta] + h'da + k'de,
+
+    [omega] + f"[xi] + g"[eta] + h"da + k"de,
+
+  where the coefficients f, g, ... &c. can be numerically calculated.
+  Now these elements, corresponding to the projection of B on the
+  spheroid of reference, must be equal severally to the astronomically
+  determined elements at B, corrected for the inclination of the
+  surfaces there. If [xi]', [eta]' be the components of the inclination
+  at that point, then we have
+
+    [xi]' = ([phi]') - [phi]' + f[xi] + g[eta] + hda + kde,
+
+    [eta]' tan [phi]' = ([alpha]') - [alpha]' + f'[xi] + g'[eta] + h'da + k'de,
+
+    [eta]' sec [phi]' = ([omega]) - [omega] + f"[xi] + g"[eta] + h"da + k"de,
+
+  where [phi]', [alpha]', [omega] are the observed elements at B. Here
+  it appears that the observation of longitude gives no additional
+  information, but is available as a check upon the azimuthal
+  observations.
+
+  If now there be a number of astronomical stations in the
+  triangulation, and we form equations such as the above for each point,
+  then we can from them determine those values of [xi], [eta], da, de,
+  which make the quantity [xi]^2 + [eta]^2 + [xi]'^2 + [eta]'^2 + ... a
+  minimum. Thus we obtain that spheroid which best represents the
+  surface covered by the triangulation.
+
+  In the _Account of the Principal Triangulation of Great Britain and
+  Ireland_ will be found the determination, from 75 equations, of the
+  spheroid best representing the surface of the British Isles. Its
+  elements are a = 20927005 [+-] 295 ft., b : a - b = 280 [+-] 8; and it
+  is so placed that at Greenwich Observatory [xi] = 1".864, [eta] =
+  -0".546.
+
+  Taking Durham Observatory as the origin, and the tangent plane to the
+  surface (determined by [xi] = -0".664, [eta] = -4".117) as the plane
+  of x and y, the former measured northwards, and z measured vertically
+  downwards, the equation to the surface is
+
+    .99524953 x^2 + .99288005 y^2 + .99763052 z^2 - 0.00671003xz - 41655070z = 0.
+
+
+  _Altitudes._
+
+  The precise determination of the altitude of his station is a matter
+  of secondary importance to the geodesist; nevertheless it is usual to
+  observe the zenith distances of all trigonometrical points. Of great
+  importance is a knowledge of the height of the base for its reduction
+  to the sea-level. Again the height of a station does influence a
+  little the observation of terrestrial angles, for a vertical line at B
+  does not lie generally in the vertical plane of A (see above). The
+  height above the sea-level also influences the geographical latitude,
+  inasmuch as the centrifugal force is increased and the magnitude and
+  direction of the attraction of the earth are altered, and the effect
+  upon the latitude is a very small term expressed by the formula h (g'-
+  g) sin 2 [phi] / ag, where g, g' are the values of gravity at the
+  equator and at the pole. This is h sin 2 [phi] / 5820 seconds, h being
+  in metres, a quantity which may be neglected, since for ordinary
+  mountain heights it amounts to only a few hundredths of a second. We
+  can assume this amount as joined with the northern component of the
+  plumb-line perturbations.
+
+  The uncertainties of terrestrial refraction render it impossible to
+  determine accurately by vertical angles the heights of distant points.
+  Generally speaking, refraction is greatest at about daybreak; from
+  that time it diminishes, being at a minimum for a couple of hours
+  before and after mid-day; later in the afternoon it again increases.
+  This at least is the general march of the phenomenon, but it is by no
+  means regular. The vertical angles measured at the station on Hart
+  Fell showed on one occasion in the month of September a refraction of
+  double the average amount, lasting from 1 P.M. to 5 P.M. The mean
+  value of the coefficient of refraction k determined from a very large
+  number of observations of terrestrial zenith distances in Great
+  Britain is .0792 [+-] .0047; and if we separate those rays which for a
+  considerable portion of their length cross the sea from those which do
+  not, the former give k = .0813 and the latter k = .0753. These values
+  are determined from high stations and long distances; when the
+  distance is short, and the rays graze the ground, the amount of
+  refraction is extremely uncertain and variable. A case is noted in the
+  Indian survey where the zenith distance of a station 10.5 miles off
+  varied from a depression of 4' 52".6 at 4.30 P.M. to an elevation of
+  2' 24".0 at 10.50 P.M.
+
+  If h, h' be the heights above the level of the sea of two stations, 90
+  deg. + [delta], 90 deg. + [delta]' their mutual zenith distances
+  ([delta] being that observed at h), s their distance apart, the earth
+  being regarded as a sphere of radius = a, then, with sufficient
+  precision,
+
+                    /   1 - 2k           \                   / 1 - 2k            \
+    h' - h = s tan ( s -------- - [delta] ), h - h' = s tan ( -------- - [delta]' ).
+                    \     2a             /                   \   2a              /
+
+  If from a station whose height is h the horizon of the sea be observed
+  to have a zenith distance 90 deg. + [delta], then the above formula
+  gives for h the value
+
+        a  tan^2 [delta]
+    h = -- -------------.
+        2     1 - 2k
+
+  Suppose the depression [delta] to be n minutes, then h = 1.054n^2 if
+  the ray be for the greater part of its length crossing the sea; if
+  otherwise, h = 1.040n^2. To take an example: the mean of eight
+  observations of the zenith distance of the sea horizon at the top of
+  Ben Nevis is 91 deg. 4' 48", or [delta] = 64.8; the ray is pretty
+  equally disposed over land and water, and hence h = 1.047n^2 = 4396
+  ft. The actual height of the hill by spirit-levelling is 4406 ft., so
+  that the error of the height thus obtained is only 10 ft.
+
+  The determination of altitudes by means of spirit-levelling is
+  undoubtedly the most exact method, particularly in its present
+  development as precise-levelling, by which there have been determined
+  in all civilized countries close-meshed nets of elevated points
+  covering the entire land.     (A. R. C; F. R. H.)
+
+
+FOOTNOTES:
+
+  [1] An arrangement acting similarly had been previously introduced by
+    Borda.
+
+  [2] _Geodetic Survey of South Africa_, vol. iii. (1905), p. viii;
+    _Les Nouveaux Appareils pour la mesure rapide des bases geod._, par
+    J. Rene Benoit et Ch. Ed. Guillaume (1906).
+
+  [3] See a paper "On the Course of Geodetic Lines on the Earth's
+    Surface" in the _Phil. Mag._ 1870; Helmert, _Theorien der hoheren
+    Geodasie_, 1. 321.
+
+  [4] Helmert, Theorien der hoheren Geodasie, 1. 232, 247.
+
+
+
+
+GEOFFREY, surnamed MARTEL (1006-1060), count of Anjou, son of the count
+Fulk Nerra (q.v.) and of the countess Hildegarde or Audegarde, was born
+on the 14th of October 1006. During his father's lifetime he was
+recognized as suzerain by Fulk l'Oison ("the Gosling"), count of
+Vendome, the son of his half-sister Adela. Fulk having revolted, he
+confiscated the countship, which he did not restore till 1050. On the
+1st of January 1032 he married Agnes, widow of William the Great, duke
+of Aquitaine, and taking arms against William the Fat, eldest son and
+successor of William the Great, defeated him and took him prisoner at
+Mont-Couer near Saint-Jouin-de-Marnes on the 20th of September 1033. He
+then tried to win recognition as dukes of Aquitaine for the sons of his
+wife Agnes by William the Great, who were still minors, but Fulk Nerra
+promptly took up arms to defend his suzerain William the Fat, from whom
+he held the Loudunois and Saintonge in fief against his son. In 1036
+Geoffrey Martel had to liberate William the Fat, on payment of a heavy
+ransom, but the latter having died in 1038, and the second son of
+William the Great, Odo, duke of Gascony, having fallen in his turn at
+the siege of Mauze (10th of March 1039) Geoffrey made peace with his
+father in the autumn of 1039, and had his wife's two sons recognized as
+dukes. About this time, also, he had interfered in the affairs of Maine,
+though without much result, for having sided against Gervais, bishop of
+Le Mans, who was trying to make himself guardian of the young count of
+Maine, Hugh, he had been beaten and forced to make terms with Gervais in
+1038. In 1040 he succeeded his father in Anjou and was able to conquer
+Touraine (1044) and assert his authority over Maine (see ANJOU). About
+1050 he repudiated Agnes, his first wife, and married Grecie, the widow
+of Bellay, lord of Montreuil-Bellay (before August 1052), whom he
+subsequently left in order to marry Adela, daughter of a certain Count
+Odo. Later he returned to Grecie, but again left her to marry Adelaide
+the German. When, however, he died on the 14th of November 1060, at the
+monastery of St Nicholas at Angers, he left no children, and transmitted
+the countship to Geoffrey the Bearded, the eldest of his nephews (see
+ANJOU).
+
+  See Louis Halphen, _Le Comte d'Anjou au XI^e siecle_ (Paris, 1906). A
+  summary biography is given by Celestin Port, _Dictionnaire historique,
+  geographique et biographique de Maine-et-Loire_ (3 vols.,
+  Paris-Angers, 1874-1878), vol. ii. pp. 252-253, and a sketch of the
+  wars by Kate Norgate, _England under the Angevin Kings_ (2 vols.,
+  London, 1887), vol. i. chs. iii. iv.     (L. H.*)
+
+
+
+
+GEOFFREY, surnamed PLANTAGENET [or PLANTEGENET] (1113-1151), count of
+Anjou, was the son of Count Fulk the Young and of Eremburge (or
+Arembourg of La Fleche); he was born on the 24th of August 1113. He is
+also called "le bel" or "the handsome," and received the surname of
+Plantagenet from the habit which he is said to have had of wearing in
+his cap a sprig of broom (_genet_). In 1127 he was made a knight, and on
+the 2nd of June 1129 married Matilda, daughter of Henry I. of England,
+and widow of the emperor Henry V. Some months afterwards he succeeded to
+his father, who gave up the countship when he definitively went to the
+kingdom of Jerusalem. The years of his government were spent in subduing
+the Angevin barons and in conquering Normandy (see ANJOU). In 1151,
+while returning from the siege of Montreuil-Bellay, he took cold, in
+consequence of bathing in the Loir at Chateau-du-Loir, and died on the
+7th of September. He was buried in the cathedral of Le Mans. By his wife
+Matilda he had three sons: Henry Plantagenet, born at Le Mans on Sunday,
+the 5th of March 1133; Geoffrey, born at Argentan on the 1st of June
+1134; and William Long-Sword, born on the 22nd of July 1136.
+
+  See Kate Norgate, _England under the Angevin Kings_ (2 vols., London,
+  1887), vol. i. chs. v.-viii.; Celestin Port, _Dictionnaire historique,
+  geographique et biographique de Maine-et-Loire_ (3 vols.,
+  Paris-Angers, 1874-1878), vol. ii. pp. 254-256. A history of Geoffrey
+  le Bel has yet to be written; there is a biography of him written in
+  the 12th century by Jean, a monk of Marmoutier, _Historia Gaufredi,
+  ducis Normannorum et comitis Andegavorum_, published by Marchegay et
+  Salmon; "Chroniques des comtes d'Anjou" (_Societe de l'histoire de
+  France_, Paris, 1856), pp. 229-310.     (L. H.*)
+
+
+
+
+GEOFFREY (1158-1186), duke of Brittany, fourth son of the English king
+Henry II. and his wife Eleanor of Aquitaine, was born on the 23rd of
+September 1158. In 1167 Henry suggested a marriage between Geoffrey and
+Constance (d. 1201), daughter and heiress of Conan IV., duke of Brittany
+(d. 1171); and Conan not only assented, perhaps under compulsion, to
+this proposal, but surrendered the greater part of his unruly duchy to
+the English king. Having received the homage of the Breton nobles,
+Geoffrey joined his brothers, Henry and Richard, who, in alliance with
+Louis VII. of France, were in revolt against their father; but he made
+his peace in 1174, afterwards helping to restore order in Brittany and
+Normandy, and aiding the new French king, Philip Augustus, to crush some
+rebellious vassals. In July 1181 his marriage with Constance was
+celebrated, and practically the whole of his subsequent life was spent
+in warfare with his brother Richard. In 1183 he made peace with his
+father, who had come to Richard's assistance; but a fresh struggle soon
+broke out for the possession of Anjou, and Geoffrey was in Paris
+treating for aid with Philip Augustus, when he died on the 19th of
+August 1186. He left a daughter, Eleanor, and his wife bore a
+posthumous son, the unfortunate Arthur.
+
+
+
+
+GEOFFREY (c. 1152-1212), archbishop of York, was a bastard son of Henry
+II., king of England. He was distinguished from his legitimate
+half-brothers by his consistent attachment and fidelity to his father.
+He was made bishop of Lincoln at the age of twenty-one (1173); but
+though he enjoyed the temporalities he was never consecrated and
+resigned the see in 1183. He then became his father's chancellor,
+holding a large number of lucrative benefices in plurality. Richard
+nominated him archbishop of York in 1189, but he was not consecrated
+till 1191, or enthroned till 1194. Geoffrey, though of high character,
+was a man of uneven temper; his history in chiefly one of quarrels, with
+the see of Canterbury, with the chancellor William Longchamp, with his
+half-brothers Richard and John, and especially with his canons at York.
+This last dispute kept him in litigation before Richard and the pope for
+many years. He led the clergy in their refusal to be taxed by John and
+was forced to fly the kingdom in 1207. He died in Normandy on the 12th
+of December 1212.
+
+  See Giraldus Cambrensis, _Vita Galfridi_; Stubbs's prefaces to _Roger
+  de Hoveden_, vols. iii. and iv. (Rolls Series).     (H. W. C. D.)
+
+
+
+
+GEOFFREY DE MONTBRAY (d. 1093), bishop of Coutances (_Constantiensis_),
+a right-hand man of William the Conqueror, was a type of the great
+feudal prelate, warrior and administrator at need. He knew, says
+Orderic, more about marshalling mailed knights than edifying
+psalm-singing clerks. Obtaining, as a young man, in 1048, the see of
+Coutances, by his brother's influence (see MOWBRAY), he raised from his
+fellow nobles and from their Sicilian spoils funds for completing his
+cathedral, which was consecrated in 1056. With bishop Odo, a warrior
+like himself, he was on the battle-field of Hastings, exhorting the
+Normans to victory; and at William's coronation it was he who called on
+them to acclaim their duke as king. His reward in England was a mighty
+fief scattered over twelve counties. He accompanied William on his visit
+to Normandy (1067), but, returning, led a royal force to the relief of
+Montacute in September 1069. In 1075 he again took the field, leading
+with Bishop Odo a vast host against the rebel earl of Norfolk, whose
+stronghold at Norwich they besieged and captured.
+
+Meanwhile the Conqueror had invested him with important judicial
+functions. In 1072 he had presided over the great Kentish suit between
+the primate and Bishop Odo, and about the same time over those between
+the abbot of Ely and his despoilers, and between the bishop of Worcester
+and the abbot of Ely, and there is some reason to think that he acted as
+a Domesday commissioner (1086), and was placed about the same time in
+charge of Northumberland. The bishop, who attended the Conqueror's
+funeral, joined in the great rising against William Rufus next year
+(1088), making Bristol, with which (as Domesday shows) he was closely
+connected and where he had built a strong castle, his base of
+operations. He burned Bath and ravaged Somerset, but had submitted to
+the king before the end of the year. He appears to have been at Dover
+with William in January 1090, but, withdrawing to Normandy, died at
+Coutances three years later. In his fidelity to Duke Robert he seems to
+have there held out for him against his brother Henry, when the latter
+obtained the Cotentin.
+
+  See E.A. Freeman, _Norman Conquest_ and _William Rufus_; J.H. Round,
+  _Feudal England_; and, for original authorities, the works of Orderic
+  Vitalis and William of Poitiers, and of Florence of Worcester; the
+  Anglo-Saxon Chronicle; William of Malmesbury's _Gesta pontificum_, and
+  Lanfranc's works, ed. Giles; Domesday Book.     (J. H. R.)
+
+
+
+
+GEOFFREY OF MONMOUTH (d. 1154), bishop of St Asaph and writer on early
+British history, was born about the year 1100. Of his early life little
+is known, except that he received a liberal education under the eye of
+his paternal uncle, Uchtryd, who was at that time archdeacon, and
+subsequently bishop, of Llandaff. In 1129 Geoffrey appears at Oxford
+among the witnesses of an Oseney charter. He subscribes himself Geoffrey
+Arturus; from this we may perhaps infer that he had already begun his
+experiments in the manufacture of Celtic mythology. A first edition of
+his _Historia Britonum_ was in circulation by the year 1139, although
+the text which we possess appears to date from 1147. This famous work,
+which the author has the audacity to place on the same level with the
+histories of William of Malmesbury and Henry of Huntingdon, professes to
+be a translation from a Celtic source; "a very old book in the British
+tongue" which Walter, archdeacon of Oxford, had brought from Brittany.
+Walter the archdeacon is a historical personage; whether his book has
+any real existence may be fairly questioned. There is nothing in the
+matter or the style of the _Historia_ to preclude us from supposing that
+Geoffrey drew partly upon confused traditions, partly on his own powers
+of invention, and to a very slight degree upon the accepted authorities
+for early British history. His chronology is fantastic and incredible;
+William of Newburgh justly remarks that, if we accepted the events which
+Geoffrey relates, we should have to suppose that they had happened in
+another world. William of Newburgh wrote, however, in the reign of
+Richard I. when the reputation of Geoffrey's work was too well
+established to be shaken by such criticisms. The fearless romancer had
+achieved an immediate success. He was patronized by Robert, earl of
+Gloucester, and by two bishops of Lincoln; he obtained, about 1140, the
+archdeaconry of Llandaff "on account of his learning"; and in 1151 was
+promoted to the see of St Asaph.
+
+Before his death the _Historia Britonum_ had already become a model and
+a quarry for poets and chroniclers. The list of imitators begins with
+Geoffrey Gaimar, the author of the _Estorie des Engles_ (c. 1147), and
+Wace, whose _Roman de Brut_ (1155) is partly a translation and partly a
+free paraphrase of the _Historia_. In the next century the influence of
+Geoffrey is unmistakably attested by the _Brut_ of Layamon, and the
+rhyming English chronicle of Robert of Gloucester. Among later
+historians who were deceived by the _Historia Britonum_ it is only
+needful to mention Higdon, Hardyng, Fabyan (1512), Holinshed (1580) and
+John Milton. Still greater was the influence of Geoffrey upon those
+writers who, like Warner in _Albion's England_ (1586), and Drayton in
+_Polyolbion_ (1613), deliberately made their accounts of English history
+as poetical as possible. The stories which Geoffrey preserved or
+invented were not infrequently a source of inspiration to literary
+artists. The earliest English tragedy, _Gorboduc_ (1565), the _Mirror
+for Magistrates_ (1587), and Shakespeare's Lear, are instances in point.
+It was, however, the Arthurian legend which of all his fabrications
+attained the greatest vogue. In the work of expanding and elaborating
+this theme the successors of Geoffrey went as far beyond him as he had
+gone beyond Nennius; but he retains the credit due to the founder of a
+great school. Marie de France, who wrote at the court of Henry II., and
+Chretien de Troyes, her French contemporary, were the earliest of the
+avowed romancers to take up the theme. The succeeding age saw the
+Arthurian story popularized, through translations of the French
+romances, as far afield as Germany and Scandinavia. It produced in
+England the _Roman du Saint Graal_ and the _Roman de Merlin_, both from
+the pen of Robert de Borron; the _Roman de Lancelot_; the _Roman de
+Tristan_, which is attributed to a fictitious Lucas de Gast. In the
+reign of Edward IV. Sir Thomas Malory paraphrased and arranged the best
+episodes of these romances in English prose. His _Morte d'Arthur_,
+printed by Caxton in 1485, epitomizes the rich mythology which
+Geoffrey's work had first called into life, and gave the Arthurian story
+a lasting place in the English imagination. The influence of the
+_Historia Britonum_ may be illustrated in another way, by enumerating
+the more familiar of the legends to which it first gave popularity. Of
+the twelve books into which it is divided only three (Bks. IX., X., XI.)
+are concerned with Arthur. Earlier in the work, however, we have the
+adventures of Brutus; of his follower Corineus, the vanquisher of the
+Cornish giant Goemagol (Gogmagog); of Locrinus and his daughter Sabre
+(immortalized in Milton's _Comus_); of Bladud the builder of Bath; of
+Lear and his daughters; of the three pairs of brothers, Ferrex and
+Porrex, Brennius and Belinus, Elidure and Peridure. The story of
+Vortigern and Rowena takes its final form in the _Historia Britonum_;
+and Merlin makes his first appearance in the prelude to the Arthur
+legend. Besides the _Historia Britonum_ Geoffrey is also credited with
+a _Life of Merlin_ composed in Latin verse. The authorship of this work
+has, however, been disputed, on the ground that the style is distinctly
+superior to that of the _Historia_. A minor composition, the _Prophecies
+of Merlin_, was written before 1136, and afterwards incorporated with
+the _Historia_, of which it forms the seventh book.
+
+  For a discussion of the manuscripts of Geoffrey's work, see Sir T.D.
+  Hardy's _Descriptive Catalogue_ (Rolls Series), i. pp. 341 ff. The
+  _Historia Britonum_ has been critically edited by San Marte (Halle,
+  1854). There is an English translation by J.A. Giles (London, 1842).
+  The _Vita Merlini_ has been edited by F. Michel and T. Wright (Paris,
+  1837). See also the _Dublin Univ. Magazine_ for April 1876, for an
+  article by T. Gilray on the literary influence of Geoffrey; G.
+  Heeger's _Trojanersage der Britten_ (1889); and La Borderie's _Etudes
+  historiques bretonnes_ (1883).     (H. W. C. D.)
+
+
+
+
+GEOFFREY OF PARIS (d. c. 1320), French chronicler, was probably the
+author of the _Chronique metrique de Philippe le Bel, or Chronique rimee
+de Geoffroi de Paris_. This work, which deals with the history of France
+from 1300 to 1316, contains 7918 verses, and is valuable as that of a
+writer who had a personal knowledge of many of the events which he
+relates. Various short historical poems have also been attributed to
+Geoffrey, but there is no certain information about either his life or
+his writings.
+
+  The _Chronique_ was published by J.A. Buchon in his _Collection des
+  chroniques_, tome ix. (Paris, 1827), and it has also been printed in
+  tome xxii. of the _Recueil des historiens des Gaules et de la France_
+  (Paris, 1865). See G. Paris, _Histoire de la litterature francaise au
+  moyen age_ (Paris, 1890); and A. Molinier, _Les Sources de l'histoire
+  de France_, tome iii. (Paris, 1903).
+
+
+
+
+GEOFFREY THE BAKER (d. c. 1360), English chronicler, is also called
+Walter of Swinbroke, and was probably a secular clerk at Swinbrook in
+Oxfordshire. He wrote a _Chronicon Angliae temporibus Edwardi II. et
+Edwardi III._, which deals with the history of England from 1303 to
+1356. From the beginning until about 1324 this work is based upon Adam
+Murimuth's _Continuatio chronicarum_, but after this date it is valuable
+and interesting, containing information not found elsewhere, and closing
+with a good account of the battle of Poitiers. The author obtained his
+knowledge about the last days of Edward II. from William Bisschop, a
+companion of the king's murderers, Thomas Gurney and John Maltravers.
+Geoffrey also wrote a _Chroniculum_ from the creation of the world until
+1336, the value of which is very slight. His writings have been edited
+with notes by Sir E.M. Thompson as the _Chronicon Galfridi le Baker de
+Swynebroke_ (Oxford, 1889). Some doubt exists concerning Geoffrey's
+share in the compilation of the _Vita et mors Edwardi II._, usually
+attributed to Sir Thomas de la More, or Moor, and printed by Camden in
+his _Anglica scripta_. It has been maintained by Camden and others that
+More wrote an account of Edward's reign in French, and that this was
+translated into Latin by Geoffrey and used by him in compiling his
+_Chronicon_. Recent scholarship, however, asserts that More was no
+writer, and that the _Vita et mors_ is an extract from Geoffrey's
+_Chronicon_, and was attributed to More, who was the author's patron. In
+the main this conclusion substantiates the verdict of Stubbs, who has
+published the _Vita et mors_ in his _Chronicles of the reigns of Edward
+I. and Edward II._ (London, 1883). The manuscripts of Geoffrey's works
+are in the Bodleian library at Oxford.
+
+
+
+
+GEOFFRIN, MARIE THERESE RODET (1699-1777), a Frenchwoman who played an
+interesting part in French literary and artistic life, was born in Paris
+in 1699. She married, on the 19th of July 1713, Pierre Francois
+Geoffrin, a rich manufacturer and lieutenant-colonel of the National
+Guard, who died in 1750. It was not till Mme Geoffrin was nearly fifty
+years of age that we begin to hear of her as a power in Parisian
+society. She had learned much from Mme de Tencin, and about 1748 began
+to gather round her a literary and artistic circle. She had every week
+two dinners, on Monday for artists, and on Wednesday for her friends the
+Encyclopaedists and other men of letters. She received many foreigners
+of distinction, Hume and Horace Walpole among others. Walpole spent much
+time in her society before he was finally attached to Mme du Deffand,
+and speaks of her in his letters as a model of common sense. She was
+indeed somewhat of a small tyrant in her circle. She had adopted the
+pose of an old woman earlier than necessary, and her coquetry, if such
+it can be called, took the form of being mother and mentor to her
+guests, many of whom were indebted to her generosity for substantial
+help. Although her aim appears to have been to have the _Encyclopedie_
+in conversation and action around her, she was extremely displeased with
+any of her friends who were so rash as to incur open disgrace. Marmontel
+lost her favour after the official censure of _Belisaire_, and her
+advanced views did not prevent her from observing the forms of religion.
+A devoted Parisian, Mme Geoffrin rarely left the city, so that her
+journey to Poland in 1766 to visit the king, Stanislas Poniatowski, whom
+she had known in his early days in Paris, was a great event in her life.
+Her experiences induced a sensible gratitude that she had been born
+"_Francaise_" and "_particuliere_." In her last illness her daughter,
+Therese, marquise de la Ferte Imbault, excluded her mother's old friends
+so that she might die as a good Christian, a proceeding wittily
+described by the old lady: "My daughter is like Godfrey de Bouillon, she
+wished to defend my tomb from the infidels." Mme Geoffrin died in Paris
+on the 6th of October 1777.
+
+  See _Correspondance inedite du roi Stanislas Auguste Poniatowski et de
+  Madame Geoffrin_, edited by the comte de Mouy (1875); P. de Segur, _Le
+  Royaume de la rue Saint-Honore, Madame Geoffrin et sa fille_ (1897);
+  A. Tornezy, _Un Bureau d'esprit au XVIII^e siecle: le salon de Madame
+  Geoffrin_ (1895); and Janet Aldis, _Madame Geoffrin, her Salon and her
+  Times, 1750-1777_ (1905).
+
+
+
+
+GEOFFROY, ETIENNE FRANCOIS (1672-1731), French chemist, born in Paris on
+the 13th of February 1672, was first an apothecary and then practised
+medicine. After studying at Montpellier he accompanied Marshal Tallard
+on his embassy to London in 1698 and thence travelled to Holland and
+Italy. Returning to Paris he became professor of chemistry at the Jardin
+du Roi and of pharmacy and medicine at the College de France, and dean
+of the faculty of medicine. He died in Paris on the 6th of January 1731.
+His name is best known in connexion with his tables of affinities
+(_tables des rapports_), which he presented to the French Academy in
+1718 and 1720. These were lists, prepared by collating observations on
+the actions of substances one upon another, showing the varying degrees
+of affinity exhibited by analogous bodies for different reagents, and
+they retained their vogue for the rest of the century, until displaced
+by the profounder conceptions introduced by C.L. Berthollet. Another of
+his papers dealt with the delusions of the philosopher's stone, but
+nevertheless he believed that iron could be artificially formed in the
+combustion of vegetable matter. His _Tractatus de materia medica_,
+published posthumously in 1741, was long celebrated.
+
+His brother CLAUDE JOSEPH, known as Geoffroy the younger (1685-1752),
+was also an apothecary and chemist who, having a considerable knowledge
+of botany, devoted himself especially to the study of the essential oils
+in plants.
+
+
+
+
+GEOFFROY, JULIEN LOUIS (1743-1814), French critic, was born at Rennes in
+1743. He studied in the school of his native town and at the College
+Louis le Grand in Paris. He took orders and fulfilled for some time the
+humble functions of an usher, eventually becoming professor of rhetoric
+at the _College Mazarin_. A bad tragedy, Caton, was accepted at the
+_Theatre Francais_, but was never acted. On the death of Elie Freron in
+1776 the other collaborators in the _Annee litteraire_ asked Geoffroy to
+succeed him, and he conducted the journal until in 1792 it ceased to
+appear. Geoffroy was a bitter critic of Voltaire and his followers, and
+made for himself many enemies. An enthusiastic royalist, he published
+with Freron's brother-in-law, the abbe Thomas Royou (1741-1792), a
+journal, _L'Ami du roi_ (1790-1792), which possibly did more harm than
+good to the king's cause by its ill-advised partisanship. During the
+Terror Geoffroy hid in the neighbourhood of Paris, only returning in
+1799. An attempt to revive the _Annee litteraire_ failed, and Geoffroy
+undertook the dramatic feuilleton of the _Journal des debats_. His
+scathing criticisms had a success of notoriety, but their popularity was
+ephemeral, and the publication of them (5 vols., 1819-1820) as _Cours de
+litterature dramatique_ proved a failure. He was also the author of a
+perfunctory _Commentaire_ on the works of Racine prefixed to Lenormant's
+edition (1808). He died in Paris on the 27th of February 1814.
+
+
+
+
+GEOFFROY SAINT-HILAIRE, ETIENNE (1772-1844), French naturalist, was the
+son of Jean Gerard Geoffroy, procurator and magistrate of Etampes,
+Seine-et-Oise, where he was born on the 15th of April 1772. Destined for
+the church he entered the college of Navarre, in Paris, where he studied
+natural philosophy under M.J. Brisson; and in 1788 he obtained one of
+the canonicates of the chapter of Sainte Croix at Etampes, and also a
+benefice. Science, however, offered him a more congenial career, and he
+gained from his father permission to remain in Paris, and to attend the
+lectures at the College de France and the Jardin des Plantes, on the
+condition that he should also read law. He accordingly took up his
+residence at Cardinal Lemoine's college, and there became the pupil and
+soon the esteemed associate of Brisson's friend, the abbe Hauy, the
+mineralogist. Having, before the close of the year 1790, taken the
+degree of bachelor in law, he became a student of medicine, and attended
+the lectures of A.F. de Fourcroy at the Jardin des Plantes, and of
+L.J.M. Daubenton at the College de France. His studies at Paris were at
+length suddenly interrupted, for, in August 1792, Hauy and the other
+professors of Lemoine's college, as also those of the college of
+Navarre, were arrested by the revolutionists as priests, and confined in
+the prison of St Firmin. Through the influence of Daubenton and others
+Geoffroy on the 14th of August obtained an order for the release of Hauy
+in the name of the Academy; still the other professors of the two
+colleges, save C.F. Lhomond, who had been rescued by his pupil J.L.
+Tallien, remained in confinement. Geoffroy, foreseeing their certain
+destruction if they remained in the hands of the revolutionists,
+determined if possible to secure their liberty by stratagem. By bribing
+one of the officials at St Firmin, and disguising himself as a
+commissioner of prisons, he gained admission to his friends, and
+entreated them to effect their escape by following him. All, however,
+dreading lest their deliverance should render the doom of their
+fellow-captives the more certain, refused the offer, and one priest
+only, who was unknown to Geoffroy, left the prison. Already on the night
+of the 2nd of September the massacre of the proscribed had begun, when
+Geoffroy, yet intent on saving the life of his friends and teachers,
+repaired to St Firmin. At 4 o'clock on the morning of the 3rd of
+September, after eight hours' waiting, he by means of a ladder assisted
+the escape of twelve ecclesiastics, not of the number of his
+acquaintance, and then the approach of dawn and the discharge of a gun
+directed at him warned him, his chief purpose unaccomplished, to return
+to his lodgings. Leaving Paris he retired to Etampes, where, in
+consequence of the anxieties of which he had lately been the prey, and
+the horrors which he had witnessed, he was for some time seriously ill.
+At the beginning of the winter of 1792 he returned to his studies in
+Paris, and in March of the following year Daubenton, through the
+interest of Bernardin de Saint Pierre, procured him the office of
+sub-keeper and assistant demonstrator of the cabinet of natural history,
+vacant by the resignation of B.G.E. Lacepede. By a law passed in June
+1793, Geoffroy was appointed one of the twelve professors of the newly
+constituted museum of natural history, being assigned the chair of
+zoology. In the same year he busied himself with the formation of a
+menagerie at that institution.
+
+In 1794 through the introduction of A.H. Tessier he entered into
+correspondence with Georges Cuvier, to whom, after the perusal of some
+of his manuscripts, he wrote: "Venez jouer parmi nous le role de Linne,
+d'un autre legislateur de l'histoire naturelle." Shortly after the
+appointment of Cuvier as assistant at the Museum d'Histoire Naturelle,
+Geoffroy received him into his house. The two friends wrote together
+five memoirs on natural history, one of which, on the classification of
+mammals, puts forward the idea of the subordination of characters upon
+which Cuvier based his zoological system. It was in a paper entitled
+"Histoire des Makis, ou singes de Madagascar," written in 1795, that
+Geoffroy first gave expression to his views on "the unity of organic
+composition," the influence of which is perceptible in all his
+subsequent writings; nature, he observes, presents us with only one plan
+of construction, the same in principle, but varied in its accessory
+parts.
+
+In 1798 Geoffroy was chosen a member of the great scientific expedition
+to Egypt, and on the capitulation of Alexandria in August 1801, he took
+part in resisting the claim made by the British general to the
+collections of the expedition, declaring that, were that demand
+persisted in, history would have to record that he also had burnt a
+library in Alexandria. Early in January 1802 Geoffroy returned to his
+accustomed labours in Paris. He was elected a member of the academy of
+sciences of that city in September 1807. In March of the following year
+the emperor, who had already recognized his national services by the
+award of the cross of the legion of honour, selected him to visit the
+museums of Portugal, for the purpose of procuring collections from them,
+and in the face of considerable opposition from the British he
+eventually was successful in retaining them as a permanent possession
+for his country. In 1809, the year after his return to France, he was
+made professor of zoology at the faculty of sciences at Paris, and from
+that period he devoted himself more exclusively than before to
+anatomical study. In 1818 he gave to the world the first part of his
+celebrated _Philosophie anatomique_, the second volume of which,
+published in 1822, and subsequent memoirs account for the formation of
+monstrosities on the principle of arrest of development, and of the
+attraction of similar parts. When, in 1830, Geoffroy proceeded to apply
+to the invertebrata his views as to the unity of animal composition, he
+found a vigorous opponent in Georges Cuvier, and the discussion between
+them, continued up to the time of the death of the latter, soon
+attracted the attention of the scientific throughout Europe. Geoffroy, a
+synthesist, contended, in accordance with his theory of unity of plan in
+organic composition, that all animals are formed of the same elements,
+in the same number, and with the same connexions: homologous parts,
+however they differ in form and size, must remain associated in the same
+invariable order. With Goethe he held that there is in nature a law of
+compensation or balancing of growth, so that if one organ take on an
+excess of development, it is at the expense of some other part; and he
+maintained that, since nature takes no sudden leaps, even organs which
+are superfluous in any given species, if they have played an important
+part in other species of the same family, are retained as rudiments,
+which testify to the permanence of the general plan of creation. It was
+his conviction that, owing to the conditions of life, the same forms had
+not been perpetuated since the origin of all things, although it was not
+his belief that existing species are becoming modified. Cuvier, who was
+an analytical observer of facts, admitted only the prevalence of "laws
+of co-existence" or "harmony" in animal organs, and maintained the
+absolute invariability of species, which he declared had been created
+with a regard to the circumstances in which they were placed, each organ
+contrived with a view to the function it had to fulfil, thus putting, in
+Geoffroy's considerations, the effect for the cause.
+
+In July 1840 Geoffroy became blind, and some months later he had a
+paralytic attack. From that time his strength gradually failed him. He
+resigned his chair at the museum in 1841, and died at Paris on the 19th
+of June 1844.
+
+  Geoffroy wrote: _Catalogue des mammiferes du Museum National
+  d'Histoire Naturelle_ (1813), not quite completed; _Philosophie
+  anatomique_--t. i., _Des organes respiratoires_ (1818), and t. ii.,
+  _Des monstruosites humaines_ (1822); _Systeme dentaire des mammiferes
+  et des oiseaux_ (1st pt., 1824); _Sur le principe de l'unite de
+  composition organique_ (1828); _Cours de l'histoire naturelle des
+  mammiferes_ (1829); _Principes de philosophie zoologique_ (1830);
+  _Etudes progressives d'un naturaliste_ (1835); _Fragments
+  biographiques_ (1832); _Notions synthetiques, historiques et
+  physiologiques de philosophie naturelle_ (1838), and other works; also
+  part of the _Description de l'Egypte par la commission des sciences_
+  (1821-1830); and, with Frederic Cuvier (1773-1838), a younger brother
+  of G. Cuvier, _Histoire naturelle des mammiferes_ (4 vols.,
+  1820-1842); besides numerous papers on such subjects as the anatomy of
+  marsupials, ruminants and electrical fishes, the vertebrate theory of
+  the skull, the opercula of fishes, teratology, palaeontology and the
+  influence of surrounding conditions in modifying animal forms.
+
+  See _Vie, travaux, et doctrine scientifique d'Etienne Geoffroy
+  Saint-Hilaire, par son fils M. Isidore Geoffroy Saint-Hilaire_ (Paris
+  and Strasburg, 1847), to which is appended a list of Geoffroy's works;
+  and Joly, in _Biog. universelle_, t. xvi. (1856).
+
+
+
+
+
+GEOFFROY SAINT-HILAIRE, ISIDORE (1805-1861), French zoologist, son of
+the preceding, was born at Paris on the 16th of December 1805. In his
+earlier years he showed an aptitude for mathematics, but eventually he
+devoted himself to the study of natural history and of medicine, and in
+1824 he was appointed assistant naturalist to his father. On the
+occasion of his taking the degree of doctor of medicine in September
+1829, he read a thesis entitled _Propositions sur la monstruosite,
+consideree chez l'homme et les animaux_; and in 1832-1837 was published
+his great teratological work, _Histoire generale et particuliere des
+anomalies de l'organisation chez l'homme et les animaux_, 3 vols. 8vo.
+with 20 plates. In 1829 he delivered for his father the second part of a
+course of lectures on ornithology, and during the three following years
+he taught zoology at the Athenee, and teratology at the Ecole pratique.
+He was elected a member of the academy of sciences at Paris in 1833, was
+in 1837 appointed to act as deputy for his father at the faculty of
+sciences in Paris, and in the following year was sent to Bordeaux to
+organize a similar faculty there. He became successively inspector of
+the academy of Paris (1840), professor of the museum on the retirement
+of his father (1841), inspector-general of the university (1844), a
+member of the royal council for public instruction (1845), and on the
+death of H.M.D. de Blainville, professor of zoology at the faculty of
+sciences (1850). In 1854 he founded the Acclimatization Society of
+Paris, of which he was president. He died at Paris on the 10th of
+November 1861.
+
+  Besides the above-mentioned works, he wrote: _Essais de zoologie
+  generale_ (1841); _Vie ... d'Etienne Geoffroy Saint-Hilaire_ (1847);
+  _Acclimatation et domestication des animaux utiles_ (1849; 4th ed.,
+  1861); _Lettres sur les substances alimentaires et particulierement
+  sur la viande de cheval_ (1856); and _Histoire naturelle generale des
+  regnes organiques_ (3 vols., 1854-1862), which was not quite
+  completed. He was the author also of various papers on zoology,
+  comparative anatomy and palaeontology.
+
+
+
+
+GEOGRAPHY (Gr. [Greek: ge], earth, and [Greek: graphein], to write), the
+exact and organized knowledge of the distribution of phenomena on the
+surface of the earth. The fundamental basis of geography is the vertical
+relief of the earth's crust, which controls all mobile distributions.
+The grander features of the relief of the lithosphere or stony crust of
+the earth control the distribution of the hydrosphere or collected
+waters which gather into the hollows, filling them up to a height
+corresponding to the volume, and thus producing the important practical
+division of the surface into land and water. The distribution of the
+mass of the atmosphere over the surface of the earth is also controlled
+by the relief of the crust, its greater or lesser density at the surface
+corresponding to the lesser or greater elevation of the surface. The
+simplicity of the zonal distribution of solar energy on the earth's
+surface, which would characterize a uniform globe, is entirely destroyed
+by the dissimilar action of land and water with regard to radiant heat,
+and by the influence of crust-forms on the direction of the resulting
+circulation. The influence of physical environment becomes clearer and
+stronger when the distribution of plant and animal life is considered,
+and if it is less distinct in the case of man, the reason is found in
+the modifications of environment consciously produced by human effort.
+Geography is a synthetic science, dependent for the data with which it
+deals on the results of specialized sciences such as astronomy, geology,
+oceanography, meteorology, biology and anthropology, as well as on
+topographical description. The physical and natural sciences are
+concerned in geography only so far as they deal with the forms of the
+earth's surface, or as regards the distribution of phenomena. The
+distinctive task of geography as a science is to investigate the control
+exercised by the crust-forms directly or indirectly upon the various
+mobile distributions. This gives to it unity and definiteness, and
+renders superfluous the attempts that have been made from time to time
+to define the limits which divide geography from geology on the one hand
+and from history on the other. It is essential to classify the
+subject-matter of geography in such a manner as to give prominence not
+only to facts, but to their mutual relations and their natural and
+inevitable order.
+
+The fundamental conception of geography is form, including the figure of
+the earth and the varieties of crustal relief. Hence mathematical
+geography (see MAP), including cartography as a practical application,
+comes first. It merges into physical geography, which takes account of the
+forms of the lithosphere (geomorphology), and also of the distribution of
+the hydrosphere and the rearrangements resulting from the workings of
+solar energy throughout the hydrosphere and atmosphere (oceanography and
+climatology). Next follows the distribution of plants and animals
+(biogeography), and finally the distribution of mankind and the various
+artificial boundaries and redistributions (anthropogeography). The
+applications of anthropogeography to human uses give rise to political and
+commercial geography, in the elucidation of which all the earlier
+departments or stages have to be considered, together with historical and
+other purely human conditions. The evolutionary idea has revolutionized
+and unified geography as it did biology, breaking down the old
+hard-and-fast partitions between the various departments, and substituting
+the study of the nature and influence of actual terrestrial environments
+for the earlier motive, the discovery and exploration of new lands.
+
+
+  HISTORY OF GEOGRAPHICAL THEORY
+
+  The earliest conceptions of the earth, like those held by the
+  primitive peoples of the present day, are difficult to discover and
+  almost impossible fully to grasp. Early generalizations, as far as
+  they were made from known facts, were usually expressed in symbolic
+  language, and for our present purpose it is not profitable to
+  speculate on the underlying truths which may sometimes be suspected in
+  the old mythological cosmogonies.
+
+
+    Early Greek ideas.
+
+    Flat earth of Homer.
+
+    Hecataeus.
+
+    Herodotus.
+
+    The idea of symmetry.
+
+  The first definite geographical theories to affect the western world
+  were those evolved, or at least first expressed, by the Greeks.[1] The
+  earliest theoretical problem of geography was the form of the earth.
+  The natural supposition that the earth is a flat disk, circular or
+  elliptical in outline, had in the time of Homer acquired a special
+  definiteness by the introduction of the idea of the ocean river
+  bounding the whole, an application of imperfectly understood
+  observations. Thales of Miletus is claimed as the first exponent of
+  the idea of a spherical earth; but, although this does not appear to
+  be warranted, his disciple Anaximander (c. 580 B.C.) put forward the
+  theory that the earth had the figure of a solid body hanging freely in
+  the centre of the hollow sphere of the starry heavens. The Pythagorean
+  school of philosophers adopted the theory of a spherical earth, but
+  from metaphysical rather than scientific reasons; their convincing
+  argument was that a sphere being the most perfect solid figure was the
+  only one worthy to circumscribe the dwelling-place of man. The
+  division of the sphere into parallel zones and some of the
+  consequences of this generalization seem to have presented themselves
+  to Parmenides (c. 450 B.C.); but these ideas did not influence the
+  Ionian school of philosophers, who in their treatment of geography
+  preferred to deal with facts demonstrable by travel rather than with
+  speculations. Thus Hecataeus, claimed by H.F. Tozer[2] as the father
+  of geography on account of his _Periodos_, or general treatise on the
+  earth, did not advance beyond the primitive conception of a circular
+  disk. He systematized the form of the land within the ring of
+  ocean--the [Greek: oikoumene], or habitable world--by recognizing two
+  continents: Europe to the north, and Asia to the south of the midland
+  sea. Herodotus, equally oblivious of the sphere, criticized and
+  ridiculed the circular outline of the _oekumene_, which he knew to be
+  longer from east to west than it was broad from north to south. He
+  also pointed out reasons for accepting a division of the land into
+  three continents--Europe, Asia and Africa. Beyond the limits of his
+  personal travels Herodotus applied the characteristically Greek theory
+  of symmetry to complete, in the unknown, outlines of lands and rivers
+  analogous to those which had been explored. Symmetry was in fact the
+  first geographical theory, and the effect of Herodotus's hypothesis
+  that the Nile must flow from west to east before turning north in
+  order to balance the Danube running from west to east before turning
+  south lingered in the maps of Africa down to the time of Mungo
+  Park.[3]
+
+
+    Aristotle and the sphere.
+
+  To Aristotle (384-322 B.C.) must be given the distinction of founding
+  scientific geography. He demonstrated the sphericity of the earth by
+  three arguments, two of which could be tested by observation. These
+  were: (1) that the earth must be spherical, because of the tendency of
+  matter to fall together towards a common centre; (2) that only a
+  sphere could always throw a circular shadow on the moon during an
+  eclipse; and (3) that the shifting of the horizon and the appearance
+  of new constellations, or the disappearance of familiar stars, as one
+  travelled from north to south, could only be explained on the
+  hypothesis that the earth was a sphere. Aristotle, too, gave greater
+  definiteness to the idea of zones conceived by Parmenides, who had
+  pictured a torrid zone uninhabitable by reason of heat, two frigid
+  zones uninhabitable by reason of cold, and two intermediate temperate
+  zones fit for human occupation. Aristotle defined the temperate zone
+  as extending from the tropic to the arctic circle, but there is some
+  uncertainty as to the precise meaning he gave to the term "arctic
+  circle." Soon after his time, however, this conception was clearly
+  established, and with so large a generalization the mental horizon was
+  widened to conceive of a geography which was a science. Aristotle had
+  himself shown that in the southern temperate zone winds similar to
+  those of the northern temperate zone should blow, but from the
+  opposite direction.
+
+
+    Fitting the oekumene to the sphere.
+
+  While the theory of the sphere was being elaborated the efforts of
+  practical geographers were steadily directed towards ascertaining the
+  outline and configuration of the _oekumene_, or habitable world, the
+  only portion of the terrestrial surface known to the ancients and to
+  the medieval peoples, and still retaining a shadow of its old monopoly
+  of geographical attention in its modern name of the "Old World." The
+  fitting of the _oekumene_ to the sphere was the second theoretical
+  problem. The circular outline had given way in geographical opinion to
+  the elliptical with the long axis lying east and west, and Aristotle
+  was inclined to view it as a very long and relatively narrow band
+  almost encircling the globe in the temperate zone. His argument as to
+  the narrowness of the sea between West Africa and East Asia, from the
+  occurrence of elephants at both extremities, is difficult to
+  understand, although it shows that he looked on the distribution of
+  animals as a problem of geography.
+
+
+    Problem of the Antipodes.
+
+  Pythagoras had speculated as to the existence of antipodes, but it was
+  not until the first approximately accurate measurements of the globe
+  and estimates of the length and breadth of the _oekumene_ were made by
+  Eratosthenes (c. 250 B.C.) that the fact that, as then known, it
+  occupied less than a quarter of the surface of the sphere was clearly
+  recognized. It was natural, if not strictly logical, that the ocean
+  river should be extended from a narrow stream to a world-embracing
+  sea, and here again Greek theory, or rather fancy, gave its modern
+  name to the greatest feature of the globe. The old instinctive idea of
+  symmetry must often have suggested other _oekumene_ balancing the
+  known world in the other quarters of the globe. The Stoic
+  philosophers, especially Crates of Mallus, arguing from the love of
+  nature for life, placed an _oekumene_ in each quarter of the sphere,
+  the three unknown world-islands being those of the Antoeci, Perioeci
+  and Antipodes. This was a theory not only attractive to the
+  philosophical mind, but eminently adapted to promote exploration. It
+  had its opponents, however, for Herodotus showed that sea-basins
+  existed cut off from the ocean, and it is still a matter of
+  controversy how far the pre-Ptolemaic geographers believed in a
+  water-connexion between the Atlantic and Indian oceans. It is quite
+  clear that Pomponius Mela (c. A.D. 40), following Strabo, held that
+  the southern temperate zone contained a habitable land, which he
+  designated by the name _Antichthones_.
+
+
+    Aristotle's geographical views.
+
+  Aristotle left no work on geography, so that it is impossible to know
+  what facts he associated with the science of the earth's surface. The
+  word geography did not appear before Aristotle, the first use of it
+  being in the [Greek: Peri kosmon], which is one of the writings
+  doubtfully ascribed to him, and H. Berger considers that the
+  expression was introduced by Eratosthenes.[4] Aristotle was certainly
+  conversant with many facts, such as the formation of deltas,
+  coast-erosion, and to a certain extent the dependence of plants and
+  animals on their physical surroundings. He formed a comprehensive
+  theory of the variations of climate with latitude and season, and was
+  convinced of the necessity of a circulation of water between the sea
+  and rivers, though, like Plato, he held that this took place by water
+  rising from the sea through crevices in the rocks, losing its
+  dissolved salts in the process. He speculated on the differences in
+  the character of races of mankind living in different climates, and
+  correlated the political forms of communities with their situation on
+  a seashore, or in the neighbourhood of natural strongholds.
+
+
+    Strabo.
+
+  Strabo (c. 50 B.C.-A.D. 24) followed Eratosthenes rather than
+  Aristotle, but with sympathies which went out more to the human
+  interests than the mathematical basis of geography. He compiled a very
+  remarkable work dealing, in large measure from personal travel, with
+  the countries surrounding the Mediterranean. He may be said to have
+  set the pattern which was followed in succeeding ages by the compilers
+  of "political geographies" dealing less with theories than with
+  facts, and illustrating rather than formulating the principles of the
+  science.
+
+
+    Ptolemy.
+
+  Claudius Ptolemaeus (c. A.D. 150) concentrated in his writings the
+  final outcome of all Greek geographical learning, and passed it across
+  the gulf of the middle ages by the hands of the Arabs, to form the
+  starting-point of the science in modern times. His geography was based
+  more immediately on the work of his predecessor, Marinus of Tyre, and
+  on that of Hipparchus, the follower and critic of Eratosthenes. It was
+  the ambition of Ptolemy to describe and represent accurately the
+  surface of the _oekumene_, for which purpose he took immense trouble
+  to collect all existing determinations of the latitude of places, all
+  estimates of longitude, and to make every possible rectification in
+  the estimates of distances by land or sea. His work was mainly
+  cartographical in its aim, and theory was as far as possible excluded.
+  The symmetrically placed hypothetical islands in the great continuous
+  ocean disappeared, and the _oekumene_ acquired a new form by the
+  representation of the Indian Ocean as a larger Mediterranean
+  completely cut off by land from the Atlantic. The _terra incognita_
+  uniting Africa and Farther Asia was an unfortunate hypothesis which
+  helped to retard exploration. Ptolemy used the word _geography_ to
+  signify the description of the whole _oekumene_ on mathematical
+  principles, while _chorography_ signified the fuller description of a
+  particular region, and _topography_ the very detailed description of a
+  smaller locality. He introduced the simile that geography represented
+  an artist's sketch of a whole portrait, while chorography corresponded
+  to the careful and detailed drawing of an eye or an ear.[5]
+
+  The Caliph al-Mam[ = u]n (c. A.D. 815), the son and successor of H[ =
+  a]r[ = u]n al-Rash[ = i]d, caused an Arabic version of Ptolemy's great
+  astronomical work ([Greek: Suntaxis megiste]) to be made, which is
+  known as the _Almagest_, the word being nothing more than the Gr.
+  [Greek: megiste] with the Arabic article _al_ prefixed. The geography
+  of Ptolemy was also known and is constantly referred to by Arab
+  writers. The Arab astronomers measured a degree on the plains of
+  Mesopotamia, thereby deducing a fair approximation to the size of the
+  earth. The caliph's librarian, Abu Jafar Muhammad Ben Musa, wrote a
+  geographical work, now unfortunately lost, entitled _Rasm el Arsi_ ("A
+  Description of the World"), which is often referred to by subsequent
+  writers as having been composed on the model of that of Ptolemy.
+
+
+    Geography in the middle ages.
+
+  The middle ages saw geographical knowledge die out in Christendom,
+  although it retained, through the Arabic translations of Ptolemy, a
+  certain vitality in Islam. The verbal interpretation of Scripture led
+  Lactantius (c. A.D. 320) and other ecclesiastics to denounce the
+  spherical theory of the earth as heretical. The wretched subterfuge of
+  Cosmas (c. A.D. 550) to explain the phenomena of the apparent
+  movements of the sun by means of an earth modelled on the plan of the
+  Jewish Tabernacle gave place ultimately to the wheel-maps--the T in an
+  O--which reverted to the primitive ignorance of the times of Homer and
+  Hecataeus.[6]
+
+  The journey of Marco Polo, the increasing trade to the East and the
+  voyages of the Arabs in the Indian Ocean prepared the way for the
+  reacceptance of Ptolemy's ideas when the sealed books of the Greek
+  original were translated into Latin by Angelus in 1410.
+
+
+    Revival of geography.
+
+  The old arguments of Aristotle and the old measurements of Ptolemy
+  were used by Toscanelli and Columbus in urging a westward voyage to
+  India; and mainly on this account did the crossing of the Atlantic
+  rank higher in the history of scientific geography than the laborious
+  feeling out of the coast-line of Africa. But not until the voyage of
+  Magellan shook the scales from the eyes of Europe did modern geography
+  begin to advance. Discovery had outrun theory; the rush of new facts
+  made Ptolemy practically obsolete in a generation, after having been
+  the fount and origin of all geography for a millennium.
+
+
+    Apianus.
+
+  The earliest evidence of the reincarnation of a sound theoretical
+  geography is to be found in the text-books by Peter Apian and
+  Sebastian Munster. Apian in his _Cosmographicus liber_, published in
+  1524, and subsequently edited and added to by Gemma Frisius under the
+  title of _Cosmographia_, based the whole science on mathematics and
+  measurement. He followed Ptolemy closely, enlarging on his distinction
+  between geography and chorography, and expressing the artistic analogy
+  in a rough diagram. This slender distinction was made much of by most
+  subsequent writers until Nathanael Carpenter in 1625 pointed out that
+  the difference between geography and chorography was simply one of
+  degree, not of kind.
+
+
+    Munster.
+
+  Sebastian Munster, on the other hand, in his _Cosmographia
+  universalis_ of 1544, paid no regard to the mathematical basis of
+  geography, but, following the model of Strabo, described the world
+  according to its different political divisions, and entered with great
+  zest into the question of the productions of countries, and into the
+  manners and costumes of the various peoples. Thus early commenced the
+  separation between what were long called mathematical and political
+  geography, the one subject appealing mainly to mathematicians, the
+  other to historians.
+
+  Throughout the 16th and 17th centuries the rapidly accumulating store
+  of facts as to the extent, outline and mountain and river systems of
+  the lands of the earth were put in order by the generation of
+  cartographers of which Mercator was the chief; but the writings of
+  Apian and Munster held the field for a hundred years without a serious
+  rival, unless the many annotated editions of Ptolemy might be so
+  considered. Meanwhile the new facts were the subject of original study
+  by philosophers and by practical men without reference to classical
+  traditions. Bacon argued keenly on geographical matters and was a
+  lover of maps, in which he observed and reasoned upon such
+  resemblances as that between the outlines of South America and Africa.
+
+
+    Cluverius.
+
+  Philip Cluver's _Introductio in geographiam universam tam veterem quam
+  novam_ was published in 1624. Geography he defined as "the description
+  of the whole earth, so far as it is known to us." It is distinguished
+  from cosmography by dealing with the earth alone, not with the
+  universe, and from chorography and topography by dealing with the
+  whole earth, not with a country or a place. The first book, of
+  fourteen short chapters, is concerned with the general properties of
+  the globe; the remaining six books treat in considerable detail of the
+  countries of Europe and of the other continents. Each country is
+  described with particular regard to its people as well as to its
+  surface, and the prominence given to the human element is of special
+  interest.
+
+
+    Carpenter.
+
+  A little-known book which appears to have escaped the attention of
+  most writers on the history of modern geography was published at
+  Oxford in 1625 by Nathanael Carpenter, fellow of Exeter College, with
+  the title _Geographie delineated forth in Two Bookes, containing the
+  Sphericall and Topicall parts thereof_. It is discursive in its style
+  and verbose; but, considering the period at which it appeared, it is
+  remarkable for the strong common sense displayed by the author, his
+  comparative freedom from prejudice, and his firm application of the
+  methods of scientific reasoning to the interpretation of phenomena.
+  Basing his work on the principles of Ptolemy, he brings together
+  illustrations from the most recent travellers, and does not hesitate
+  to take as illustrative examples the familiar city of Oxford and his
+  native county of Devon. He divides geography into _The Spherical
+  Part_, or that for the study of which mathematics alone is required,
+  and _The Topical Part_, or the description of the physical relations
+  of parts of the earth's surface, preferring this division to that
+  favoured by the ancient geographers--into general and special. It is
+  distinguished from other English geographical books of the period by
+  confining attention to the principles of geography, and not describing
+  the countries of the world.
+
+
+    Varenius.
+
+  A much more important work in the history of geographical method is
+  the _Geographia generalis_ of Bernhard Varenius, a German medical
+  doctor of Leiden, who died at the age of twenty-eight in 1650, the
+  year of the publication of his book. Although for a time it was lost
+  sight of on the continent, Sir Isaac Newton thought so highly of this
+  book that he prepared an annotated edition which was published in
+  Cambridge in 1672, with the addition of the plates which had been
+  planned by Varenius, but not produced by the original publishers. "The
+  reason why this great man took so much care in correcting and
+  publishing our author was, because he thought him necessary to be read
+  by his audience, the young gentlemen of Cambridge, while he was
+  delivering lectures on the same subject from the Lucasian Chair."[7]
+  The treatise of Varenius is a model of logical arrangement and terse
+  expression; it is a work of science and of genius; one of the few of
+  that age which can still be studied with profit. The English
+  translation renders the definition thus: "Geography is that part of
+  _mixed mathematics_ which explains the state of the earth and of its
+  parts, depending on quantity, viz. its figure, place, magnitude and
+  motion, with the celestial appearances, &c. By some it is taken in too
+  limited a sense, for a bare description of the several countries; and
+  by others too extensively, who along with such a description would
+  have their political constitution."
+
+  Varenius was reluctant to include the human side of geography in his
+  system, and only allowed it as a concession to custom, and in order to
+  attract readers by imparting interest to the sterner details of the
+  science. His division of geography was into two parts--(i.) General or
+  universal, dealing with the earth in general, and explaining its
+  properties without regard to particular countries; and (ii.) Special
+  or particular, dealing with each country in turn from the
+  chorographical or topographical point of view. General geography was
+  divided into--(1) the _Absolute_ part, dealing with the form,
+  dimensions, position and substance of the earth, the distribution of
+  land and water, mountains, woods and deserts, hydrography (including
+  all the waters of the earth) and the atmosphere; (2) the _Relative_
+  part, including the celestial properties, i.e. latitude, climate
+  zones, longitude, &c.; and (3) the _Comparative_ part, which
+  "considers the particulars arising from comparing one part with
+  another"; but under this head the questions discussed were longitude,
+  the situation and distances of places, and navigation. Varenius does
+  not treat of special geography, but gives a scheme for it under three
+  heads--(1) _Terrestrial_, including position, outline, boundaries,
+  mountains, mines, woods and deserts, waters, fertility and fruits, and
+  living creatures; (2) _Celestial_, including appearance of the heavens
+  and the climate; (3) _Human_, but this was added out of deference to
+  popular usage.
+
+  This system of geography founded a new epoch, and the book--translated
+  into English, Dutch and French--was the unchallenged standard for more
+  than a century. The framework was capable of accommodating itself to
+  new facts, and was indeed far in advance of the knowledge of the
+  period. The method included a recognition of the causes and effects of
+  phenomena as well as the mere fact of their occurrence, and for the
+  first time the importance of the vertical relief of the land was
+  fairly recognized.
+
+  The physical side of geography continued to be elaborated after
+  Varenius's methods, while the historical side was developed
+  separately. Both branches, although enriched by new facts, remained
+  stationary so far as method is concerned until nearly the end of the
+  18th century. The compilation of "geography books" by uninstructed
+  writers led to the pernicious habit, which is not yet wholly overcome,
+  of reducing the general or "physical" part to a few pages of
+  concentrated information, and expanding the particular or "political"
+  part by including unrevised travellers' stories and uncritical
+  descriptions of the various countries of the world. Such books were in
+  fact not geography, but merely compressed travel.
+
+
+    Bergman.
+
+  The next marked advance in the theory of geography may be taken as the
+  nearly simultaneous studies of the physical earth carried out by the
+  Swedish chemist, Torbern Bergman, acting under the impulse of
+  Linnaeus, and by the German philosopher, Immanuel Kant. Bergman's
+  _Physical Description of the Earth_ was published in Swedish in 1766,
+  and translated into English in 1772 and into German in 1774. It is a
+  plain, straightforward description of the globe, and of the various
+  phenomena of the surface, dealing only with definitely ascertained
+  facts in the natural order of their relationships, but avoiding any
+  systematic classification or even definitions of terms.
+
+
+    Kant.
+
+  The problems of geography had been lightened by the destructive
+  criticism of the French cartographer D'Anville (who had purged the map
+  of the world of the last remnants of traditional fact unverified by
+  modern observations) and rendered richer by the dawn of the new era of
+  scientific travel, when Kant brought his logical powers to bear upon
+  them. Kant's lectures on physical geography were delivered in the
+  university of Konigsberg from 1765 onwards.[8] Geography appealed to
+  him as a valuable educational discipline, the joint foundation with
+  anthropology of that "knowledge of the world" which was the result of
+  reason and experience. In this connexion he divided the communication
+  of experience from one person to another into two categories--the
+  narrative or historical and the descriptive or geographical; both
+  history and geography being viewed as descriptions, the former a
+  description in order of time, the latter a description in order of
+  space.
+
+  Physical geography he viewed as a summary of nature, the basis not
+  only of history but also of "all the other possible geographies," of
+  which he enumerates five, viz. (1) _Mathematical geography_, which
+  deals with the form, size and movements of the earth and its place in
+  the solar system; (2) _Moral geography_, or an account of the
+  different customs and characters of mankind according to the region
+  they inhabit; (3) _Political geography_, the divisions according to
+  their organized governments; (4) _Mercantile geography_, dealing with
+  the trade in the surplus products of countries; (5) _Theological
+  geography_, or the distribution of religions. Here there is a clear
+  and formal statement of the interaction and causal relation of all the
+  phenomena of distribution on the earth's surface, including the
+  influence of physical geography upon the various activities of mankind
+  from the lowest to the highest. Notwithstanding the form of this
+  classification, Kant himself treats mathematical geography as
+  preliminary to, and therefore not dependent on, physical geography.
+  Physical geography itself is divided into two parts: a general, which
+  has to do with the earth and all that belongs to it--water, air and
+  land; and a particular, which deals with special products of the
+  earth--mankind, animals, plants and minerals. Particular importance is
+  given to the vertical relief of the land, on which the various
+  branches of human geography are shown to depend.
+
+
+    Humboldt.
+
+  Alexander von Humboldt (1769-1859) was the first modern geographer to
+  become a great traveller, and thus to acquire an extensive stock of
+  first-hand information on which an improved system of geography might
+  be founded. The impulse given to the study of natural history by the
+  example of Linnaeus; the results brought back by Sir Joseph Banks, Dr
+  Solander and the two Forsters, who accompanied Cook in his voyages of
+  discovery; the studies of De Saussure in the Alps, and the lists of
+  desiderata in physical geography drawn up by that investigator,
+  combined to prepare the way for Humboldt. The theory of geography was
+  advanced by Humboldt mainly by his insistence on the great principle
+  of the unity of nature. He brought all the "observable things," which
+  the eager collectors of the previous century had been heaping together
+  regardless of order or system, into relation with the vertical relief
+  and the horizontal forms of the earth's surface. Thus he demonstrated
+  that the forms of the land exercise a directive and determining
+  influence on climate, plant life, animal life and on man himself. This
+  was no new idea; it had been familiar for centuries in a less definite
+  form, deduced from a priori considerations, and so far as regards the
+  influence of surrounding circumstances upon man, Kant had already
+  given it full expression. Humboldt's concrete illustrations and the
+  remarkable power of his personality enabled him to enforce these
+  principles in a way that produced an immediate and lasting effect. The
+  treatises on physical geography by Mrs Mary Somerville and Sir John
+  Herschel (the latter written for the eighth edition of the
+  _Encyclopaedia Britannica_) showed the effect produced in Great
+  Britain by the stimulus of Humboldt's work.
+
+
+    Ritter.
+
+  Humboldt's contemporary, Carl Ritter (1779-1859), extended and
+  disseminated the same views, and in his interpretation of "Comparative
+  Geography" he laid stress on the importance of forming conclusions,
+  not from the study of one region by itself, but from the comparison of
+  the phenomena of many places. Impressed by the influence of
+  terrestrial relief and climate on human movements, Ritter was led
+  deeper and deeper into the study of history and archaeology. His
+  monumental _Vergleichende Geographie_, which was to have made the
+  whole world its theme, died out in a wilderness of detail in
+  twenty-one volumes before it had covered more of the earth's surface
+  than Asia and a portion of Africa. Some of his followers showed a
+  tendency to look on geography rather as an auxiliary to history than
+  as a study of intrinsic worth.
+
+
+    Geography as a natural science.
+
+  During the rapid development of physical geography many branches of
+  the study of nature, which had been included in the cosmography of the
+  early writers, the physiography of Linnaeus and even the _Erdkunde_ of
+  Ritter, had been so much advanced by the labours of specialists that
+  their connexion was apt to be forgotten. Thus geology, meteorology,
+  oceanography and anthropology developed into distinct sciences. The
+  absurd attempt was, and sometimes is still, made by geographers to
+  include all natural science in geography; but it is more common for
+  specialists in the various detailed sciences to think, and sometimes
+  to assert, that the ground of physical geography is now fully occupied
+  by these sciences. Political geography has been too often looked on
+  from both sides as a mere summary of guide-book knowledge, useful in
+  the schoolroom, a poor relation of physical geography that it was
+  rarely necessary to recognize.
+
+  The science of geography, passed on from antiquity by Ptolemy,
+  re-established by Varenius and Newton, and systematized by Kant,
+  included within itself definite aspects of all those terrestrial
+  phenomena which are now treated exhaustively under the heads of
+  geology, meteorology, oceanography and anthropology; and the inclusion
+  of the requisite portions of the perfected results of these sciences
+  in geography is simply the gathering in of fruit matured from the seed
+  scattered by geography itself.
+
+  The study of geography was advanced by improvements in cartography
+  (see MAP), not only in the methods of survey and projection, but in
+  the representation of the third dimension by means of contour lines
+  introduced by Philippe Buache in 1737, and the more remarkable because
+  less obvious invention of isotherms introduced by Humboldt in 1817.
+
+
+    The teleological argument in geography.
+
+  The "argument from design" had been a favourite form of reasoning
+  amongst Christian theologians, and, as worked out by Paley in his
+  _Natural Theology_, it served the useful purpose of emphasizing the
+  fitness which exists between all the inhabitants of the earth and
+  their physical environment. It was held that the earth had been
+  created so as to fit the wants of man in every particular. This
+  argument was tacitly accepted or explicitly avowed by almost every
+  writer on the theory of geography, and Carl Ritter distinctly
+  recognized and adopted it as the unifying principle of his system. As
+  a student of nature, however, he did not fail to see, and as professor
+  of geography he always taught, that man was in very large measure
+  conditioned by his physical environment. The apparent opposition of
+  the observed fact to the assigned theory he overcame by looking upon
+  the forms of the land and the arrangement of land and sea as
+  instruments of Divine Providence for guiding the destiny as well as
+  for supplying the requirements of man. This was the central theme of
+  Ritter's philosophy; his religion and his geography were one, and the
+  consequent fervour with which he pursued his mission goes far to
+  account for the immense influence he acquired in Germany.
+
+
+    The theory of evolution in geography.
+
+  The evolutionary theory, more than hinted at in Kant's "Physical
+  Geography," has, since the writings of Charles Darwin, become the
+  unifying principle in geography. The conception of the development of
+  the plan of the earth from the first cooling of the surface of the
+  planet throughout the long geological periods, the guiding power of
+  environment on the circulation of water and of air, on the
+  distribution of plants and animals, and finally on the movements of
+  man, give to geography a philosophical dignity and a scientific
+  completeness which it never previously possessed. The influence of
+  environment on the organism may not be quite so potent as it was once
+  believed to be, in the writings of Buckle, for instance,[9] and
+  certainly man, the ultimate term in the series, reacts upon and
+  greatly modifies his environment; yet the fact that environment does
+  influence all distributions is established beyond the possibility of
+  doubt. In this way also the position of geography, at the point where
+  physical science meets and mingles with mental science, is explained
+  and justified. The change which took place during the 19th century in
+  the substance and style of geography may be well seen by comparing the
+  eight volumes of Malte-Brun's _Geographie universelle_ (Paris,
+  1812-1829) with the twenty-one volumes of Reclus's _Geographie
+  universelle_ (Paris, 1876-1895).
+
+  In estimating the influence of recent writers on geography it is usual
+  to assign to Oscar Peschel (1826-1875) the credit of having corrected
+  the preponderance which Ritter gave to the historical element, and of
+  restoring physical geography to its old pre-eminence.[10] As a matter
+  of fact, each of the leading modern exponents of theoretical
+  geography--such as Ferdinand von Richthofen, Hermann Wagner, Friedrich
+  Ratzel, William M. Davis, A. Penck, A. de Lapparent and Elisee
+  Reclus--has his individual point of view, one devoting more attention
+  to the results of geological processes, another to anthropological
+  conditions, and the rest viewing the subject in various blendings of
+  the extreme lights.
+
+  The two conceptions which may now be said to animate the theory of
+  geography are the genetic, which depends upon processes of origin, and
+  the morphological, which depends on facts of form and distribution.
+
+
+  PROGRESS OF GEOGRAPHICAL DISCOVERY
+
+  Exploration and geographical discovery must have started from more
+  than one centre, and to deal justly with the matter one ought to treat
+  of these separately in the early ages before the whole civilized world
+  was bound together by the bonds of modern intercommunication. At the
+  least there should be some consideration of four separate systems of
+  discovery--the Eastern, in which Chinese and Japanese explorers
+  acquired knowledge of the geography of Asia, and felt their way
+  towards Europe and America; the Western, in which the dominant races
+  of the Mexican and South American plateaus extended their knowledge of
+  the American continent before Columbus; the Polynesian, in which the
+  conquering races of the Pacific Islands found their way from group to
+  group; and the Mediterranean. For some of these we have no certain
+  information, and regarding others the tales narrated in the early
+  records are so hard to reconcile with present knowledge that they are
+  better fitted to be the battle-ground of scholars championing rival
+  theories than the basis of definite history. So it has come about that
+  the only practicable history of geographical exploration starts from
+  the Mediterranean centre, the first home of that civilization which
+  has come to be known as European, though its field of activity has
+  long since overspread the habitable land of both temperate zones,
+  eastern Asia alone in part excepted.
+
+  From all centres the leading motives of exploration were probably the
+  same--commercial intercourse, warlike operations, whether resulting in
+  conquest or in flight, religious zeal expressed in pilgrimages or
+  missionary journeys, or, from the other side, the avoidance of
+  persecution, and, more particularly in later years, the advancement of
+  knowledge for its own sake. At different times one or the other motive
+  predominated.
+
+  Before the 14th century B.C. the warrior kings of Egypt had carried
+  the power of their arms southward from the delta of the Nile well-nigh
+  to its source, and eastward to the confines of Assyria. The
+  hieroglyphic inscriptions of Egypt and the cuneiform inscriptions of
+  Assyria are rich in records of the movements and achievements of
+  armies, the conquest of towns and the subjugation of peoples; but
+  though many of the recorded sites have been identified, their
+  discovery by wandering armies was isolated from their subsequent
+  history and need not concern us here.
+
+
+    The Phoenicians.
+
+  The Phoenicians are the earliest Mediterranean people in the
+  consecutive chain of geographical discovery which joins pre-historic
+  time with the present. From Sidon, and later from its more famous
+  rival Tyre, the merchant adventurers of Phoenicia explored and
+  colonized the coasts of the Mediterranean and fared forth into the
+  ocean beyond. They traded also on the Red sea, and opened up regular
+  traffic with India as well as with the ports of the south and west, so
+  that it was natural for Solomon to employ the merchant navies of Tyre
+  in his oversea trade. The western emporium known in the scriptures as
+  Tarshish was probably situated in the south of Spain, possibly at
+  Cadiz, although some writers contend that it was Carthage in North
+  Africa. Still more diversity of opinion prevails as to the southern
+  gold-exporting port of Ophir, which some scholars place in Arabia,
+  others at one or another point on the east coast of Africa. Whether
+  associated with the exploitation of Ophir (q.v.) or not the first
+  great voyage of African discovery appears to have been accomplished by
+  the Phoenicians sailing the Red Sea. Herodotus (himself a notable
+  traveller in the 5th century B.C.) relates that the Egyptian king
+  Necho of the XXVIth Dynasty (c. 600 B.C.) built a fleet on the Red
+  Sea, and confided it to Phoenician sailors with the orders to sail
+  southward and return to Egypt by the Pillars of Hercules and the
+  Mediterranean sea. According to the tradition, which Herodotus quotes
+  sceptically, this was accomplished; but the story is too vague to be
+  accepted as more than a possibility.
+
+  The great Phoenician colony of Carthage, founded before 800 B.C.,
+  perpetuated the commercial enterprise of the parent state, and
+  extended the sphere of practical trade to the ocean shores of Africa
+  and Europe. The most celebrated voyage of antiquity undertaken for the
+  express purpose of discovery was that fitted out by the senate of
+  Carthage under the command of Hanno, with the intention of founding
+  new colonies along the west coast of Africa. According to Pliny, the
+  only authority on this point, the period of the voyage was that of the
+  greatest prosperity of Carthage, which may be taken as somewhere
+  between 570 and 480 B.C. The extent of this voyage is doubtful, but it
+  seems probable that the farthest point reached was on the east-running
+  coast which bounds the Gulf of Guinea on the north. Himilco, a
+  contemporary of Hanno, was charged with an expedition along the west
+  coast of Iberia northward, and as far as the uncertain references to
+  this voyage can be understood, he seems to have passed the Bay of
+  Biscay and possibly sighted the coast of England.
+
+
+    The Greeks.
+
+  The sea power of the Greek communities on the coast of Asia Minor and
+  in the Archipelago began to be a formidable rival to the Phoenician
+  soon after the time of Hanno and Himilco, and peculiar interest
+  attaches to the first recorded Greek voyage beyond the Pillars of
+  Hercules. Pytheas, a navigator of the Phocean colony of Massilia
+  (Marseilles), determined the latitude of that port with considerable
+  precision by the somewhat clumsy method of ascertaining the length of
+  the longest day, and when, about 330 B.C., he set out on exploration
+  to the northward in search of the lands whence came gold, tin and
+  amber, he followed this system of ascertaining his position from time
+  to time. If on each occasion he himself made the observations his
+  voyage must have extended over six years; but it is not impossible
+  that he ascertained the approximate length of the longest day in some
+  cases by questioning the natives. Pytheas, whose own narrative is not
+  preserved, coasted the Bay of Biscay, sailed up the English Channel
+  and followed the coast of Britain to its most northerly point. Beyond
+  this he spoke of a land called _Thule_, which, if his estimate of the
+  length of the longest day is correct, may have been Shetland, but was
+  possibly Iceland; and from some confused statements as to a sea which
+  could not be sailed through, it has been assumed that Pytheas was the
+  first of the Greeks to obtain direct knowledge of the Arctic regions.
+  During this or a second voyage Pytheas entered the Baltic, discovered
+  the coasts where amber is obtained and returned to the Mediterranean.
+  It does not seem that any maritime trade followed these discoveries,
+  and indeed it is doubtful whether his contemporaries accepted the
+  truth of Pytheas's narrative; Strabo four hundred years later
+  certainly did not, but the critical studies of modern scholars have
+  rehabilitated the Massilian explorer.
+
+
+    Alexander the Great.
+
+  The Greco-Persian wars had made the remoter parts of Asia Minor more
+  than a name to the Greek geographers before the time of Alexander the
+  Great, but the campaigns of that conqueror from 329 to 325 B.C. opened
+  up the greater Asia to the knowledge of Europe. His armies crossed the
+  plains beyond the Caspian, penetrated the wild mountain passes
+  north-west of India, and did not turn back until they had entered on
+  the Indo-Gangetic plain. This was one of the few great epochs of
+  geographical discovery.
+
+  The world was henceforth viewed as a very large place stretching far
+  on every side beyond the Midland or Mediterranean Sea, and the land
+  journey of Alexander resulted in a voyage of discovery in the outer
+  ocean from the mouth of the Indus to that of the Tigris, thus opening
+  direct intercourse between Grecian and Hindu civilization. The Greeks
+  who accompanied Alexander described with care the towns and villages,
+  the products and the aspect of the country. The conqueror also
+  intended to open up trade by sea between Europe and India, and the
+  narrative of his general Nearchus records this famous voyage of
+  discovery, the detailed accounts of the chief pilot Onesicritus being
+  lost. At the beginning of October 326 B.C. Nearchus left the Indus
+  with his fleet, and the anchorages sought for each night are carefully
+  recorded. He entered the Persian Gulf, and rejoined Alexander at Susa,
+  when he was ordered to prepare another expedition for the
+  circumnavigation of Arabia. Alexander died at Babylon in 323 B.C., and
+  the fleet was dispersed without making the voyage.
+
+  The dynasties founded by Alexander's generals, Seleucus, Antiochus and
+  Ptolemy, encouraged the same spirit of enterprise which their master
+  had fostered, and extended geographical knowledge in several
+  directions. Seleucus Nicator established the Greco-Bactrian empire and
+  continued the intercourse with India. Authentic information respecting
+  the great valley of the Ganges was supplied by Megasthenes, an
+  ambassador sent by Seleucus, who reached the remote city of
+  Patali-putra, the modern Patna.
+
+
+    The Ptolemies.
+
+  The Ptolemies in Egypt showed equal anxiety to extend the bounds of
+  geographical knowledge. Ptolemy Euergetes (247-222 B.C.) rendered the
+  greatest service to geography by the protection and encouragement of
+  Eratosthenes, whose labours gave the first approximate knowledge of
+  the true size of the spherical earth. The second Euergetes and his
+  successor Ptolemy Lathyrus (118-115 B.C.) furnished Eudoxus with a
+  fleet to explore the Arabian sea. After two successful voyages,
+  Eudoxus, impressed with the idea that Africa was surrounded by ocean
+  on the south, left the Egyptian service, and proceeded to Cadiz and
+  other Mediterranean centres of trade seeking a patron who would
+  finance an expedition for the purpose of African discovery; and we
+  learn from Strabo that the veteran explorer made at least two voyages
+  southward along the coast of Africa. The Ptolemies continued to send
+  fleets annually from their Red Sea ports of Berenice and Myos Hormus
+  to Arabia, as well as to ports on the coasts of Africa and India.
+
+
+    The Romans.
+
+  The Romans did not encourage navigation and commerce with the same
+  ardour as their predecessors; still the luxury of Rome, which gave
+  rise to demands for the varied products of all the countries of the
+  known world, led to an active trade both by ships and caravans. But it
+  was the military genius of Rome, and the ambition for universal
+  empire, which led, not only to the discovery, but also to the survey
+  of nearly all Europe, and of large tracts in Asia and Africa. Every
+  new war produced a new survey and itinerary of the countries which
+  were conquered, and added one more to the imperishable roads that led
+  from every quarter of the known world to Rome. In the height of their
+  power the Romans had surveyed and explored all the coasts of the
+  Mediterranean, Italy, Greece, the Balkan Peninsula, Spain, Gaul,
+  western Germany and southern Britain. In Africa their empire included
+  Egypt, Carthage, Numidia and Mauritania. In Asia they held Asia Minor
+  and Syria, had sent expeditions into Arabia, and were acquainted with
+  the more distant countries formerly invaded by Alexander, including
+  Persia, Scythia, Bactria and India. Roman intercourse with India
+  especially led to the extension of geographical knowledge.
+
+  Before the Roman legions were sent into a new region to extend the
+  limits of the empire, it was usual to send out exploring expeditions
+  to report as to the nature of the country. It is narrated by Pliny and
+  Seneca that the emperor Nero sent out two centurions on such a mission
+  towards the source of the Nile (probably about A.D. 60), and that the
+  travellers pushed southwards until they reached vast marshes through
+  which they could not make their way either on foot or in boats. This
+  seems to indicate that they had penetrated to about 9 deg. N. Shortly
+  before A.D. 79 Hippalus took advantage of the regular alternation of
+  the monsoons to make the voyage from the Red Sea to India across the
+  open ocean out of sight of land. Even though this sea-route was known,
+  the author of the _Periplus of the Erythraean Sea_, published after
+  the time of Pliny, recites the old itinerary around the coast of the
+  Arabian Gulf. It was, however, in the reigns of Severus and his
+  immediate successors that Roman intercourse with India was at its
+  height, and from the writings of Pausanias (c. 174) it appears that
+  direct communication between Rome and China had already taken place.
+
+  After the division of the Roman empire, Constantinople became the last
+  refuge of learning, arts and taste; while Alexandria continued to be
+  the emporium whence were imported the commodities of the East. The
+  emperor Justinian (483-565), in whose reign the greatness of the
+  Eastern empire culminated, sent two Nestorian monks to China, who
+  returned with eggs of the silkworm concealed in a hollow cane, and
+  thus silk manufactures were established in the Peloponnesus and the
+  Greek islands. It was also in the reign of Justinian that Cosmas
+  Indicopleustes, an Egyptian merchant, made several voyages, and
+  afterwards composed his [Greek: Christianike topographia] (Christian
+  Topography), containing, in addition to his absurd cosmogony, a
+  tolerable description of India.
+
+
+    The Arabs.
+
+  The great outburst of Mahommedan conquest in the 7th century was
+  followed by the Arab civilization, having its centres at Bagdad and
+  Cordova, in connexion with which geography again received a share of
+  attention. The works of the ancient Greek geographers were translated
+  into Arabic, and starting with a sound basis of theoretical knowledge,
+  exploration once more made progress. From the 9th to the 13th century
+  intelligent Arab travellers wrote accounts of what they had seen and
+  heard in distant lands. The earliest Arabian traveller whose
+  observations have come down to us is the merchant Sulaiman, who
+  embarked in the Persian Gulf and made several voyages to India and
+  China, in the middle of the 9th century. Abu Zaid also wrote on India,
+  and his work is the most important that we possess before the
+  epoch-making discoveries of Marco Polo. Masudi, a great traveller who
+  knew from personal experience all the countries between Spain and
+  China, described the plains, mountains and seas, the dynasties and
+  peoples, in his _Meadows of Gold_, an abstract made by himself of his
+  larger work _News of the Time_. He died in 956, and was known, from
+  the comprehensiveness of his survey, as the Pliny of the East. Amongst
+  his contemporaries were Istakhri, who travelled through all the
+  Mahommedan countries and wrote his _Book of Climates_ in 950, and Ibn
+  Haukal, whose _Book of Roads and Kingdoms_, based on the work of
+  Istakhri, was written in 976. Idrisi, the best known of the Arabian
+  geographical authors, after travelling far and wide in the first half
+  of the 12th century, settled in Sicily, where he wrote a treatise
+  descriptive of an armillary sphere which he had constructed for Roger
+  II., the Norman king, and in this work he incorporated all accessible
+  results of contemporary travel.
+
+
+    The Northmen.
+
+  The Northmen of Denmark and Norway, whose piratical adventures were
+  the terror of all the coasts of Europe, and who established themselves
+  in Great Britain and Ireland, in France and Sicily, were also
+  geographical explorers in their rough but practical way during the
+  darkest period of the middle ages. All Northmen were not bent on
+  rapine and plunder; many were peaceful merchants. Alfred the Great,
+  king of the Saxons in England, not only educated his people in the
+  learning of the past ages; he inserted in the geographical works he
+  translated many narratives of the travel of his own time. Thus he
+  placed on record the voyages of the merchant Ulfsten in the Baltic,
+  including particulars of the geography of Germany. And in particular
+  he told of the remarkable voyage of Other, a Norwegian of Helgeland,
+  who was the first authentic Arctic explorer, the first to tell of the
+  rounding of the North Cape and the sight of the midnight sun. This
+  voyage of the middle of the 9th century deserves to be held in happy
+  memory, for it unites the first Norwegian polar explorer with the
+  first English collector of travels. Scandinavian merchants brought the
+  products of India to England and Ireland. From the 8th to the 11th
+  century a commercial route from India passed through Novgorod to the
+  Baltic, and Arabian coins found in Sweden, and particularly in the
+  island of Gotland, prove how closely the enterprise of the Northmen
+  and of the Arabs intertwined. Five-sixths of these coins preserved at
+  Stockholm were from the mints of the Samanian dynasty, which reigned
+  in Khorasan and Transoxiana from about A.D. 900 to 1000. It was the
+  trade with the East that originally gave importance to the city of
+  Visby in Gotland.
+
+  In the end of the 9th century Iceland was colonized from Norway; and
+  about 985 the intrepid viking, Eric the Red, discovered Greenland, and
+  induced some of his Icelandic countrymen to settle on its inhospitable
+  shores. His son, Leif Ericsson, and others of his followers were
+  concerned in the discovery of the North American coast (see VINLAND),
+  which, but for the isolation of Iceland from the centres of European
+  awakening, would have had momentous consequences. As things were, the
+  importance of this discovery passed unrecognized. The story of two
+  Venetians, Nicolo and Antonio Zeno, who gave a vague account of
+  voyages in the northern seas in the end of the 13th century, is no
+  longer to be accepted as history.
+
+
+    Close of the dark ages.
+
+  At length the long period of barbarism which accompanied and followed
+  the fall of the Roman empire drew to a close in Europe. The Crusades
+  had a favourable influence on the intellectual state of the Western
+  nations. Interesting regions, known only by the scant reports of
+  pilgrims, were made the objects of attention and study; while
+  religious zeal, and the hope of gain, combined with motives of mere
+  curiosity, induced several persons to travel by land into remote
+  regions of the East, far beyond the countries to which the operations
+  of the crusaders extended. Among these was Benjamin of Tudela, who set
+  out from Spain in 1160, travelled by land to Constantinople, and
+  having visited India and some of the eastern islands, returned to
+  Europe by way of Egypt after an absence of thirteen years.
+
+
+    Asiatic journeys.
+
+  Joannes de Plano Carpini, a Franciscan monk, was the head of one of
+  the missions despatched by Pope Innocent to call the chief and people
+  of the Tatars to a better mind. He reached the headquarters of Batu,
+  on the Volga, in February 1246; and, after some stay, went on to the
+  camp of the great khan near Karakorum in central Asia, and returned
+  safely in the autumn of 1247. A few years afterwards, a Fleming named
+  Rubruquis was sent on a similar mission, and had the merit of being
+  the first traveller of this era who gave a correct account of the
+  Caspian Sea. He ascertained that it had no outlet. At nearly the same
+  time Hayton, king of Armenia, made a journey to Karakorum in 1254, by
+  a route far to the north of that followed by Carpini and Rubruquis. He
+  was treated with honour and hospitality, and returned by way of
+  Samarkand and Tabriz, to his own territory. The curious narrative of
+  King Hayton was translated by Klaproth.
+
+  While the republics of Italy, and above all the state of Venice, were
+  engaged in distributing the rich products of India and the Far East
+  over the Western world, it was impossible that motives of curiosity,
+  as well as a desire of commercial advantage, should not be awakened to
+  such a degree as to impel some of the merchants to visit those remote
+  lands. Among these were the brothers Polo, who traded with the East
+  and themselves visited Tatary. The recital of their travels fired the
+  youthful imagination of young Marco Polo, son of Nicolo, and he set
+  out for the court of Kublai Khan, with his father and uncle, in 1265.
+  Marco remained for seventeen years in the service of the Great Khan,
+  and was employed on many important missions. Besides what he learnt
+  from his own observation, he collected much information from others
+  concerning countries which he did not visit. He returned to Europe
+  possessed of a vast store of knowledge respecting the eastern parts of
+  the world, and, being afterwards made a prisoner by the Genoese, he
+  dictated the narrative of his travels during his captivity. The work
+  of Marco Polo is the most valuable narrative of travels that appeared
+  during the middle ages, and despite a cold reception and many denials
+  of the accuracy of the record, its substantial truthfulness has been
+  abundantly proved.
+
+  Missionaries continued to do useful geographical work. Among them were
+  John of Monte Corvino, a Franciscan monk, Andrew of Perugia, John
+  Marignioli and Friar Jordanus, who visited the west coast of India,
+  and above all Friar Odoric of Pordenone. Odoric set out on his travels
+  about 1318, and his journeys embraced parts of India, the Malay
+  Archipelago, China and even Tibet, where he was the first European to
+  enter Lhasa, not yet a forbidden city.
+
+  Ibn Batuta, the great Arab traveller, is separated by a wide space of
+  time from his countrymen already mentioned, and he finds his proper
+  place in a chronological notice after the days of Marco Polo, for he
+  did not begin his wanderings until 1325, his career thus coinciding in
+  time with the fabled journeyings of Sir John Mandeville. While Arab
+  learning flourished during the darkest ages of European ignorance, the
+  last of the Arab geographers lived to see the dawn of the great period
+  of the European awakening. Ibn Batuta went by land from Tangier to
+  Cairo, then visited Syria, and performed the pilgrimages to Medina and
+  Mecca. After exploring Persia, and again residing for some time at
+  Mecca, he made a voyage down the Red sea to Yemen, and travelled
+  through that country to Aden. Thence he visited the African coast,
+  touching at Mombasa and Quiloa, and then sailed across to Ormuz and
+  the Persian Gulf. He crossed Arabia from Bahrein to Jidda, traversed
+  the Red sea and the desert to Syene, and descended the Nile to Cairo.
+  After this he revisited Syria and Asia Minor, and crossed the Black
+  sea, the desert from Astrakhan to Bokhara, and the Hindu Kush. He was
+  in the service of Muhammad Tughluk, ruler of Delhi, about eight years,
+  and was sent on an embassy to China, in the course of which the
+  ambassadors sailed down the west coast of India to Calicut, and then
+  visited the Maldive Islands and Ceylon. Ibn Batuta made the voyage
+  through the Malay Archipelago to China, and on his return he proceeded
+  from Malabar to Bagdad and Damascus, ultimately reaching Fez, the
+  capital of his native country, in November 1349. After a journey into
+  Spain he set out once more for Central Africa in 1352, and reached
+  Timbuktu and the Niger, returning to Fez in 1353. His narrative was
+  committed to writing from his dictation.
+
+
+    Spanish exploration.
+
+  The European country which had come the most completely under the
+  influence of Arab culture now began to send forth explorers to distant
+  lands, though the impulse came not from the Moors but from Italian
+  merchant navigators in Spanish service. The peaceful reign of Henry
+  III. of Castile is famous for the attempts of that prince to extend
+  the diplomatic relations of Spain to the remotest parts of the earth.
+  He sent embassies to all the princes of Christendom and to the Moors.
+  In 1403 the Spanish king sent a knight of Madrid, Ruy Gonzalez de
+  Clavijo, to the distant court of Timur, at Samarkand. He returned in
+  1406, and wrote a valuable narrative of his travels.
+
+  Italians continued to make important journeys in the East during the
+  15th century. Among them was Nicolo Conti, who passed through Persia,
+  sailed along the coast of Malabar, visited Sumatra, Java and the south
+  of China, returned by the Red sea, and got home to Venice in 1444
+  after an absence of twenty-five years. He related his adventures to
+  Poggio Bracciolini, secretary to Pope Eugenius IV.; and the narrative
+  contains much interesting information. One of the most remarkable of
+  the Italian travellers was Ludovico di Varthema, who left his native
+  land in 1502. He went to Egypt and Syria, and for the sake of visiting
+  the holy cities became a Mahommedan. He was the first European who
+  gave an account of the interior of Yemen. He afterwards visited and
+  described many places in Persia, India and the Malay Archipelago,
+  returning to Europe in a Portuguese ship after an absence of five
+  years.
+
+
+    Portuguese exploration--Prince Henry the Navigator.
+
+  In the 15th century the time was approaching when the discovery of the
+  Cape of Good Hope was to widen the scope of geographical enterprise.
+  This great event was preceded by the general utilization in Europe of
+  the polarity of the magnetic needle in the construction of the
+  mariner's compass. Portugal took the lead along this new path, and
+  foremost among her pioneers stands Prince Henry the Navigator
+  (1394-1460), who was a patron both of exploration and of the study of
+  geographical theory. The great westward projection of the coast of
+  Africa, and the islands to the north-west of that continent, were the
+  principal scene of the work of the mariners sent out at his expense;
+  but his object was to push onward and reach India from the Atlantic.
+  The progress of discovery received a check on his death, but only for
+  a time. In 1462 Pedro de Cintra extended Portuguese exploration along
+  the African coast and discovered Sierra Leone. Fernan Gomez followed
+  in 1469, and opened trade with the Gold Coast; and in 1484 Diogo Cao
+  discovered the mouth of the Congo. The king of Portugal next
+  despatched Bartolomeu Diaz in 1486 to continue discoveries southwards;
+  while, in the following year, he sent Pedro de Covilhao and Affonso de
+  Payva to discover the country of Prester John. Diaz succeeded in
+  rounding the southern point of Africa, which he named Cabo
+  Tormentoso--the Cape of Storms--but King Joao II., foreseeing the
+  realization of the long-sought passage to India, gave it the
+  stimulating and enduring name of the Cape of Good Hope. Payva died at
+  Cairo; but Covilhao, having heard that a Christian ruler reigned in
+  the mountains of Ethiopia, penetrated into Abyssinia in 1490. He
+  delivered the letter which Joao II. had addressed to Prester John to
+  the Negus Alexander of Abyssinia, but he was detained by that prince
+  and never allowed to leave the country.
+
+
+    Columbus.
+
+  The Portuguese, following the lead of Prince Henry, continued to look
+  for the road to India by the Cape of Good Hope. The same end was
+  sought by Christopher Columbus, following the suggestion of
+  Toscanelli, and under-estimating the diameter of the globe, by sailing
+  due west. The voyages of Columbus (1492-1498) resulted in the
+  discovery of the West Indies and North America which barred the way to
+  the Far East. In 1493 the pope, Alexander VI., issued a bull
+  instituting the famous "line of demarcation" running from N. to S. 100
+  leagues W. of the Azores, to the west of which the Spaniards were
+  authorized to explore and to the east of which the Portuguese received
+  the monopoly of discovery. The direct line of Portuguese exploration
+  resulted in the discovery of the Cape route to India by Vasco da Gama
+  (1498), and in 1500 to the independent discovery of South America by
+  Pedro Alvarez Cabral. The voyages of Columbus and of Vasco da Gama
+  were so important that it is unnecessary to detail their results in
+  this place. See COLUMBUS, CHRISTOPHER; GAMA, VASCO DA.
+
+
+    Vasco da Gama.
+
+  The three voyages of Vasco da Gama (who died on the scene of his
+  labours, at Cochin, in 1524) revolutionized the commerce of the East.
+  Until then the Venetians held the carrying trade of India, which was
+  brought by the Persian Gulf and Red sea into Syria and Egypt, the
+  Venetians receiving the products of the East at Alexandria and Beirut
+  and distributing them over Europe. This commerce was a great source of
+  wealth to Venice; but after the discovery of the new passage round the
+  Cape, and the conquests of the Portuguese, the trade of the East
+  passed into other hands.
+
+
+    Spaniards in America.
+
+  The discoveries of Columbus awakened a spirit of enterprise in Spain
+  which continued in full force for a century; adventurers flocked
+  eagerly across the Atlantic, and discovery followed discovery in rapid
+  succession. Many of the companions of Columbus continued his work.
+  Vicente Yanez Pinzon in 1500 reached the mouth of the Amazon. In the
+  same year Alonso de Ojeda, accompanied by Juan de la Cosa, from whose
+  maps we learn much of the discoveries of the 16th century navigators,
+  and by a Florentine named Amerigo Vespucci, touched the coast of South
+  America somewhere near Surinam, following the shore as far as the Gulf
+  of Maracaibo. Vespucci afterwards made three voyages to the Brazilian
+  coast; and in 1504 he wrote an account of his four voyages, which was
+  widely circulated, and became the means of procuring for its author at
+  the hands of the cartographer Waldseemuller in 1507 the
+  disproportionate distinction of giving his name to the whole
+  continent. In 1508 Alonso de Ojeda obtained the government of the
+  coast of South America from Cabo de la Vela to the Gulf of Darien;
+  Ojeda landed at Cartagena in 1510, and sustained a defeat from the
+  natives, in which his lieutenant, Juan de la Cosa, was killed. After
+  another reverse on the east side of the Gulf of Darien Ojeda returned
+  to Hispaniola and died there. The Spaniards in the Gulf of Darien were
+  left by Ojeda under the command of Francisco Pizarro, the future
+  conqueror of Peru. After suffering much from famine and disease,
+  Pizarro resolved to leave, and embarked the survivors in small
+  vessels, but outside the harbour they met a ship which proved to be
+  that of Martin Fernandez Enciso, Ojeda's partner, coming with
+  provisions and reinforcements. One of the crew of Enciso's ship, Vasco
+  Nunez de Balboa, the future discoverer of the Pacific Ocean, induced
+  his commander to form a settlement on the other side of the Gulf of
+  Darien. The soldiers became discontented and deposed Enciso, who was a
+  man of learning and an accomplished cosmographer. His work _Suma de
+  Geografia_, which was printed in 1519, is the first Spanish book which
+  gives an account of America. Vasco Nunez, the new commander, entered
+  upon a career of conquest in the neighbourhood of Darien, which ended
+  in the discovery of the Pacific Ocean on the 25th of September 1513.
+  Vasco Nunez was beheaded in 1517 by Pedrarias de Avila, who was sent
+  out to supersede him. This was one of the greatest calamities that
+  could have happened to South America; for the discoverer of the South
+  sea was on the point of sailing with a little fleet into his unknown
+  ocean, and a humane and judicious man would probably have been the
+  conqueror of Peru, instead of the cruel and ignorant Pizarro. In the
+  year 1519 Panama was founded by Pedrarias; and the conquest of Peru by
+  Pizarro followed a few years afterwards. Hernan Cortes overran and
+  conquered Mexico from 1518 to 1521, and the discovery and conquest of
+  Guatemala by Alvarado, the invasion of Florida by De Soto, and of
+  Nueva Granada by Quesada, followed in rapid succession. The first
+  detailed account of the west coast of South America was written by a
+  keenly observant old soldier, Pedro de Cieza de Leon, who was
+  travelling in South America from 1533 to 1550, and published his story
+  at Seville in 1553.
+
+
+    Pacific Ocean.
+
+  The great desire of the Spanish government at that time was to find a
+  westward route to the Moluccas. For this purpose Juan Diaz de Solis
+  was despatched in October 1515, and in January 1516 he discovered the
+  mouth of the Rio de la Plata. He was, however, killed by the natives,
+  and his ships returned. In the following year the Portuguese
+  Ferdinando Magalhaes, familiarly known as Magellan, laid before
+  Charles V., at Valladolid, a scheme for reaching the Spice Islands by
+  sailing westward. He started on the 21st of September 1519, entered
+  the strait which now bears his name in October 1520, worked his way
+  through between Patagonia and Tierra del Fuego, and entered on the
+  vast Pacific which he crossed without sighting any of its innumerable
+  island groups. This was unquestionably the greatest of the voyages
+  which followed from the impulse of Prince Henry, and it was rendered
+  possible only by the magnificent courage of the commander in spite of
+  rebellion, mutiny and starvation. It was the 6th of March 1521 when he
+  reached the Ladrone Islands. Thence Magellan proceeded to the
+  Philippines, and there his career ended in an unimportant encounter
+  with hostile natives. Eventually a Biscayan named Sebastian del Cano,
+  sailing home by way of the Cape of Good Hope, reached San Lucar in
+  command of the "Victoria" on the 6th of September 1522, with eighteen
+  survivors; this one ship of the squadron which sailed on the quest
+  succeeded in accomplishing the first circumnavigation of the globe.
+  Del Cano was received with great distinction by the emperor, who
+  granted him a globe for his crest, and the motto _Primus circumdedisti
+  me_.
+
+
+    Portuguese in Africa and the East.
+
+  While the Spaniards were circumnavigating the world and completing
+  their knowledge of the coasts of Central and South America, the
+  Portuguese were actively engaged on similar work as regards Africa and
+  the East Indies.
+
+  With Abyssinia the mission of Covilhao led to further intercourse. In
+  April 1520 Vasco da Gama, as viceroy of the Indies, took a fleet into
+  the Red sea, and landed an embassy consisting of Dom Rodriguez de Lima
+  and Father Francisco Alvarez, a priest whose detailed narrative is the
+  earliest and not the least interesting account we possess of
+  Abyssinia. It was not until 1526 that the embassy was dismissed; and
+  not many years afterwards the negus entreated the help of the
+  Portuguese against Mahommedan invaders, and the viceroy sent an
+  expeditionary force, commanded by his brother Cristoforo da Gama, with
+  450 musketeers. Da Gama was taken prisoner and killed, but his
+  followers enabled the Christians of Abyssinia to regain their power,
+  and a Jesuit mission remained in the country. The Portuguese also
+  established a close connexion with the kingdom of Congo on the west
+  side of Africa, and obtained much information respecting the interior
+  of the continent. Duarte Lopez, a Portuguese settled in the country,
+  was sent on a mission to Rome by the king of Congo, and Pope Sixtus V.
+  caused him to recount to his chamberlain, Felipe Pigafetta, all he had
+  learned during the nine years he had been in Africa, from 1578 to
+  1587. This narrative, under the title of _Description of the Kingdom
+  of Congo_, was published at Rome by Pigafetta in 1591. A map was
+  attached on which several great equatorial lakes are shown, and the
+  empire of Monomwezi or Unyamwezi is laid down. The most valuable work
+  on Africa about this time is, however, that written by the Moor Leo
+  Africanus in the early part of the 16th century. Leo travelled
+  extensively in the north and west of Africa, and was eventually taken
+  by pirates and sold to a master who presented him to Pope Leo X. At
+  the pope's desire he translated his work on Africa into Italian.
+
+  In Further India and the Malay Archipelago the Portuguese acquired
+  predominating influence at sea, establishing factories on the Malabar
+  coast, in the Persian Gulf, at Malacca, and in the Spice Islands, and
+  extending their commercial enterprises from the Red sea to China.
+  Their missionaries were received at the court of Akbar, and Benedict
+  Goes, a native of the Azores, was despatched on a journey overland
+  from Agra to China. He started in 1603, and, after traversing the
+  least-known parts of Central Asia, he reached the confines of China.
+  He appears to have ascended from Kabul to the plateau of the Pamir,
+  and thence onwards by Yarkand, Khotan and Aksu. He died on the journey
+  in March 1607; and thus, as one of the brethren pronounced his
+  epitaph, "seeking Cathay he found heaven."
+
+
+    English, Dutch and French.
+
+  The activity and love of adventure, which became a passion for two or
+  three generations in Spain and Portugal, spread to other countries. It
+  was the spirit of the age; and England, Holland and France were fired
+  by it. English enterprise was first aroused by John and Sebastian
+  Cabot, father and son, who came from Venice and settled at Bristol in
+  the time of Henry VII. The Cabots received a patent in 1496,
+  empowering them to seek unknown lands; and John Cabot discovered
+  Newfoundland and part of the coast of America. Sebastian afterwards
+  made a voyage to Rio de la Plata in the service of Spain, but he
+  returned to England in 1548 and received a pension from Edward VI. At
+  his suggestion a voyage was undertaken for the discovery of a
+  north-east passage to Cathay, with Sir Hugh Willoughby as
+  captain-general of the fleet and Richard Chancellor as pilot-major.
+  They sailed in May 1553, but Willoughby and all his crew perished on
+  the Lapland coast. Chancellor, however, was more fortunate. He reached
+  the White Sea, performed the journey overland to Moscow, where he was
+  well received, and may be said to have been the founder of the trade
+  between Russia and England. He returned to Archangel and brought his
+  ship back in safety to England. On a second voyage, in 1556,
+  Chancellor was drowned; and three subsequent voyages, led by Stephen
+  Burrough, Arthur Pet and Charles Jackman, in small craft of 50 tons
+  and under, carried on an examination of the straits which lead into
+  the Kara sea.
+
+  The French followed closely on the track of John Cabot, and Norman and
+  Breton fishermen frequented the banks of Newfoundland at the beginning
+  of the 16th century. In 1524 Francis I. sent Giovanni da Verazzano of
+  Florence on an expedition of discovery to the coast of North America;
+  and the details of his voyage were embodied in a letter addressed by
+  him to the king of France from Dieppe, in July 1524. In 1534 Jacques
+  Cartier set out to continue the discoveries of Verazzano, and visited
+  Newfoundland and the Gulf of St Lawrence. In the following year he
+  made another voyage, discovered the island of Anticosti, and ascended
+  the St Lawrence to Hochelaga, now Montreal. He returned, after passing
+  two winters in Canada; and on another occasion he also failed to
+  establish a colony. Admiral de Coligny made several unsuccessful
+  endeavours to form a colony in Florida under Jean Ribault of Dieppe,
+  Rene de Laudonniere and others, but the settlers were furiously
+  assailed by the Spaniards and the attempt was abandoned.
+
+
+    The Elizabethan era.
+
+  The reign of Elizabeth is famous for the gallant enterprises that were
+  undertaken by sea and land to discover and bring to light the unknown
+  parts of the earth. The great promoter of geographical discovery in
+  the Elizabethan period was Richard Hakluyt (1553-1616), who was active
+  in the formation of the two companies for colonizing Virginia in 1606;
+  and devoted his life to encouraging and recording similar
+  undertakings. He published much, and left many valuable papers at his
+  death, most of which, together with many other narratives, were
+  published in 1622 in the great work of the Rev. Samuel Purchas,
+  entitled _Hakluytus Posthumus, or Purchas his Pilgrimes_.
+
+  It is from these works that our knowledge of the gallant deeds of the
+  English and other explorers of the Elizabethan age is mainly derived.
+  The great and splendidly illustrated collections of voyages and
+  travels of Theodorus de Bry and Hulsius served a similar useful
+  purpose on the continent of Europe. One important object of English
+  maritime adventurers of those days was to discover a route to Cathay
+  by the north-west, a second was to settle Virginia, and a third was to
+  raid the Spanish settlements in the West Indies. Nor was the trade to
+  Muscovy and Turkey neglected; while latterly a resolute and successful
+  attempt was made to establish direct commercial relations with India.
+
+  The conception of the north-western route to Cathay now leads the
+  story of exploration, for the first time as far as important and
+  sustained efforts are concerned, towards the Arctic seas. This part of
+  the story is fully told under the heading of POLAR REGIONS, and only
+  the names of Martin Frobisher (1576), John Davis (1585), Henry Hudson
+  (1607) and William Baffin (1616) need be mentioned here in order to
+  preserve the complete conspectus of the history of discovery. The
+  Dutch emulated the British in the Arctic seas during this period,
+  directing their efforts mainly towards the discovery of a north-east
+  passage round the northern end of Novaya Zemlya; and William Barents
+  or Barendsz (1594-1597) is the most famous name in this connexion, his
+  boat voyage along the coast of Novaya Zemlya after losing his ship and
+  wintering in a high latitude, being one of the most remarkable
+  achievements in polar annals.
+
+  Many English voyages were also made to Guinea and the West Indies, and
+  twice English vessels followed in the track of Magellan, and
+  circumnavigated the globe. In 1577 Francis Drake, who had previously
+  served with Hawkins in the West Indies, undertook his celebrated
+  voyage round the world. Reaching the Pacific through the Strait of
+  Magellan, Drake proceeded northward along the west coast of America,
+  resolved to attempt the discovery of a northern passage from the
+  Pacific to the Atlantic. The coast from the southern extremity of the
+  Californian peninsula to Cape Mendocino had been discovered by Juan
+  Rodriguez Cabrillo and Francisco de Ulloa in 1539. Drake's discoveries
+  extended from Cape Mendocino to 48 deg. N., in which latitude he gave
+  up his quest, sailed across the Pacific and reached the Philippine
+  Islands, returning home round the Cape of Good Hope in 1580.
+
+  Thomas Cavendish, emulous of Drake's example, fitted out three vessels
+  for an expedition to the South sea in 1586. He took the same route as
+  Drake along the west coast of America. From Cape San Lucas Cavendish
+  steered across the Pacific, seeing no land until he reached the
+  Ladrone Islands. He returned to England in 1588. The third English
+  voyage into the Pacific was not so fortunate. Sir Richard Hawkins
+  (1593) on reaching the bay of Atacames, in 1 deg.N. in 1594, was
+  attacked by a Spanish fleet, and, after a desperate naval engagement,
+  was forced to surrender. Hawkins declared his object to be discovery
+  and the survey of unknown lands, and his voyage, though terminating in
+  disaster, bore good fruit. _The Observations of Sir Richard Hawkins in
+  his Voyage into the South Sea_, published in 1622, are very valuable.
+  It was long before another British ship entered the Pacific Ocean. Sir
+  John Narborough took two ships through the Strait of Magellan in 1670
+  and touched on the coast of Chile, but it was not until 1685 that
+  Dampier sailed over the part of the Pacific where Hawkins met his
+  defeat.
+
+  The exploring enterprise of the Spanish nation did not wane after the
+  conquest of Peru and Mexico, and the acquisition of the vast empire of
+  the Indies. It was spurred into renewed activity by the audacity of
+  Sir John Hawkins in the West Indies, and by the appearance of Drake,
+  Cavendish and Richard Hawkins in the Pacific.
+
+  In the interior of South America the Spanish conquerors had explored
+  the region of the Andes from the isthmus of Panama to Chile. Pedro de
+  Valdivia in 1540 made an expedition into the country of the Araucanian
+  Indians of Chile, and was the first to explore the eastern base of
+  the Andes in what is now Argentine Patagonia. In 1541 Francisco de
+  Orellana discovered the whole course of the Amazon from its source in
+  the Andes to the Atlantic. A second voyage on the Amazon was made in
+  1561 by the mad pirate Lope de Aguirre; but it was not until 1639 that
+  a full account was written of the great river by Father Cristoval de
+  Acuna, who ascended it from its mouth and reached the city of Quito.
+
+
+    Spaniards in the Pacific.
+
+  The voyage of Drake across the Pacific was preceded by that of Alvaro
+  de Mendana, who was despatched from Peru in 1567 to discover the great
+  Antarctic continent which was believed to extend far northward into
+  the South sea, the search for which now became one of the leading
+  motives of exploration. After a voyage of eighty days across the
+  Pacific, Mendana discovered the Solomon Islands; and the expedition
+  returned in safety to Callao. The appearance of Drake on the Peruvian
+  coast led to an expedition being fitted out at Callao, to go in chase
+  of him, under the command of Pedro Sarmiento. He sailed from Callao in
+  October 1579, and made a careful survey of the Strait of Magellan,
+  with the object of fortifying that entrance to the South sea. The
+  colony which he afterwards took out from Spain was a complete failure,
+  and is only remembered now from the name of "Port Famine," which
+  Cavendish gave to the site at which he found the starving remnant of
+  Sarmiento's settlers. In June 1595 Mendana sailed from the coast of
+  Peru in command of a second expedition to colonize the Solomon
+  Islands. After discovering the Marquesas, he reached the island of
+  Santa Cruz of evil memory, where he and many of the settlers died. His
+  young widow took command of the survivors and brought them safely to
+  Manila. The viceroys of Peru still persevered in their attempts to
+  plant a colony in the hypothetical southern continent. Pedro Fernandez
+  de Quiros, who was pilot under Mendana and Luis Vaez de Torres, were
+  sent in command of two ships to continue the work of exploration. They
+  sailed from Callao in December 1605, and discovered several islands of
+  the New Hebrides group. They anchored in a bay of a large island which
+  Quiros named "Australia del Espiritu Santo." From this place Quiros
+  returned to America, but Torres continued the voyage, passed through
+  the strait between Australia and New Guinea which bears his name, and
+  explored and mapped the southern and eastern coasts of New Guinea.
+
+  The Portuguese, in the early part of the 17th century (1578-1640),
+  were under the dominion of Spain, and their enterprise was to some
+  extent damped; but their missionaries extended geographical knowledge
+  in Africa. Father Francisco Paez acquired great influence in
+  Abyssinia, and explored its highlands from 1600 to 1622. Fathers
+  Mendez and Lobo traversed the deserts between the coast of the Red sea
+  and the mountains, became acquainted with Lake Tsana, and discovered
+  the sources of the Blue Nile in 1624-1633.
+
+
+    Rivalry in the East.
+
+  But the attention of the Portuguese was mainly devoted to vain
+  attempts to maintain their monopoly of the trade of India against the
+  powerful rivalry of the English and Dutch. The English enterprises
+  were persevering, continuous and successful. James Lancaster made a
+  voyage to the Indian Ocean from 1591 to 1594; and in 1599 the
+  merchants and adventurers of London resolved to form a company, with
+  the object of establishing a trade with the East Indies. On the 31st
+  of December 1599 Queen Elizabeth granted the charter of incorporation
+  to the East India Company, and Sir James Lancaster, one of the
+  directors, was appointed general of their first fleet. He was
+  accompanied by John Davis, the great Arctic navigator, as pilot-major.
+  This voyage was eminently successful. The ships touched at Achin in
+  Sumatra and at Java, returning with full ladings of pepper in 1603.
+  The second voyage was commanded by Sir Henry Middleton; but it was in
+  the third voyage, under Keelinge and Hawkins, that the mainland of
+  India was first reached in 1607. Captain Hawkins landed at Surat and
+  travelled overland to Agra, passing some time at the court of the
+  Great Mogul. In the voyage of Sir Edward Michelborne in 1605, John
+  Davis lost his life in a fight with a Japanese junk. The eighth
+  voyage, led by Captain Saris, extended the operations of the company
+  to Japan; and in 1613 the Japanese government granted privileges to
+  the company; but the British retired in 1623, giving up their factory.
+  The chief result of this early intercourse between Great Britain and
+  Japan was the interesting series of letters written by William Adams
+  from 1611 to 1617. From the tenth voyage of the East India Company,
+  commanded by Captain Best, who left England in 1612, dates the
+  establishment of permanent British factories on the coast of India. It
+  was Captain Best who secured a regular _firman_ for trade from the
+  Great Mogul. From that time a fleet was despatched every year, and the
+  company's operations greatly increased geographical knowledge of India
+  and the Eastern Archipelago. British visits to Eastern countries, at
+  this time, were not confined to the voyages of the company. Journeys
+  were also made by land, and, among others, the entertaining author of
+  the _Crudities_, Thomas Coryate, of Odcombe in Somersetshire, wandered
+  on foot from France to India, and died (1617) in the company's factory
+  at Surat. In 1561 Anthony Jenkinson arrived in Persia with a letter
+  from Queen Elizabeth to the shah. He travelled through Russia to
+  Bokhara, and returned by the Caspian and Volga. In 1579 Christopher
+  Burroughs built a ship at Nizhniy Novgorod and traded across the
+  Caspian to Baku; and in 1598 Sir Anthony and Robert Shirley arrived in
+  Persia, and Robert was afterwards sent by the shah to Europe as his
+  ambassador. He was followed by a Spanish mission under Garcia de
+  Silva, who wrote an interesting account of his travels; and to Sir
+  Dormer Cotton's mission, in 1628, we are indebted for Sir Thomas
+  Herbert's charming narrative. In like manner Sir Thomas Roe's mission
+  to India resulted not only in a large collection of valuable reports
+  and letters of his own, but also in the detailed account of his
+  chaplain Terry. But the most learned and intelligent traveller in the
+  East, during the 17th century, was the German, Engelbrecht Kaempfer,
+  who accompanied an embassy to Persia, in 1684, and was afterwards a
+  surgeon in the service of the Dutch East India Company. He was in the
+  Persian Gulf, India and Java, and resided for more than two years in
+  Japan, of which he wrote a history.
+
+
+    Dutch exploration, 16th-17th centuries.
+
+  The Dutch nation, as soon as it was emancipated from Spanish tyranny,
+  displayed an amount of enterprise, which, for a long time, was fully
+  equal to that of the British. The Arctic voyages of Barents were
+  quickly followed by the establishment of a Dutch East India Company;
+  and the Dutch, ousting the Portuguese, not only established factories
+  on the mainland of India and in Japan, but acquired a preponderating
+  influence throughout the Malay Archipelago. In 1583 Jan Hugen van
+  Linschoten made a voyage to India with a Portuguese fleet, and his
+  full and graphic descriptions of India, Africa, China and the Malay
+  Archipelago must have been of no small use to his countrymen in their
+  distant voyages. The first of the Dutch Indian voyages was performed
+  by ships which sailed in April 1595, and rounded the Cape of Good
+  Hope. A second large Dutch fleet sailed in 1598; and, so eager was the
+  republic to extend her commerce over the world that another fleet,
+  consisting of five ships of Rotterdam, was sent in the same year by
+  way of Magellan's Strait, under Jacob Mahu as admiral, with William
+  Adams as pilot. Mahu died on the passage out, and was succeeded by
+  Simon de Cordes, who was killed on the coast of Chile. In September
+  1599 the fleet had entered the Pacific. The ships were then steered
+  direct for Japan, and anchored off Bungo in April 1600. In the same
+  year, 1598, a third expedition was despatched under Oliver van Noort,
+  a native of Utrecht, but the voyage contributed nothing to geography.
+  The Dutch Company in 1614 again resolved to send a fleet to the
+  Moluccas by the westward route, and Joris Spilbergen was appointed to
+  the command as admiral, with a commission from the States-General. He
+  was furnished with four ships of Amsterdam, two of Rotterdam and one
+  from Zeeland. On the 6th of May 1615 Spilbergen entered the Pacific
+  Ocean, and touched at several places on the coast of Chile and Peru,
+  defeating the Spanish fleet in a naval engagement off Chilca. After
+  plundering Payta and making requisitions at Acapulco, the Dutch fleet
+  crossed the Pacific and reached the Moluccas in March 1616.
+
+  The Dutch now resolved to discover a passage into the Pacific to the
+  south of Tierra del Fuego, the insular nature of which had been
+  ascertained by Sir Francis Drake. The vessels fitted out for this
+  purpose were the "Eendracht," of 360 tons, commanded by Jacob Lemaire,
+  and the "Hoorn," of 110 tons, under Willem Schouten. They sailed from
+  the Texel on the 14th of June 1615, and by the 20th of January 1616
+  they were south of the entrance of Magellan's Strait. Passing through
+  the strait of Lemaire they came to the southern extremity of Tierra
+  del Fuego, which was named Cape Horn, in honour of the town of Hoorn
+  in West Friesland, of which Schouten was a native. They passed the
+  cape on the 31st of January, encountering the usual westerly winds.
+  The great merit of this discovery of a second passage into the South
+  sea lies in the fact that it was not accidental or unforeseen, but was
+  due to the sagacity of those who designed the voyage. On the 1st of
+  March the Dutch fleet sighted the island of Juan Fernandez; and,
+  having crossed the Pacific, the explorers sailed along the north coast
+  of New Guinea and arrived at the Moluccas on the 17th of September
+  1616.
+
+  There were several early indications of the existence of the great
+  Australian continent, and the Dutch endeavoured to obtain further
+  knowledge concerning the country and its extent; but only its northern
+  and western coasts had been visited before the time of Governor van
+  Diemen. Dirk Hartog had been on the west coast in latitude 26 deg. 30'
+  S. in 1616. Pelsert struck on a reef called "Houtman's Abrolhos" on
+  the 4th of June 1629. In 1697 the Dutch captain Vlamingh landed on the
+  west coast of Australia, then called New Holland, in 31 deg. 43' S.,
+  and named the Swan river from the black swans he discovered there. In
+  1642 the governor and council of Batavia fitted out two ships to
+  prosecute the discovery of the south land, then believed to be part of
+  a vast Antarctic continent, and entrusted the command to Captain Abel
+  Jansen Tasman. This voyage proved to be the most important to
+  geography that had been undertaken since the first circumnavigation of
+  the globe. Tasman sailed from Batavia in 1642, and on the 24th of
+  November sighted high land in 42 deg. 30' S., which was named van
+  Diemen's Land, and after landing there proceeded to the discovery of
+  the western coast of New Zealand; at first called Staten Land, and
+  supposed to be connected with the Antarctic continent from which this
+  voyage proved New Holland to be separated. He then reached Tongatabu,
+  one of the Friendly Islands of Cook; and returned by the north coast
+  of New Guinea to Batavia. In 1644 Tasman made a second voyage to
+  effect a fuller discovery of New Guinea.
+
+
+    French in North America.
+
+  The French directed their enterprise more in the direction of North
+  America than of the Indies. One of their most distinguished explorers
+  was Samuel Champlain, a captain in the navy, who, after a remarkable
+  journey through Mexico and the West Indies from 1599 to 1602,
+  established his historic connexion with Canada, to the geographical
+  knowledge of which he made a very large addition.
+
+
+    Missionaries in the East.
+
+  The principles and methods of surveying and position finding had by
+  this time become well advanced, and the most remarkable example of the
+  early application of these improvements is to be found in the survey
+  of China by Jesuit missionaries. They first prepared a map of the
+  country round Peking, which was submitted to the emperor Kang-hi, and,
+  being satisfied with the accuracy of the European method of surveying,
+  he resolved to have a survey made of the whole empire on the same
+  principles. This great work was begun in July 1708, and the completed
+  maps were presented to the emperor in 1718. The records preserved in
+  each city were examined, topographical information was diligently
+  collected, and the Jesuit fathers checked their triangulation by
+  meridian altitudes of the sun and pole star and by a system of
+  remeasurements. The result was a more accurate map of China than
+  existed, at that time, of any country in Europe. Kang-hi next ordered
+  a similar map to be made of Tibet, the survey being executed by two
+  lamas who were carefully trained as surveyors by the Jesuits at
+  Peking. From these surveys were constructed the well-known maps which
+  were forwarded to Duhalde, and which D'Anville utilized for his atlas.
+
+
+    The 18th century.
+
+    Asia.
+
+  Several European missionaries had previously found their way from
+  India to Tibet. Antonio Andrada, in 1624, was the first European to
+  enter Tibet since the visit of Friar Odoric in 1325. The next journey
+  was that of Fathers Grueber and Dorville about 1660, who succeeded in
+  passing from China, through Tibet, into India. In 1715 Fathers
+  Desideri and Freyre made their way from Agra, across the Himalayas, to
+  Lhasa, and the Capuchin Friar Orazio della Penna resided in that city
+  from 1735 until 1747. But the most remarkable journey in this
+  direction was performed by a Dutch traveller named Samuel van de
+  Putte. He left Holland in 1718, went by land through Persia to India,
+  and eventually made his way to Lhasa, where he resided for a long
+  time. He went thence to China, returned to Lhasa, and was in India in
+  time to be an eye-witness of the sack of Delhi by Nadir Shah in 1737.
+  In 1743 he left India and died at Batavia on the 27th of September
+  1745. The premature death of this illustrious traveller is the more to
+  be lamented because his vast knowledge died with him. Two English
+  missions sent by Warren Hastings to Tibet, one led by George Bogle in
+  1774, and the other by Captain Turner in 1783, complete Tibetan
+  exploration in the 18th century.
+
+  From Persia much new information was supplied by Jean Chardin, Jean
+  Tavernier, Charles Hamilton, Jean de Thevenot and Father Jude
+  Krusinski, and by English traders on the Caspian. In 1738 John Elton
+  traded between Astrakhan and the Persian port of Enzeli on the
+  Caspian, and undertook to build a fleet for Nadir Shah. Another
+  English merchant, named Jonas Hanway, arrived at Astrabad from Russia,
+  and travelled to the camp of Nadir at Kazvin. One lasting and valuable
+  result of Hanway's wanderings was a charming book of travels. In 1700
+  Guillaume Delisle published his map of the continents of the Old
+  World; and his successor D'Anville produced his map of India in 1752.
+  D'Anville's map contained all that was then known, but ten years
+  afterwards Major Rennell began his surveying labours, which extended
+  over the period from 1763 to 1782. His survey covered an area 900 m.
+  long by 300 wide, from the eastern confines of Bengal to Agra, and
+  from the Himalayas to Calpi. Rennell was indefatigable in collecting
+  geographical information; his Bengal atlas appeared in 1781, his
+  famous map of India in 1788 and the memoir in 1792. Surveys were also
+  made along the Indian coasts.
+
+  Arabia received very careful attention, in the 18th century, from the
+  Danish scientific mission, which included Carsten Niebuhr among its
+  members. Niebuhr landed at Loheia, on the coast of Yemen, in December
+  1762, and went by land to Sana. All the other members of the mission
+  died, but he proceeded from Mokha to Bombay. He then made a journey
+  through Persia and Syria to Constantinople, returning to Copenhagen in
+  1767. His valuable work, the _Description of Arabia_, was published in
+  1772, and was followed in 1774-1778 by two volumes of travels in Asia.
+  The great traveller survived until 1815, when he died at the age of
+  eighty-two.
+
+
+    Africa.
+
+  James Bruce of Kinnaird, the contemporary of Niebuhr, was equally
+  devoted to Eastern travel; and his principal geographical work was the
+  tracing of the Blue Nile from its source to its junction with the
+  White Nile. Before the death of Bruce an African Association was
+  formed, in 1788, for collecting information respecting the interior of
+  that continent, with Major Rennell and Sir Joseph Banks as leading
+  members. The association first employed John Ledyard (who had
+  previously made an extraordinary journey into Siberia) to cross Africa
+  from east to west on the parallel of the Niger, and William Lucas to
+  cross the Sahara to Fezzan. Lucas went from Tripoli to Mesurata,
+  obtained some information respecting Fezzan and returned in 1789. One
+  of the chief problems the association wished to solve was that of the
+  existence and course of the river Niger, which was believed by some
+  authorities to be identical with the Congo. Mungo Park, then an
+  assistant surgeon of an Indiaman, volunteered his services, which were
+  accepted by the association, and in 1795 he succeeded in reaching the
+  town of Segu on the Niger, but was prevented from continuing his
+  journey to Timbuktu. Five years later he accepted an offer from the
+  government to command an expedition into the interior of Africa, the
+  plan being to cross from the Gambia to the Niger and descend the
+  latter river to the sea. After losing most of his companions he
+  himself and the rest perished in a rapid on the Niger at Busa, having
+  been attacked from the shore by order of a chief who thought he had
+  not received suitable presents. His work, however, had established the
+  fact that the Niger was not identical with the Congo.
+
+  While the British were at work in the direction of the Niger, the
+  Portuguese were not unmindful of their old exploring fame. In 1798 Dr
+  F.J.M. de Lacerda, an accomplished astronomer, was appointed to
+  command a scientific expedition of discovery to the north of the
+  Zambesi. He started in July, crossed the Muchenja Mountains, and
+  reached the capital of the Cazembe, where he died of fever. Lacerda
+  left a valuable record of his adventurous journey; but with Mungo Park
+  and Lacerda the history of African exploration in the 18th century
+  closes.
+
+
+    South America.
+
+  In South America scientific exploration was active during this period.
+  The great geographical event of the century, as regards that
+  continent, was the measurement of an arc of the meridian. The
+  undertaking was proposed by the French Academy as part of an
+  investigation with the object of ascertaining the length of the degree
+  near the equator and near the pole respectively so as to determine the
+  figure of the earth. A commission left Paris in 1735, consisting of
+  Charles Marie de la Condamine, Pierre Bouguer, Louis Godin and Joseph
+  de Jussieu the naturalist. Spain appointed two accomplished naval
+  officers, the brothers Ulloa, as coadjutors. The operations were
+  carried on during eight years on a plain to the south of Quito; and,
+  in addition to his memoir on this memorable measurement, La Condamine
+  collected much valuable geographical information during a voyage down
+  the Amazon. The arc measured was 3 deg. 7' 3" in length; and the work
+  consisted of two measured bases connected by a series of triangles,
+  one north and the other south of the equator, on the meridian of
+  Quito. Contemporaneously, in 1738, Pierre Louis Moreau de Maupertuis,
+  Alexis Claude Clairaut, Charles Etienne Louis Camus, Pierre Charles
+  Lemonnier and the Swedish physicist Celsius measured an arc of the
+  meridian in Lapland.
+
+
+    The Pacific Ocean.
+
+  The British and French governments despatched several expeditions of
+  discovery into the Pacific and round the world during the 18th
+  century. They were preceded by the wonderful and romantic voyages of
+  the buccaneers. The narratives of such men as Woodes Rogers, Edward
+  Davis, George Shelvocke, Clipperton and William Dampier, can never
+  fail to interest, while they are not without geographical value. The
+  works of Dampier are especially valuable, and the narratives of
+  William Funnell and Lionel Wafer furnished the best accounts then
+  extant of the Isthmus of Darien. Dampier's literary ability eventually
+  secured for him a commission in the king's service; and he was sent on
+  a voyage of discovery, during which he explored part of the coasts of
+  Australia and New Guinea, and discovered the strait which bears his
+  name between New Guinea and New Britain, returning in 1701. In 1721
+  Jacob Roggewein was despatched on a voyage of some importance across
+  the Pacific by the Dutch West India Company, during which he
+  discovered Easter Island on the 6th of April 1722.
+
+  The voyage of Lord Anson to the Pacific in 1740-1744 was of a
+  predatory character, and he lost more than half his men from scurvy;
+  while it is not pleasant to reflect that at the very time when the
+  French and Spaniards were measuring an arc of the meridian at Quito,
+  the British under Anson were pillaging along the coast of the Pacific
+  and burning the town of Payta. But a romantic interest attaches to the
+  wreck of the "Wager," one of Anson's fleet, on a desert island near
+  Chiloe, for it bore fruit in the charming narrative of Captain John
+  Byron, which will endure for all time. In 1764 Byron himself was sent
+  on a voyage of discovery round the world, which led immediately after
+  his return to the despatch of another to complete his work, under the
+  command of Captain Samuel Wallis.
+
+  The expedition, consisting of the "Dolphin" commanded by Wallis, and
+  the "Swallow" under Captain Philip Carteret, sailed in September 1766,
+  but the ships were separated on entering the Pacific from the Strait
+  of Magellan. Wallis discovered Tahiti on the 19th of June 1767, and he
+  gave a detailed account of that island. He returned to England in May
+  1768. Carteret discovered the Charlotte and Gloucester Islands, and
+  Pitcairn Island on the 2nd of July 1767; revisited the Santa Cruz
+  group, which was discovered by Mendana and Quiros; and discovered the
+  strait separating New Britain from New Ireland. He reached Spithead
+  again in February 1769. Wallis and Carteret were followed very closely
+  by the French expedition of Bougainville, which sailed from Nantes in
+  November 1766. Bougainville had first to perform the unpleasant task
+  of delivering up the Falkland Islands, where he had encouraged the
+  formation of a French settlement, to the Spaniards. He then entered
+  the Pacific, and reached Tahiti in April 1768. Passing through the New
+  Hebrides group he touched at Batavia, and arrived at St Malo after an
+  absence of two years and four months.
+
+
+    Captain Cook.
+
+  The three voyages of Captain James Cook form an era in the history of
+  geographical discovery. In 1767 he sailed for Tahiti, with the object
+  of observing the transit of Venus, accompanied by two naturalists, Sir
+  Joseph Banks and Dr Solander, a pupil of Linnaeus, as well as by two
+  astronomers. The transit was observed on the 3rd of June 1769. After
+  exploring Tahiti and the Society group, Cook spent six months
+  surveying New Zealand, which he discovered to be an island, and the
+  coast of New South Wales from latitude 38 deg. S. to the northern
+  extremity. The belief in a vast Antarctic continent stretching far
+  into the temperate zone had never been abandoned, and was vehemently
+  asserted by Charles Dalrymple, a disappointed candidate nominated by
+  the Royal Society for the command of the Transit expedition of 1769.
+  In 1772 the French explorer Yves Kerguelen de Tremarec had discovered
+  the land that bears his name in the South Indian Ocean without
+  recognizing it to be an island, and naturally believed it to be part
+  of the southern continent.
+
+  Cook's second voyage was mainly intended to settle the question of the
+  existence of such a continent once for all, and to define the limits
+  of any land that might exist in navigable seas towards the Antarctic
+  circle. James Cook at his first attempt reached a south latitude of 57
+  deg. 15'. On a second cruise from the Society Islands, in 1773, he,
+  first of all men, crossed the Antarctic circle, and was stopped by ice
+  in 71 deg. 10' S. During the second voyage Cook visited Easter Island,
+  discovered several islands of the New Hebrides and New Caledonia; and
+  on his way home by Cape Horn, in March 1774, he discovered the
+  Sandwich Island group and described South Georgia. He proved
+  conclusively that any southern continent that might exist lay under
+  the polar ice. The third voyage was intended to attempt the passage
+  from the Pacific to the Atlantic by the north-east. The "Resolution"
+  and "Discovery" sailed in 1776, and Cook again took the route by the
+  Cape of Good Hope. On reaching the North American coast, he proceeded
+  northward, fixed the position of the western extremity of America and
+  surveyed Bering Strait. He was stopped by the ice in 70 deg. 41' N.,
+  and named the farthest visible point on the American shore Icy Cape.
+  He then visited the Asiatic shore and discovered Cape North. Returning
+  to Hawaii, Cook was murdered by the natives. On the 14th of February
+  1779, his second, Captain Edward Clerke, took command, and proceeding
+  to Petropavlovsk in the following summer, he again examined the edge
+  of the ice, but only got as far as 70 deg. 33' N. The ships returned
+  to England in October 1780.
+
+  In 1785 the French government carefully fitted out an expedition of
+  discovery at Brest, which was placed under the command of Francois La
+  Perouse, an accomplished and experienced officer. After touching at
+  Concepcion in Chile and at Easter Island, La Perouse proceeded to
+  Hawaii and thence to the coast of California, of which he has given a
+  very interesting account. He then crossed the Pacific to Macao, and in
+  July 1787 he proceeded to explore the Gulf of Tartary and the shores
+  of Sakhalin, remaining some time at Castries Bay, so named after the
+  French minister of marine. Thence he went to the Kurile Islands and
+  Kamchatka, and sailed from the far north down the meridian to the
+  Navigator and Friendly Islands. He was in Botany Bay in January 1788;
+  and sailing thence, the explorer, his ship and crew were never seen
+  again. Their fate was long uncertain. In September 1791 Captain
+  Antoine d'Entrecasteaux sailed from Brest with two vessels to seek for
+  tidings. He visited the New Hebrides, Santa Cruz, New Caledonia and
+  Solomon Islands, and made careful though rough surveys of the
+  Louisiade Archipelago, islands north of New Britain and part of New
+  Guinea. D'Entrecasteaux died on board his ship on the 20th of July
+  1793, without ascertaining the fate of La Perouse. Captain Peter
+  Dillon at length ascertained, in 1828, that the ships of La Perouse
+  had been wrecked on the island of Vanikoro during a hurricane.
+
+  The work of Captain Cook bore fruit in many ways. His master, Captain
+  William Bligh, was sent in the "Bounty" to convey breadfruit plants
+  from Tahiti to the West Indies. He reached Tahiti in October 1788, and
+  in April 1789 a mutiny broke out, and he, with several officers and
+  men, was thrust into an open boat in mid-ocean. During the remarkable
+  voyage he then made to Timor, Bligh passed amongst the northern
+  islands of the New Hebrides, which he named the Banks Group, and made
+  several running surveys. He reached England in March 1790. The
+  "Pandora," under Captain Edwards, was sent out in search of the
+  "Bounty," and discovered the islands of Cherry and Mitre, east of the
+  Santa Cruz group, but she was eventually lost on a reef in Torres
+  Strait. In 1796-1797 Captain Wilson, in the missionary ship "Duff,"
+  discovered the Gambier and other islands, and rediscovered the islands
+  known to and seen by Quiros, but since called the Duff Group. Another
+  result of Captain Cook's work was the colonization of Australia. On
+  the 18th of January 1788 Admiral Phillip and Captain Hunter arrived in
+  Botany Bay in the "Supply" and "Sirius," followed by six transports,
+  and established a colony at Port Jackson. Surveys were then undertaken
+  in several directions. In 1795 and 1796 Matthew Flinders and George
+  Bass were engaged on exploring work in a small boat called the "Tom
+  Thumb." In 1797 Bass, who had been a surgeon, made an expedition
+  southwards, continued the work of Cook from Ram Head, and explored the
+  strait which bears his name, and in 1798 he and Flinders were
+  surveying on the east coast of Van Diemen's land.
+
+  Yet another outcome of Captain Cook's work was the voyage of George
+  Vancouver, who had served as a midshipman in Cook's second and third
+  voyages. The Spaniards under Quadra had begun a survey of
+  north-western America and occupied Nootka Sound, which their
+  government eventually agreed to surrender. Captain Vancouver was sent
+  out to receive the cession, and to survey the coast from Cape
+  Mendocino northwards. He commanded the old "Discovery," and was at
+  work during the seasons of 1792, 1793 and 1794, wintering at Hawaii.
+  Returning home in 1795, he completed his narrative and a valuable
+  series of charts.
+
+
+    Arctic regions.
+
+  The 18th century saw the Arctic coast of North America reached at two
+  points, as well as the first scientific attempt to reach the North
+  Pole. The Hudson Bay Company had been incorporated in 1670, and its
+  servants soon extended their operations over a wide area to the north
+  and west of Canada. In 1741 Captain Christopher Middleton was ordered
+  to solve the question of a passage from Hudson Bay to the westward.
+  Leaving Fort Churchill in July 1742, he discovered the Wager river and
+  Repulse Bay. He was followed by Captain W. Moor in 1746, and Captain
+  Coats in 1751, who examined the Wager Inlet up to the end. In November
+  1769 Samuel Hearne was sent by the Hudson Bay Company to discover the
+  sea on the north side of America, but was obliged to return. In
+  February 1770 he set out again from Fort Prince of Wales; but, after
+  great hardships, he was again forced to return to the fort. He started
+  once more in December 1771, and at length reached the Coppermine
+  river, which he surveyed to its mouth, but his observations are
+  unreliable. With the same object Alexander Mackenzie, with a party of
+  Canadians, set out from Fort Chippewyan on the 3rd of June 1789, and
+  descending the great river which now bears the explorer's name reached
+  the Arctic sea.
+
+  In February 1773 the Royal Society submitted a proposal to the king
+  for an expedition towards the North Pole. The expedition was fitted
+  out under Captains Constantine Phipps and Skeffington Lutwidge, and
+  the highest latitude reached was 80 deg. 48' N., but no opening was
+  discovered in the heavy Polar pack. The most important Arctic work in
+  the 18th century was performed by the Russians, for they succeeded in
+  delineating the whole of the northern coast of Siberia. Some of this
+  work was possibly done at a still earlier date. The Cossack Simon
+  Dezhneff is thought to have made a voyage, in the summer of 1648, from
+  the river Kolyma, through Bering Strait (which was rediscovered by
+  Vitus Bering in 1728) to Anadyr. Between 1738 and 1750 Manin and
+  Sterlegoff made their way in small sloops from the mouth of the
+  Yenesei as far north as 75 deg. 15' N. The land from Taimyr to Cape
+  Chelyuskin, the most northern extremity of Siberia, was mapped in many
+  years of patient exploration by Chelyuskin, who reached the extreme
+  point (77 deg. 34' N.) in May 1742. To the east of Cape Chelyuskin the
+  Russians encountered greater difficulties. They built small vessels at
+  Yakutsk on the Lena, 900 m. from its mouth, whence the first
+  expedition was despatched under Lieut. Prontschichev in 1735. He
+  sailed from the mouth of the Lena to the mouth of the Olonek, where he
+  wintered, and on the 1st of September 1736 he got as far as 77 deg.
+  29' N., within 5 m. of Cape Chelyuskin. Both he and his young wife
+  died of scurvy, and the vessel returned. A second expedition, under
+  Lieut. Laptyev, started from the Lena in 1739, but encountered masses
+  of drift ice in Chatanga bay, and with this ended the voyages to the
+  westward of the Lena. Several attempts were also made to navigate the
+  sea from the Lena to the Kolyma. In 1736 Lieut. Laptyev sailed, but
+  was stopped by the drift ice in August, and in 1739, during another
+  trial, he reached the mouth of the Indigirka, where he wintered. In
+  the season of 1740 he continued his voyage to beyond the Kolyma,
+  wintering at Nizhni Kolymsk. In September 1740 Vitus Bering sailed
+  from Okhotsk on a second Arctic voyage with George William Steller on
+  board as naturalist. In June 1741 he named the magnificent peak on the
+  coast of North America Mount St Elias and explored the Aleutian
+  Islands. In November the ship was wrecked on Bering Island; and the
+  gallant Dane, worn out with scurvy, died there on the 8th of December
+  1741. In March 1770 a merchant named Liakhov saw a large herd of
+  reindeer coming from the north to the Siberian coast, which induced
+  him to start in a sledge in the direction whence they came. Thus he
+  reached the New Siberian or Liakhov Islands, and for years afterwards
+  the seekers for fossil ivory resorted to them. The Russian Captain
+  Vassili Chitschakov in 1765 and 1766 made two persevering attempts to
+  penetrate the ice north of Spitsbergen, and reached 80 deg. 30' N.,
+  while Russian parties twice wintered at Bell Sound.
+
+
+    Geographical societies.
+
+  In reviewing the progress of geographical discovery thus far, it has
+  been possible to keep fairly closely to a chronological order. But in
+  the 19th century and after exploring work was so generally and
+  steadily maintained in all directions, and was in so many cases
+  narrowed down from long journeys to detailed surveys within relatively
+  small areas, that it becomes desirable to cover the whole period at
+  one view for certain great divisions of the world. (See AFRICA; ASIA;
+  AUSTRALIA; POLAR REGIONS; &c.) Here, however, may be noticed the
+  development of geographical societies devoted to the encouragement of
+  exploration and research. The first of the existing geographical
+  societies was that of Paris, founded in 1825 under the title of La
+  Societe de Geographie. The Berlin Geographical Society (Gesellschaft
+  fur Erdkunde) is second in order of seniority, having been founded in
+  1827. The Royal Geographical Society, which was founded in London in
+  1830, comes third on the list; but it may be viewed as a direct result
+  of the earlier African Association founded in 1788. Sir John Barrow,
+  Sir John Cam Hobhouse (Lord Broughton), Sir Roderick Murchison, Mr
+  Robert Brown and Mr Bartle Frere formed the foundation committee of
+  the Royal Geographical Society, and the first president was Lord
+  Goderich. The action of the society in supplying practical instruction
+  to intending travellers, in astronomy, surveying and the various
+  branches of science useful to collectors, has had much to do with
+  advancement of discovery. Since the war of 1870 many geographical
+  societies have been established on the continent of Europe. At the
+  close of the 19th century there were upwards of 100 such societies in
+  the world, with more than 50,000 members, and over 150 journals were
+  devoted entirely to geographical subjects.[11] The great development
+  of photography has been a notable aid to explorers, not only by
+  placing at their disposal a faithful and ready means of recording the
+  features of a country and the types of inhabitants, but by supplying a
+  method of quick and accurate topographical surveying.
+
+
+  THE PRINCIPLES OF GEOGRAPHY
+
+  As regards the scope of geography, the order of the various
+  departments and their inter-relation, there is little difference of
+  opinion, and the principles of geography[12] are now generally
+  accepted by modern geographers. The order in which the various
+  subjects are treated in the following sketch is the natural succession
+  from fundamental to dependent facts, which corresponds also to the
+  evolution of the diversities of the earth's crust and of its
+  inhabitants.
+
+
+    Mathematical geography.
+
+  The fundamental geographical conceptions are mathematical, the
+  relations of space and form. The figure and dimensions of the earth
+  are the first of these. They are ascertained by a combination of
+  actual measurement of the highest precision on the surface and angular
+  observations of the positions of the heavenly bodies. The science of
+  geodesy is part of mathematical geography, of which the arts of
+  surveying and cartography are applications. The motions of the earth
+  as a planet must be taken into account, as they render possible the
+  determination of position and direction by observations of the
+  heavenly bodies. The diurnal rotation of the earth furnishes two fixed
+  points or poles, the axis joining which is fixed or nearly so in its
+  direction in space. The rotation of the earth thus fixes the
+  directions of north and south and defines those of east and west. The
+  angle which the earth's axis makes with the plane in which the planet
+  revolves round the sun determines the varying seasonal distribution of
+  solar radiation over the surface and the mathematical zones of
+  climate. Another important consequence of rotation is the deviation
+  produced in moving bodies relatively to the surface. In the form known
+  as Ferrell's Law this runs: "If a body moves in any direction on the
+  earth's surface, there is a deflecting force which arises from the
+  earth's rotation which tends to deflect it to the right in the
+  northern hemisphere but to the left in the southern hemisphere." The
+  deviation is of importance in the movement of air, of ocean currents,
+  and to some extent of rivers.[13]
+
+
+    Physical geography.
+
+  In popular usage the words "physical geography" have come to mean
+  geography viewed from a particular standpoint rather than any special
+  department of the subject. The popular meaning is better conveyed by
+  the word physiography, a term which appears to have been introduced by
+  Linnaeus, and was reinvented as a substitute for the cosmography of
+  the middle ages by Professor Huxley. Although the term has since been
+  limited by some writers to one particular part of the subject, it
+  seems best to maintain the original and literal meaning. In the
+  stricter sense, physical geography is that part of geography which
+  involves the processes of contemporary change in the crust and the
+  circulation of the fluid envelopes. It thus draws upon physics for the
+  explanation of the phenomena with the space-relations of which it is
+  specially concerned. Physical geography naturally falls into three
+  divisions, dealing respectively with the surface of the
+  lithosphere--geomorphology; the hydrosphere--oceanography; and the
+  atmosphere--climatology. All these rest upon the facts of mathematical
+  geography, and the three are so closely inter-related that they cannot
+  be rigidly separated in any discussion.
+
+
+    Geomorphology.
+
+  Geomorphology is the part of geography which deals with terrestrial
+  relief, including the submarine as well as the subaerial portions of
+  the crust. The history of the origin of the various forms belongs to
+  geology, and can be completely studied only by geological methods. But
+  the relief of the crust is not a finished piece of sculpture; the
+  forms are for the most part transitional, owing their characteristic
+  outlines to the process by which they are produced; therefore the
+  geographer must, for strictly geographical purposes, take some account
+  of the processes which are now in action modifying the forms of the
+  crust. Opinion still differs as to the extent to which the
+  geographer's work should overlap that of the geologist.
+
+  The primary distinction of the forms of the crust is that between
+  elevations and depressions. Granting that the geoid or mean surface of
+  the ocean is a uniform spheroid, the distribution of land and water
+  approximately indicates a division of the surface of the globe into
+  two areas, one of elevation and one of depression. The increasing
+  number of measurements of the height of land in all continents and
+  islands, and the very detailed levellings in those countries which
+  have been thoroughly surveyed, enable the average elevation of the
+  land above sea-level to be fairly estimated, although many vast gaps
+  in accurate knowledge remain, and the estimate is not an exact one.
+  The only part of the sea-bed the configuration of which is at all well
+  known is the zone bordering the coasts where the depth is less than
+  about 100 fathoms or 200 metres, i.e. those parts which sailors speak
+  of as "in soundings." Actual or projected routes for telegraph cables
+  across the deep sea have also been sounded with extreme accuracy in
+  many cases; but beyond these lines of sounding the vast spaces of the
+  ocean remain unplumbed save for the rare researches of scientific
+  expeditions, such as those of the "Challenger," the "Valdivia," the
+  "Albatross" and the "Scotia." Thus the best approximation to the
+  average depth of the ocean is little more than an expert guess; yet a
+  fair approximation is probable for the features of sub-oceanic relief
+  are so much more uniform than those of the land that a smaller number
+  of fixed points is required to determine them.
+
+
+    Crustal relief.
+
+  The chief element of uncertainty as to the largest features of the
+  relief of the earth's crust is due to the unexplored area in the
+  Arctic region and the larger regions of the Antarctic, of which we
+  know nothing. We know that the earth's surface if unveiled of water
+  would exhibit a great region of elevation arranged with a certain
+  rough radiate symmetry round the north pole, and extending southwards
+  in three unequal arms which taper to points in the south. A depression
+  surrounds the little-known south polar region in a continuous ring and
+  extends northwards in three vast hollows lying between the arms of the
+  elevated area. So far only is it possible to speak with certainty, but
+  it is permissible to take a few steps into the twilight of dawning
+  knowledge and indicate the chief subdivisions which are likely to be
+  established in the great crust-hollow and the great crust-heap. The
+  boundary between these should obviously be the mean surface of the
+  sphere.
+
+  Sir John Murray deduced the mean height of the land of the globe as
+  about 2250 ft. above sea-level, and the mean depth of the oceans as
+  2080 fathoms or 12,480 ft. below sea-level.[14] Calculating the area
+  of the land at 55,000,000 sq. m. (or 28.6% of the surface), and that
+  of the oceans as 137,200,000 sq. m. (or 71.4% of the surface), he
+  found that the volume of the land above sea-level was 23,450,000 cub.
+  m., the volume of water below sea-level 323,800,000, and the total
+  volume of the water equal to about 1/666th of the volume of the whole
+  globe. From these data, as revised by A. Supan,[15] H.R. Mill
+  calculated the position of mean sphere-level at about 10,000 ft. or
+  1700 fathoms below sea-level. He showed that an imaginary spheroidal
+  shell, concentric with the earth and cutting the slope between the
+  elevated and depressed areas at the contour-line of 1700 fathoms,
+  would not only leave above it a volume of the crust equal to the
+  volume of the hollow left below it, but would also divide the surface
+  of the earth so that the area of the elevated region was equal to that
+  of the depressed region.[16]
+
+
+    Areas of the crust according to Murray.
+
+  A similar observation was made almost simultaneously by Romieux,[17]
+  who further speculated on the equilibrium between the weight of the
+  elevated land mass and that of the total waters of the ocean, and
+  deduced some interesting relations between them. Murray, as the result
+  of his study, divided the earth's surface into three zones--the
+  _continental area_ containing all dry land, the _transitional area_
+  including the submarine slopes down to 1000 fathoms, and the _abysmal
+  area_ consisting of the floor of the ocean beyond that depth; and Mill
+  proposed to take the line of mean-sphere level, instead of the
+  empirical depth of 1000 fathoms, as the boundary between the
+  transitional and abysmal areas.
+
+  An elaborate criticism of all the existing data regarding the volume
+  relations of the vertical relief of the globe was made in 1894 by
+  Professor Hermann Wagner, whose recalculations of volumes and mean
+  heights--the best results which have yet been obtained--led to the
+  following conclusions.[18]
+
+
+    Areas of the crust according to Wagner.
+
+  The area of the dry land was taken as 28.3% of the surface of the
+  globe, and that of the oceans as 71.7%. The mean height deduced for
+  the land was 2300 ft. above sea-level, the mean depth of the sea
+  11,500 ft. below, while the position of mean-sphere level comes out as
+  7500 ft. (1250 fathoms) below sea-level. From this it would appear
+  that 43% of the earth's surface was above and 57% below the mean
+  level. It must be noted, however, that since 1895 the soundings of
+  Nansen in the north polar area, of the "Valdivia," "Belgica," "Gauss"
+  and "Scotia" in the Southern Ocean, and of various surveying ships in
+  the North and South Pacific, have proved that the mean depth of the
+  ocean is considerably greater than had been supposed, and mean-sphere
+  level must therefore lie deeper than the calculations of 1895 show;
+  possibly not far from the position deduced from the freer estimate of
+  1888. The whole of the available data were utilized by the prince of
+  Monaco in 1905 in the preparation of a complete bathymetrical map of
+  the oceans on a uniform scale, which must long remain the standard
+  work for reference on ocean depths.
+
+  By the device of a hypsographic curve co-ordinating the vertical
+  relief and the areas of the earth's surface occupied by each zone of
+  elevation, according to the system introduced by Supan,[19] Wagner
+  showed his results graphically.
+
+  This curve with the values reduced from metres to feet is reproduced
+  below.
+
+  Wagner subdivides the earth's surface, according to elevation, into
+  the following five regions:
+
+
+    _Wagner's Divisions of the Earth's Crust:_
+
+    +---------------------+-----------+-------------+-------------+
+    |        Name.        |Per cent of|    From     |     To      |
+    |                     |  Surface. |             |             |
+    +---------------------+-----------+-------------+-------------+
+    | Depressed area      |     3     |   Deepest.  |-16,400 feet.|
+    | Oceanic plateau     |    54     |-16,400 feet.|- 7,400   "  |
+    | Continental slope   |     9     |- 7,400   "  |-   660   "  |
+    | Continental plateau |    28     |-   660   "  |+ 3,000   "  |
+    | Culminating area    |     6     |+ 3,300   "  |   Highest.  |
+    +---------------------+-----------+-------------+-------------+
+
+  [Illustration]
+
+  The continental plateau might for purposes of detailed study be
+  divided into the _continental shelf_ from -660 ft. to sea-level, and
+  _lowlands_ from sea-level to +660 ft. (corresponding to the mean level
+  of the whole globe).[20] _Uplands_ reaching from 660 ft. to 2300 (the
+  approximate mean level of the land), and _highlands_, from 2300
+  upwards, might also be distinguished.
+
+
+    Arrangement of world-ridges and hollows.
+
+  A striking fact in the configuration of the crust is that each
+  continent, or elevated mass of the crust, is diametrically opposite to
+  an ocean basin or great depression; the only partial exception being
+  in the case of southern South America, which is antipodal to eastern
+  Asia. Professor C. Lapworth has generalized the grand features of
+  crustal relief in a scheme of attractive simplicity. He sees
+  throughout all the chaos of irregular crust-forms the recurrence of a
+  certain harmony, a succession of folds or waves which build up all the
+  minor features.[21] One great series of crust waves from east to west
+  is crossed by a second great series of crust waves from north to
+  south, giving rise by their interference to six great elevated masses
+  (the continents), arranged in three groups, each consisting of a
+  northern and a southern member separated by a minor depression. These
+  elevated masses are divided from one another by similar great
+  depressions.
+
+
+    Lapworth's fold-theory.
+
+  He says: "The surface of each of our great continental masses of land
+  resembles that of a long and broad arch-like form, of which we see the
+  simplest type in the New World. The surface of the North American arch
+  is sagged downwards in the middle into a central depression which lies
+  between two long marginal plateaus, and these plateaus are finally
+  crowned by the wrinkled crests which form its two modern mountain
+  systems. The surface of each of our ocean floors exactly resembles
+  that of a continent turned upside down. Taking the Atlantic as our
+  simplest type, we may say that the surface of an ocean basin resembles
+  that of a mighty trough or syncline, buckled up more or less centrally
+  in a medial ridge, which is bounded by two long and deep marginal
+  hollows, in the cores of which still deeper grooves sink to the
+  profoundest depths. This complementary relationship descends even to
+  the minor features of the two. Where the great continental sag sinks
+  below the ocean level, we have our gulfs and our Mediterraneans, seen
+  in our type continent, as the Mexican Gulf and Hudson Bay. Where the
+  central oceanic buckle attains the water-line we have our oceanic
+  islands, seen in our type ocean, as St Helena and the Azores. Although
+  the apparent crust-waves are neither equal in size nor symmetrical in
+  form, this complementary relationship between them is always
+  discernible. The broad Pacific depression seems to answer to the broad
+  elevation of the Old World--the narrow trough of the Atlantic to the
+  narrow continent of America."
+
+
+    Suess's theory.
+
+  The most thorough discussion of the great features of terrestrial
+  relief in the light of their origin is that by Professor E. Suess,[22]
+  who points out that the plan of the earth is the result of two
+  movements of the crust--one, subsidence over wide areas, giving rise
+  to oceanic depressions and leaving the continents protuberant; the
+  other, folding along comparatively narrow belts, giving rise to
+  mountain ranges. This theory of crust blocks dropped by subsidence is
+  opposed to Lapworth's theory of vast crust-folds, but geology is the
+  science which has to decide between them.
+
+  Geomorphology is concerned, however, in the suggestions which have
+  been made as to the cause of the distribution of heap and hollow in
+  the larger features of the crust. Elie de Beaumont, in his
+  speculations on the relation between the direction of mountain ranges
+  and their geological age and character, was feeling towards a
+  comprehensive theory of the forms of crustal relief; but his ideas
+  were too geometrical, and his theory that the earth is a spheroid
+  built up on a rhombic dodecahedron, the pentagonal faces of which
+  determined the direction of mountain ranges, could not be proved.[23]
+  The "tetrahedral theory" brought forward by Lowthian Green,[24] that
+  the form of the earth is a spheroid based on a regular tetrahedron, is
+  more serviceable, because it accounts for three very interesting facts
+  of the terrestrial plan--(1) the antipodal position of continents and
+  ocean basins; (2) the triangular outline of the continents; and (3)
+  the excess of sea in the southern hemisphere. Recent investigations
+  have recalled attention to the work of Lowthian Green, but the
+  question is still in the controversial stage.[25] The study of tidal
+  strain in the earth's crust by Sir George Darwin has led that
+  physicist to indicate the possibility of the triangular form and
+  southerly direction of the continents being a result of the
+  differential or tidal attraction of the sun and moon. More recently
+  Professor A.E.H. Love has shown that the great features of the relief
+  of the lithosphere may be expressed by spherical harmonics of the
+  first, second and third degrees, and their formation related to
+  gravitational action in a sphere of unequal density.[26]
+
+  In any case it is fully recognized that the plan of the earth is so
+  clear as to leave no doubt as to its being due to some general cause
+  which should be capable of detection.
+
+
+    The continents.
+
+  If the level of the sea were to become coincident with the mean level
+  of the lithosphere, there would result one tri-radiate land-mass of
+  nearly uniform outline and one continuous sheet of water broken by
+  few islands. The actual position of sea-level lies so near the summit
+  of the crust-heap that the varied relief of the upper portion leads to
+  the formation of a complicated coast-line and a great number of
+  detached portions of land. The hydrosphere is, in fact, continuous,
+  and the land is all in insular masses: the largest is the Old World of
+  Europe, Asia and Africa; the next in size, America; the third,
+  possibly, Antarctica; the fourth, Australia; the fifth, Greenland.
+  After this there is a considerable gap before New Guinea, Borneo,
+  Madagascar, Sumatra and the vast multitude of smaller islands
+  descending in size by regular gradations to mere rocks. The contrast
+  between island and mainland was natural enough in the days before the
+  discovery of Australia, and the mainland of the Old World was
+  traditionally divided into three continents. These "continents,"
+  "parts of the earth," or "quarters of the globe," proved to be
+  convenient divisions; America was added as a fourth, and subsequently
+  divided into two, while Australia on its discovery was classed
+  sometimes as a new continent, sometimes merely as an island, sometimes
+  compromisingly as an island-continent, according to individual
+  opinion. The discovery of the insularity of Greenland might again give
+  rise to the argument as to the distinction between island and
+  continent. Although the name of continent was not applied to large
+  portions of land for any physical reasons, it so happens that there is
+  a certain physical similarity or homology between them which is not
+  shared by the smaller islands or peninsulas.
+
+
+    Homology of continents.
+
+  The typical continental form is triangular as regards its sea-level
+  outline. The relief of the surface typically includes a central plain,
+  sometimes dipping below sea-level, bounded by lateral highlands or
+  mountain ranges, loftier on one side than on the other, the higher
+  enclosing a plateau shut in by mountains. South America and North
+  America follow this type most closely; Eurasia (the land mass of
+  Europe and Asia) comes next, while Africa and Australia are farther
+  removed from the type, and the structure of Antarctica and Greenland
+  is unknown.
+
+  If the continuous, unbroken, horizontal extent of land in a continent
+  is termed its _trunk_,[27] and the portions cut up by inlets or
+  channels of the sea into islands and peninsulas the _limbs_, it is
+  possible to compare the continents in an instructive manner.
+
+  The following table is from the statistics of Professor H. Wagner,[28]
+  his metric measurements being transposed into British units:
+
+
+    _Comparison of the Continents._
+
+    +---------------+-------+-------+-------+------+--------+------+------+
+    |               |       |       |       | Area |        |      |      |
+    |               | Area  | Mean  |  Area |penin-|  Area  | Area | Area |
+    |               | total |height,| trunk,|sulas,|islands,|limbs,|limbs,|
+    |               |  mil. | feet. |  mil. | mil. |  mil.  | mil. | per  |
+    |               | sq. m.|       | sq. m.|sq. m.| sq. m. |sq. m.| cent.|
+    +---------------+-------+-------+-------+------+--------+------+------+
+    | Old World     | 35.8  | 2360  |       |      |        |      |      |
+    | New World     | 16.2  | 2230  |       |      |        |      |      |
+    | Eurasia       | 20.85 | 2620  | 15.42 | 4.09 |  1.34  | 5.43 | 26   |
+    | Africa        | 11.46 | 2130  | 11.22 |  ..  |  0.24  | 0.24 |  2.1 |
+    | North America |  9.26 | 2300  |  6.92 | 0.78 |  1.56  | 2.34 | 25   |
+    | South America |  6.84 | 1970  |  6.76 | 0.02 |  0.06  | 0.08 |  1.1 |
+    | Australia     |  3.43 | 1310  |  2.77 | 0.16 |  0.50  | 0.66 | 19   |
+    | Asia          | 17.02 | 3120  | 12.93 | 3.05 |  1.04  | 4.09 | 24   |
+    | Europe        |  3.83 |  980  |  2.49 | 1.04 |  0.30  | 1.34 | 35   |
+    +---------------+-------+-------+-------+------+--------+------+------+
+
+
+    Islands.
+
+  The usual classification of islands is into continental and oceanic.
+  The former class includes all those which rise from the continental
+  shelf, or show evidence in the character of their rocks of having at
+  one time been continuous with a neighbouring continent. The latter
+  rise abruptly from the oceanic abysses. Oceanic islands are divided
+  according to their geological character into volcanic islands and
+  those of organic origin, including coral islands. More elaborate
+  subdivisions according to structure, origin and position have been
+  proposed.[29] In some cases a piece of land is only an island at high
+  water, and by imperceptible gradation the form passes into a
+  peninsula. The typical peninsula is connected with the mainland by a
+  relatively narrow isthmus; the name is, however, extended to any limb
+  projecting from the trunk of the mainland, even when, as in the Indian
+  peninsula, it is connected by its widest part.
+
+
+    Coasts.
+
+  Small peninsulas are known as promontories or headlands, and the
+  extremity as a cape. The opposite form, an inlet of the sea, is known
+  when wide as a gulf, bay or bight, according to size and degree of
+  inflection, or as a fjord or ria when long and narrow. It is
+  convenient to employ a specific name for a projection of a coast-line
+  less pronounced than a peninsula, and for an inlet less pronounced
+  than a bay or bight; outcurve and incurve may serve the turn. The
+  varieties of coast-lines were reduced to an exact classification by
+  Richthofen, who grouped them according to the height and slope of the
+  land into cliff-coasts (_Steilkusten_)--narrow beach coasts with
+  cliffs, wide beach coasts with cliffs, and low coasts, subdividing
+  each group according as the coast-line runs parallel to or crosses the
+  line of strike of the mountains, or is not related to mountain
+  structure. A further subdivision depends on the character of the
+  inter-relation of land and sea along the shore producing such types as
+  a fjord-coast, ria-coast or lagoon-coast. This extremely elaborate
+  subdivision may be reduced, as Wagner points out, to three types--the
+  continental coast where the sea comes up to the solid rock-material of
+  the land; the marine coast, which is formed entirely of soft material
+  sorted out by the sea; and the composite coast, in which both forms
+  are combined.
+
+
+    Coast-lines.
+
+  On large-scale maps it is necessary to show two coast-lines, one for
+  the highest, the other for the lowest tide; but in small-scale maps a
+  single line is usually wider than is required to represent the whole
+  breadth of the inter-tidal zone. The measurement of a coast-line is
+  difficult, because the length will necessarily be greater when
+  measured on a large-scale map where minute irregularities can be taken
+  into account. It is usual to distinguish between the general
+  coast-line measured from point to point of the headlands disregarding
+  the smaller bays, and the detailed coast-line which takes account of
+  every inflection shown by the map employed, and follows up river
+  entrances to the point where tidal action ceases. The ratio between
+  these two coast-lines represents the "coastal development" of any
+  region.
+
+
+    Submarine forms.
+
+  While the forms of the sea-bed are not yet sufficiently well known to
+  admit of exact classification, they are recognized to be as a rule
+  distinct from the forms of the land, and the importance of using a
+  distinctive terminology is felt. Efforts have been made to arrive at a
+  definite international agreement on this subject, and certain terms
+  suggested by a committee were adopted by the Eighth International
+  Geographical Congress at New York in 1904.[30] The forms of the ocean
+  floor include the "shelf," or shallow sea margin, the "depression," a
+  general term applied to all submarine hollows, and the "elevation." A
+  depression when of great extent is termed a "basin," when it is of a
+  more or less round form with approximately equal diameters, a "trough"
+  when it is wide and elongated with gently sloping borders, and a
+  "trench" when narrow and elongated with steeply sloping borders, one
+  of which rises higher than the other. The extension of a trough or
+  basin penetrating the land or an elevation is termed an "embayment"
+  when wide, and a "gully" when long and narrow; and the deepest part of
+  a depression is termed a "deep." A depression of small extent when
+  steep-sided is termed a "caldron," and a long narrow depression
+  crossing a part of the continental border is termed a "furrow." An
+  elevation of great extent which rises at a very gentle angle from a
+  surrounding depression is termed a "rise," one which is relatively
+  narrow and steep-sided a "ridge," and one which is approximately equal
+  in length and breadth but steep-sided a "plateau," whether it springs
+  direct from a depression or from a rise. An elevation of small extent
+  is distinguished as a "dome" when it is more than 100 fathoms from the
+  surface, a "bank" when it is nearer the surface than 100 fathoms but
+  deeper than 6 fathoms, and a "shoal" when it comes within 6 fathoms of
+  the surface and so becomes a serious danger to shipping. The highest
+  point of an elevation is termed a "height," if it does not form an
+  island or one of the minor forms.
+
+
+    Land forms.
+
+  The forms of the dry land are of infinite variety, and have been
+  studied in great detail.[31] From the descriptive or topographical
+  point of view, geometrical form alone should be considered; but the
+  origin and geological structure of land forms must in many cases be
+  taken into account when dealing with the function they exercise in the
+  control of mobile distributions. The geographers who have hitherto
+  given most attention to the forms of the land have been trained as
+  geologists, and consequently there is a general tendency to make
+  origin or structure the basis of classification rather than form
+  alone.
+
+
+    The six elementary land forms.
+
+  The fundamental form-elements may be reduced to the six proposed by
+  Professor Penck as the basis of his double system of classification by
+  form and origin.[32] These may be looked upon as being all derived by
+  various modifications or arrangements of the single form-unit, the
+  _slope_ or inclined plane surface. No one form occurs alone, but
+  always grouped together with others in various ways to make up
+  districts, regions and lands of distinctive characters. The
+  form-elements are:
+
+  1. The _plain_ or gently inclined uniform surface.
+
+  2. The _scarp_ or steeply inclined slope; this is necessarily of small
+  extent except in the direction of its length.
+
+  3. The _valley_, composed of two lateral parallel slopes inclined
+  towards a narrow strip of plain at a lower level which itself slopes
+  downwards in the direction of its length. Many varieties of this
+  fundamental form may be distinguished.
+
+  4. The _mount_, composed of a surface falling away on every side from
+  a particular place. This place may either be a point, as in a volcanic
+  cone, or a line, as in a mountain range or ridge of hills.
+
+  5. The _hollow_ or form produced by a land surface sloping inwards
+  from all sides to a particular lowest place, the converse of a mount.
+
+  6. The _cavern_ or space entirely surrounded by a land surface.
+
+
+    Geology and land forms.
+
+  These forms never occur scattered haphazard over a region, but always
+  in an orderly subordination depending on their mode of origin. The
+  dominant forms result from crustal movements, the subsidiary from
+  secondary reactions during the action of the primitive forms on mobile
+  distributions. The geological structure and the mineral composition of
+  the rocks are often the chief causes determining the character of the
+  land forms of a region. Thus the scenery of a limestone country
+  depends on the solubility and permeability of the rocks, leading to
+  the typical Karst-formations of caverns, swallow-holes and underground
+  stream courses, with the contingent phenomena of dry valleys and
+  natural bridges. A sandy beach or desert owes its character to the
+  mobility of its constituent sand-grains, which are readily drifted and
+  piled up in the form of dunes. A region where volcanic activity has
+  led to the embedding of dykes or bosses of hard rock amongst softer
+  strata produces a plain broken by abrupt and isolated eminences.[33]
+
+
+    Classification of mountains.
+
+  It would be impracticable to go fully into the varieties of each
+  specific form; but, partly as an example of modern geographical
+  classification, partly because of the exceptional importance of
+  mountains amongst the features of the land, one exception may be made.
+  The classification of mountains into types has usually had regard
+  rather to geological structure than to external form, so that some
+  geologists would even apply the name of a mountain range to a region
+  not distinguished by relief from the rest of the country if it bear
+  geological evidence of having once been a true range. A mountain may
+  be described (it cannot be defined) as an elevated region of irregular
+  surface rising comparatively abruptly from lower ground. The actual
+  elevation of a summit above sea-level does not necessarily affect its
+  mountainous character; a gentle eminence, for instance, rising a few
+  hundred feet above a tableland, even if at an elevation of say 15,000
+  ft., could only be called a hill.[34] But it may be said that any
+  abrupt slope of 2000 ft. or more in vertical height may justly be
+  called a mountain, while abrupt slopes of lesser height may be called
+  hills. Existing classifications, however, do not take account of any
+  difference in kind between mountain and hills, although it is common
+  in the German language to speak of _Hugelland_, _Mittelgebirge_ and
+  _Hochgebirge_ with a definite significance.
+
+  The simple classification employed by Professor James Geikie[35] into
+  mountains of accumulation, mountains of elevation and mountains of
+  circumdenudation, is not considered sufficiently thorough by German
+  geographers, who, following Richthofen, generally adopt a
+  classification dependent on six primary divisions, each of which is
+  subdivided. The terms employed, especially for the subdivisions,
+  cannot be easily translated into other languages, and the English
+  equivalents in the following table are only put forward tentatively:--
+
+    RICHTHOFEN'S CLASSIFICATION OF MOUNTAINS[36]
+
+      I. _Tektonische Gebirge_--Tectonic mountains.
+        (a) _Bruchgebirge oder Schollengebirge_--Block mountains.
+            1. _Einseitige Schollengebirge oder Schollenrandgebirge_--
+                  Scarp or tilted block mountains.
+                    (i.) _Tafelscholle_--Table blocks.
+                   (ii.) _Abrasionsscholle_--Abraded blocks.
+                  (iii.) _Transgressionsscholle_--Blocks of unconformable
+                              strata.
+            2. _Flexurgebirge_--Flexure mountains.
+            3. _Horstgebirge_--Symmetrical block mountains.
+        (b) _Faltungsgebirge_--Fold mountains.
+            1. _Homoomorphe Faltungsgebirge_--Homomorphic fold mountains.
+            2. _Heteromorphe Faltungsgebirge_--Heteromorphic fold
+                    mountains.
+
+     II. _Rumpfgebirge oder Abrasionsgebirge_--Trunk or abraded mountains.
+
+    III. _Ausbruchsgebirge_--Eruptive mountains.
+
+     IV. _Aufschuttungsgebirge_--Mountains of accumulation.
+
+      V. _Flachboden_--Plateaux.
+        (a) _Abrasionsplatten_--Abraded plateaux.
+        (b) _Marines Flachland_--Plain of marine erosion.
+        (c) _Schichtungstafelland_--Horizontally stratified tableland.
+        (d) _Ubergusstafelland_--Lava plain.
+        (e) _Stromflachland_--River plain.
+        (f) _Flachboden der atmospharischen Aufschuttung_--Plains of
+                   aeolian formation.
+
+     VI. _Erosionsgebirge_--Mountains of erosion.
+
+
+    Mountain forms.
+
+  From the morphological point of view it is more important to
+  distinguish the associations of forms, such as the _mountain mass_ or
+  group of mountains radiating from a centre, with the valleys furrowing
+  their flanks spreading towards every direction; the _mountain chain_
+  or line of heights, forming a long narrow ridge or series of ridges
+  separated by parallel valleys; the _dissected plateau_ or highland,
+  divided into mountains of circumdenudation by a system of deeply-cut
+  valleys; and the _isolated peak_, usually a volcanic cone or a hard
+  rock mass left projecting after the softer strata which embedded it
+  have been worn away (Monadnock of Professor Davis).
+
+
+    Distribution of mountains.
+
+  The geographical distribution of mountains is intimately associated
+  with the great structural lines of the continents of which they form
+  the culminating region. Lofty lines of fold mountains form the
+  "backbones" of North America in the Rocky Mountains and the west coast
+  systems, of South America in the Cordillera of the Andes, of Europe in
+  the Pyrenees, Alps, Carpathians and Caucasus, and of Asia in the
+  mountains of Asia Minor, converging on the Pamirs and diverging thence
+  in the Himalaya and the vast mountain systems of central and eastern
+  Asia. The remarkable line of volcanoes around the whole coast of the
+  Pacific and along the margin of the Caribbean and Mediterranean seas
+  is one of the most conspicuous features of the globe.
+
+
+    Functions of land forms.
+
+    Land waste.
+
+    Glaciers.
+
+  If land forms may be compared to organs, the part they serve in the
+  economy of the earth may, without straining the term, be characterized
+  as functions. The first and simplest function of the land surface is
+  that of guiding loose material to a lower level. The downward pull of
+  gravity suffices to bring about the fall of such material, but the
+  path it will follow and the distance it will travel before coming to
+  rest depend upon the land form. The loose material may, and in an arid
+  region does, consist only of portions of the higher parts of the
+  surface detached by the expansion and contraction produced by heating
+  and cooling due to radiation. Such broken material rolling down a
+  uniform scarp would tend to reduce its steepness by the loss of
+  material in the upper part and by the accumulation of a mound or scree
+  against the lower part of the slope. But where the side is not a
+  uniform scarp, but made up of a series of ridges and valleys, the
+  tendency will be to distribute the detritus in an irregular manner,
+  directing it away from one place and collecting it in great masses in
+  another, so that in time the land form assumes a new appearance. Snow
+  accumulating on the higher portions of the land, when compacted into
+  ice and caused to flow downwards by gravity, gives rise, on account of
+  its more coherent character, to continuous glaciers, which mould
+  themselves to the slopes down which they are guided, different
+  ice-streams converging to send forward a greater volume. Gradually
+  coming to occupy definite beds, which are deepened and polished by the
+  friction, they impress a characteristic appearance on the land, which
+  guides them as they traverse it, and, although the ice melts at lower
+  levels, vast quantities of clay and broken stones are brought down and
+  deposited in terminal moraines where the glacier ends.
+
+
+    Rain.
+
+    River systems.
+
+    Adjustment of rivers to land.
+
+  Rain is by far the most important of the inorganic mobile
+  distributions upon which land forms exercise their function of
+  guidance and control. The precipitation of rain from the aqueous
+  vapour of the atmosphere is caused in part by vertical movements of
+  the atmosphere involving heat changes and apparently independent of
+  the surface upon which precipitation occurs; but in greater part it is
+  dictated by the form and altitude of the land surface and the
+  direction of the prevailing winds, which itself is largely influenced
+  by the land. It is on the windward faces of the highest ground, or
+  just beyond the summit of less dominant heights upon the leeward side,
+  that most rain falls, and all that does not evaporate or percolate
+  into the ground is conducted back to the sea by a route which depends
+  only on the form of the land. More mobile and more searching than ice
+  or rock rubbish, the trickling drops are guided by the deepest lines
+  of the hillside in their incipient flow, and as these lines converge,
+  the stream, gaining strength, proceeds in its torrential course to
+  carve its channel deeper and entrench itself in permanent occupation.
+  Thus the stream-bed, from which at first the water might be blown away
+  into a new channel by a gale of wind, ultimately grows to be the
+  strongest line of the landscape. As the main valley deepens, the
+  tributary stream-beds are deepened also, and gradually cut their way
+  headwards, enlarging the area whence they draw their supplies. Thus
+  new land forms are created--valleys of curious complexity, for
+  example--by the "capture" and diversion of the water of one river by
+  another, leading to a change of watershed.[37] The minor tributaries
+  become more numerous and more constant, until the system of torrents
+  has impressed its own individuality on the mountain side. As the river
+  leaves the mountain, ever growing by the accession of tributaries, it
+  ceases, save in flood time, to be a formidable instrument of
+  destruction; the gentler slope of the land surface gives to it only
+  power sufficient to transport small stones, gravel, sand and
+  ultimately mud. Its valley banks are cut back by the erosion of minor
+  tributaries, or by rain-wash if the climate be moist, or left steep
+  and sharp while the river deepens its bed if the climate be arid. The
+  outline of the curve of a valley's sides ultimately depends on the
+  angle of repose of the detritus which covers them, if there has been
+  no subsequent change, such as the passage of a glacier along the
+  valley, which tends to destroy the regularity of the cross-section.
+  The slope of the river bed diminishes until the plain compels the
+  river to move slowly, swinging in _meanders_ proportioned to its size,
+  and gradually, controlled by the flattening land, ceasing to transport
+  material, but raising its banks and silting up its bed by the dropped
+  sediment, until, split up and shoaled, its distributaries struggle
+  across its delta to the sea. This is the typical river of which there
+  are infinite varieties, yet every variety would, if time were given,
+  and the land remained unchanged in level relatively to the sea,
+  ultimately approach to the type. Movements of the land either of
+  subsidence or elevation, changes in the land by the action of erosion
+  in cutting back an escarpment or cutting through a col, changes in
+  climate by affecting the rainfall and the volume of water, all tend to
+  throw the river valley out of harmony with the actual condition of its
+  stream. There is nothing more striking in geography than the
+  perfection of the adjustment of a great river system to its valleys
+  when the land has remained stable for a very lengthened period. Before
+  full adjustment has been attained the river bed may be broken in
+  places by waterfalls or interrupted by lakes; after adjustment the bed
+  assumes a permanent outline, the slope diminishing more and more
+  gradually, without a break in its symmetrical descent. Excellent
+  examples of the indecisive drainage of a new land surface, on which
+  the river system has not had time to impress itself, are to be seen in
+  northern Canada and in Finland, where rivers are separated by scarcely
+  perceptible divides, and the numerous lakes frequently belong to more
+  than one river system.
+
+
+    The geographical cycle.
+
+  The action of rivers on the land is so important that it has been made
+  the basis of a system of physical geography by Professor W.M. Davis,
+  who classifies land surfaces in terms of the three factors--structure,
+  process and time.[38] Of these time, during which the process is
+  acting on the structure, is the most important. A land may thus be
+  characterized by its position in the "geographical cycle", or cycle of
+  erosion, as young, mature or old, the last term being reached when the
+  base-level of erosion is attained, and the land, however varied its
+  relief may have been in youth or maturity, is reduced to a nearly
+  uniform surface or peneplain. By a re-elevation of a peneplain the
+  rivers of an old land surface may be restored to youthful activity,
+  and resume their shaping action, deepening the old valleys and
+  initiating new ones, starting afresh the whole course of the
+  geographical cycle. It is, however, not the action of the running
+  water on the land, but the function exercised by the land on the
+  running water, that is considered here to be the special province of
+  geography. At every stage of the geographical cycle the land forms, as
+  they exist at that stage, are concerned in guiding the condensation
+  and flow of water in certain definite ways. Thus, for example, in a
+  mountain range at right angles to a prevailing sea-wind, it is the
+  land forms which determine that one side of the range shall be richly
+  watered and deeply dissected by a complete system of valleys, while
+  the other side is dry, indefinite in its valley systems, and sends
+  none of its scanty drainage to the sea. The action of rain, ice and
+  rivers conspires with the movement of land waste to strip the layer of
+  soil from steep slopes as rapidly as it forms, and to cause it to
+  accumulate on the flat valley bottoms, on the graceful flattened cones
+  of alluvial fans at the outlet of the gorges of tributaries, or in the
+  smoothly-spread surface of alluvial plains.
+
+  The whole question of the regime of rivers and lakes is sometimes
+  treated under the name hydrography, a name used by some writers in the
+  sense of marine surveying, and by others as synonymous with
+  oceanography. For the study of rivers alone the name potamology[39]
+  has been suggested by Penck, and the subject being of much practical
+  importance has received a good deal of attention.[40]
+
+
+    Lakes and internal drainage.
+
+  The study of lakes has also been specialized under the name of
+  limnology (see LAKE).[41] The existence of lakes in hollows of the
+  land depends upon the balance between precipitation and evaporation. A
+  stream flowing into a hollow will tend to fill it up, and the water
+  will begin to escape as soon as its level rises high enough to reach
+  the lowest part of the rim. In the case of a large hollow in a very
+  dry climate the rate of evaporation may be sufficient to prevent the
+  water from ever rising to the lip, so that there is no outflow to the
+  sea, and a basin of internal drainage is the result. This is the case,
+  for instance, in the Caspian sea, the Aral and Balkhash lakes, the
+  Tarim basin, the Sahara, inner Australia, the great basin of the
+  United States and the Titicaca basin. These basins of internal
+  drainage are calculated to amount to 22% of the land surface. The
+  percentages of the land surface draining to the different oceans are
+  approximately--Atlantic, 34.3%; Arctic sea, 16.5%; Pacific, 14.4%;
+  Indian Ocean, 12.8%.[42]
+
+
+    Terminology of river systems.
+
+  The parts of a river system have not been so clearly defined as is
+  desirable, hence the exaggerated importance popularly attached to "the
+  source" of a river. A well-developed river system has in fact many
+  equally important and widely-separated sources, the most distant from
+  the mouth, the highest, or even that of largest initial volume not
+  being necessarily of greater geographical interest than the rest. The
+  whole of the land which directs drainage towards one river is known as
+  its basin, catchment area or drainage area--sometimes, by an incorrect
+  expression, as its valley or even its watershed. The boundary line
+  between one drainage area and others is rightly termed the watershed,
+  but on account of the ambiguity which has been tolerated it is better
+  to call it water-parting or, as in America, divide. The only other
+  important term which requires to be noted here is _talweg_, a word
+  introduced from the German into French and English, and meaning the
+  deepest line along the valley, which is necessarily occupied by a
+  stream unless the valley is dry.
+
+  The functions of land forms extend beyond the control of the
+  circulation of the atmosphere, the hydrosphere and the water which is
+  continually being interchanged between them; they are exercised with
+  increased effect in the higher departments of biogeography and
+  anthropogeography.
+
+
+    Biogeography.
+
+  The sum of the organic life on the globe is termed by some geographers
+  the biosphere, and it has been estimated that the whole mass of living
+  substance in existence at one time would cover the surface of the
+  earth to a depth of one-fifth of an inch.[43] The distribution of
+  living organisms is a complex problem, a function of many factors,
+  several of which are yet but little known. They include the biological
+  nature of the organism and its physical environment, the latter
+  involving conditions in which geographical elements, direct or
+  indirect, preponderate. The direct geographical elements are the
+  arrangement of land and sea (continents and islands standing in sharp
+  contrast) and the vertical relief of the globe, which interposes
+  barriers of a less absolute kind between portions of the same land
+  area or oceanic depression. The indirect geographical elements, which,
+  as a rule, act with and intensify the direct, are mainly climatic; the
+  prevailing winds, rainfall, mean and extreme temperatures of every
+  locality depending on the arrangement of land and sea and of land
+  forms. Climate thus guided affects the weathering of rocks, and so
+  determines the kind and arrangement of soil. Different species of
+  organisms come to perfection in different climates; and it may be
+  stated as a general rule that a species, whether of plant or animal,
+  once established at one point, would spread over the whole zone of the
+  climate congenial to it unless some barrier were interposed to its
+  progress. In the case of land and fresh-water organisms the sea is the
+  chief barrier; in the case of marine organisms, the land. Differences
+  in land forms do not exert great influence on the distribution of
+  living creatures directly, but indirectly such land forms as mountain
+  ranges and internal drainage basins are very potent through their
+  action on soil and climate. A snow-capped mountain ridge or an arid
+  desert forms a barrier between different forms of life which is often
+  more effective than an equal breadth of sea. In this way the surface
+  of the land is divided into numerous natural regions, the flora and
+  fauna of each of which include some distinctive species not shared by
+  the others. The distribution of life is discussed in the various
+  articles in this _Encyclopaedia_ dealing with biological, botanical
+  and zoological subjects.[44]
+
+
+    Floral zones.
+
+  The classification of the land surface into areas inhabited by
+  distinctive groups of plants has been attempted by many
+  phyto-geographers, but without resulting in any scheme of general
+  acceptance. The simplest classification is perhaps that of Drude
+  according to climatic zones, subdivided according to continents. This
+  takes account of--(1) the _Arctic-Alpine_ zone, including all the
+  vegetation of the region bordering on perpetual snow; (2) the _Boreal_
+  zone, including the temperate lands of North America, Europe and Asia,
+  all of which are substantially alike in botanical character; (3) the
+  _Tropical_ zone, divided sharply into (a) the tropical zone of the New
+  World, and (b) the tropical zone of the Old World, the forms of which
+  differ in a significant degree; (4) the _Austral_ zone, comprising all
+  continental land south of the equator, and sharply divided into three
+  regions the floras of which are strikingly distinct--(a) South
+  American, (b) South African and (c) Australian; (5) the _Oceanic_,
+  comprising all oceanic islands, the flora of which consists
+  exclusively of forms whose seeds could be drifted undestroyed by ocean
+  currents or carried by birds. To these might be added the antarctic,
+  which is still very imperfectly known. Many subdivisions and
+  transitional zones have been suggested by different authors.
+
+
+    Vegetation areas.
+
+  From the point of view of the economy of the globe this classification
+  by species is perhaps less important than that by mode of life and
+  physiological character in accordance with environment. The following
+  are the chief areas of vegetational activity usually recognized: (1)
+  The ice-deserts of the arctic and antarctic and the highest mountain
+  regions, where there is no vegetation except the lowest forms, like
+  that which causes "red snow." (2) The tundra or region of intensely
+  cold winters, forbidding tree-growth, where mosses and lichens cover
+  most of the ground when unfrozen, and shrubs occur of species which in
+  other conditions are trees, here stunted to the height of a few
+  inches. A similar zone surrounds the permanent snow on lofty mountains
+  in all latitudes. The tundra passes by imperceptible gradations into
+  the moor, bog and heath of warmer climates. (3) The temperate forests
+  of evergreen or deciduous trees, according to circumstances, which
+  occupy those parts of both temperate zones where rainfall and sunlight
+  are both abundant. (4) The grassy steppes or prairies where the
+  rainfall is diminished and temperatures are extreme, and grass is the
+  prevailing form of vegetation. These pass imperceptibly into--(5) the
+  arid desert, where rainfall is at a minimum, and the only plants are
+  those modified to subsist with the smallest supply of water. (6) The
+  tropical forest, which represents the maximum of plant luxuriance,
+  stimulated by the heaviest rainfall, greatest heat and strongest
+  light. These divisions merge one into the other, and admit of almost
+  indefinite subdivision, while they are subject to great modifications
+  by human interference in clearing and cultivating. Plants exhibit the
+  controlling power of environment to a high degree, and thus vegetation
+  is usually in close adjustment to the bolder geographical features of
+  a region.
+
+
+    Faunal realms.
+
+  The divisions of the earth into faunal regions by Dr P.L. Sclater have
+  been found to hold good for a large number of groups of animals as
+  different in their mode of life as birds and mammals, and they may
+  thus be accepted as based on nature. They are six in number: (1)
+  _Palaearctic_, including Europe, Asia north of the Himalaya, and
+  Africa north of the Sahara; (2) _Ethiopian_, consisting of Africa
+  south of the Atlas range, and Madagascar; (3) _Oriental_, including
+  India, Indo-China and the Malay Archipelago north of Wallace's line,
+  which runs between Bali and Lombok; (4) _Australian_, including
+  Australia, New Zealand, New Guinea and Polynesia; (5) _Nearctic_ or
+  North America, north of Mexico; and (6) _Neotropical_ or South
+  America. Each of these divisions is the home of a special fauna, many
+  species of which are confined to it alone; in the Australian region,
+  indeed, practically the whole fauna is peculiar and distinctive,
+  suggesting a prolonged period of complete biological isolation. In
+  some cases, such as the Ethiopian and Neotropical and the Palaearctic
+  and Nearctic regions, the faunas, although distinct, are related,
+  several forms on opposite sides of the Atlantic being analogous, e.g.
+  the lion and puma, ostrich and rhea. Where two of the faunal realms
+  meet there is usually, though not always, a mixing of faunas. These
+  facts have led some naturalists to include the Palaearctic and
+  Nearctic regions in one, termed _Holarctic_, and to suggest
+  transitional regions, such as the _Sonoran_, between North and South
+  America, and the _Mediterranean_, between Europe and Africa, or to
+  create sub-regions, such as Madagascar and New Zealand. Oceanic
+  islands have, as a rule, distinctive faunas and floras which resemble,
+  but are not identical with, those of other islands in similar
+  positions.
+
+
+    Biological distribution as a means of geographical research.
+
+  The study of the evolution of faunas and the comparison of the faunas
+  of distant regions have furnished a trustworthy instrument of
+  pre-historic geographical research, which enables earlier geographical
+  relations of land and sea to be traced out, and the approximate
+  period, or at least the chronological order of the larger changes, to
+  be estimated. In this way, for example, it has been suggested that a
+  land, "Lemuria," once connected Madagascar with the Malay Archipelago,
+  and that a northern extension of the antarctic land once united the
+  three southern continents.
+
+  The distribution of fossils frequently makes it possible to map out
+  approximately the general features of land and sea in long-past
+  geological periods, and so to enable the history of crustal relief to
+  be traced.[45]
+
+
+    Reaction of organisms on environment.
+
+  While the tendency is for the living forms to come into harmony with
+  their environment and to approach the state of equilibrium by
+  successive adjustments if the environment should happen to change, it
+  is to be observed that the action of organisms themselves often tends
+  to change their environment. Corals and other quick-growing calcareous
+  marine organisms are the most powerful in this respect by creating new
+  land in the ocean. Vegetation of all sorts acts in a similar way,
+  either in forming soil and assisting in breaking up rocks, in filling
+  up shallow lakes, and even, like the mangrove, in reclaiming wide
+  stretches of land from the sea. Plant life, utilizing solar light to
+  combine the inorganic elements of water, soil and air into living
+  substance, is the basis of all animal life. This is not by the supply
+  of food alone, but also by the withdrawal of carbonic acid from the
+  atmosphere, by which vegetation maintains the composition of the air
+  in a state fit for the support of animal life. Man in the primitive
+  stages of culture is scarcely to be distinguished from other animals
+  as regards his subjection to environment, but in the higher grades of
+  culture the conditions of control and reaction become much more
+  complicated, and the department of anthropogeography is devoted to
+  their consideration.
+
+
+    Anthropogeography.
+
+  The first requisites of all human beings are food and protection, in
+  their search for which men are brought into intimate relations with
+  the forms and productions of the earth's surface. The degree of
+  dependence of any people upon environment varies inversely as the
+  degree of culture or civilization, which for this purpose may perhaps
+  be defined as the power of an individual to exercise control over the
+  individual and over the environment for the benefit of the community.
+  The development of culture is to a certain extent a question of race,
+  and although forming one species, the varieties of man differ in
+  almost imperceptible gradations with a complexity defying
+  classification (see ANTHROPOLOGY). Professor Keane groups man round
+  four leading types, which may be named the black, yellow, red and
+  white, or the Ethiopic, Mongolic, American and Caucasic. Each may be
+  subdivided, though not with great exactness, into smaller groups,
+  either according to physical characteristics, of which the form of the
+  head is most important, or according to language.
+
+
+    Types of man.
+
+  The black type is found only in tropical or sub-tropical countries,
+  and is usually in a primitive condition of culture, unless educated by
+  contact with people of the white type. They follow the most primitive
+  forms of religion (mainly fetishism), live on products of the woods or
+  of the chase, with the minimum of work, and have only a loose
+  political organization. The red type is peculiar to America,
+  inhabiting every climate from polar to equatorial, and containing
+  representatives of many stages of culture which had apparently
+  developed without the aid or interference of people of any other race
+  until the close of the 15th century. The yellow type is capable of a
+  higher culture, cherishes higher religious beliefs, and inhabits as a
+  rule the temperate zone, although extending to the tropics on one side
+  and to the arctic regions on the other. The white type, originating in
+  the north temperate zone, has spread over the whole world. They have
+  attained the highest culture, profess the purest forms of monotheistic
+  religion, and have brought all the people of the black type and many
+  of those of the yellow under their domination.
+
+  The contrast between the yellow and white types has been softened by
+  the remarkable development of the Japanese following the assimilation
+  of western methods.
+
+  The actual number of human inhabitants in the world has been
+  calculated as follows:
+
+                            By Continents.[46]
+
+    Asia                       875,000,000
+    Europe                     392,000,000
+    Africa                     170,000,000
+    America                    143,000,000
+    Australia and Polynesia      7,000,000
+                             -------------
+      Total                  1,587,000,000
+
+                               By Race.[47]
+
+    White (Caucasic)           770,000,000
+    Yellow (Mong.)             540,000,000
+    Black (Ethiopic)           175,000,000
+    Red (American)              22,000,000
+                             -------------
+      Total                  1,507,000,000
+
+  In round numbers the population of the world is about 1,600,000,000,
+  and, according to an estimate by Ravenstein,[48] the maximum
+  population which it will be possible for the earth to maintain is 6000
+  millions, a number which, if the average rate of increase in 1891
+  continued, would be reached within 200 years.
+
+  While highly civilized communities are able to evade many of the
+  restrictions of environment, to overcome the barriers to
+  intercommunication interposed by land or sea, to counteract the
+  adverse influence of climate, and by the development of trade even to
+  inhabit countries which cannot yield a food-supply, the mass of
+  mankind is still completely under the control of those conditions
+  which in the past determined the distribution and the mode of life of
+  the whole human race.
+
+
+    Influence of environment on man.
+
+  In tropical forests primitive tribes depend on the collection of wild
+  fruits, and in a minor degree on the chase of wild animals, for their
+  food. Clothing is unnecessary; hence there is little occasion for
+  exercising the mental faculties beyond the sense of perception to
+  avoid enemies, or the inventive arts beyond what is required for the
+  simplest weapons and the most primitive fortifications. When the
+  pursuit of game becomes the chief occupation of a people there is of
+  necessity a higher development of courage, skill, powers of
+  observation and invention; and these qualities are still further
+  enhanced in predatory tribes who take by force the food, clothing and
+  other property prepared or collected by a feebler people. The
+  fruit-eating savage cannot stray beyond his woods which bound his life
+  as the water bounds that of a fish; the hunter is free to live on the
+  margin of forests or in open country, while the robber or warrior from
+  some natural stronghold of the mountains sweeps over the adjacent
+  plains and carries his raids into distant lands. Wide grassy steppes
+  lead to the organization of the people as nomads whose wealth consists
+  in flocks and herds, and their dwellings are tents. The nomad not only
+  domesticates and turns to his own use the gentler and more powerful
+  animals, such as sheep, cattle, horses, camels, but even turns some
+  predatory creatures, like the dog, into a means of defending their
+  natural prey. They hunt the beasts of prey destructive to their
+  flocks, and form armed bands for protection against marauders or for
+  purposes of aggression on weaker sedentary neighbours. On the fertile
+  low grounds along the margins of rivers or in clearings of forests,
+  agricultural communities naturally take their rise, dwelling in
+  villages and cultivating the wild grains, which by careful nurture and
+  selection have been turned into rich cereals. The agriculturist as a
+  rule is rooted to the soil. The land he tills he holds, and acquires a
+  closer connexion with a particular patch of ground than either the
+  hunter or the herdsman. In the temperate zone, where the seasons are
+  sharply contrasted, but follow each other with regularity, foresight
+  and self-denial were fostered, because if men did not exercise these
+  qualities seed-time or harvest might pass into lost opportunities and
+  the tribes would suffer. The more extreme climates of arid regions on
+  the margins of the tropics, by the unpredictable succession of
+  droughts and floods, confound the prevision of uninstructed people,
+  and make prudence and industry qualities too uncertain in their
+  results to be worth cultivating. Thus the civilization of agricultural
+  peoples of the temperate zone grew rapidly, yet in each community a
+  special type arose adapted to the soil, the crop and the climate. On
+  the seashore fishing naturally became a means of livelihood, and
+  dwellers by the sea, in virtue of the dangers to which they are
+  exposed from storm and unseaworthy craft, are stimulated to a higher
+  degree of foresight, quicker observation, prompter decision and more
+  energetic action in emergencies than those who live inland. The
+  building and handling of vessels also, and the utilization of such
+  uncontrollable powers of nature as wind and tide, helped forward
+  mechanical invention. To every type of coast there may be related a
+  special type of occupation and even of character; the deep and gloomy
+  fjord, backed by almost impassable mountains, bred bold mariners whose
+  only outlet for enterprise was seawards towards other lands--the
+  _viks_ created the vikings. On the gently sloping margin of the
+  estuary of a great river a view of tranquil inland life was equally
+  presented to the shore-dweller, and the ocean did not present the only
+  prospect of a career. Finally the mountain valley, with its patches of
+  cultivable soil on the alluvial fans of tributary torrents, its narrow
+  pastures on the uplands only left clear of snow in summer, its
+  intensified extremes of climates and its isolation, almost equal to
+  that of an island, has in all countries produced a special type of
+  brave and hardy people, whose utmost effort may bring them comfort,
+  but not wealth, by honest toil, who know little of the outer world,
+  and to whom the natural outlet for ambition is marauding on the
+  fertile plains. The highlander and viking, products of the valleys
+  raised high amid the mountains or half-drowned in the sea, are
+  everywhere of kindred spirit.
+
+  It is in some such manner as these that the natural conditions of
+  regions, which must be conformed to by prudence and utilized by labour
+  to yield shelter and food, have led to the growth of peoples differing
+  in their ways of life, thought and speech. The initial differences so
+  produced are confirmed and perpetuated by the same barriers which
+  divide the faunal or floral regions, the sea, mountains, deserts and
+  the like, and much of the course of past history and present politics
+  becomes clear when the combined results of differing race and
+  differing environment are taken into account.[49]
+
+
+    Density of population.
+
+  The specialization which accompanies the division of labour has
+  important geographical consequences, for it necessitates communication
+  between communities and the interchange of their products. Trade
+  makes it possible to work mineral resources in localities where food
+  can only be grown with great difficulty and expense, or which are even
+  totally barren and waterless, entirely dependent on supplies from
+  distant sources.
+
+  The population which can be permanently supported by a given area of
+  land differs greatly according to the nature of the resources and the
+  requirements of the people. Pastoral communities are always scattered
+  very thinly over large areas; agricultural populations may be almost
+  equally sparse where advanced methods of agriculture and labour-saving
+  machinery are employed; but where a frugal people are situated on a
+  fertile and inexhaustible soil, such as the deltas and river plains of
+  Egypt, India and China, an enormous population may be supported on a
+  small area. In most cases, however, a very dense population can only
+  be maintained in regions where mineral resources have fixed the site
+  of great manufacturing industries. The maximum density of population
+  which a given region can support is very difficult to determine; it
+  depends partly on the race and standard of culture of the people,
+  partly on the nature and origin of the resources on which they depend,
+  partly on the artificial burdens imposed and very largely on the
+  climate. Density of population is measured by the average number of
+  people residing on a unit of area; but in order to compare one part of
+  the world with another the average should, strictly speaking, be taken
+  for regions of equal size or of equal population; and the portions of
+  the country which are permanently uninhabitable ought to be excluded
+  from the calculation.[50] Considering the average density of
+  population within the political limits of countries, the following
+  list is of some value; the figures for a few smaller divisions of
+  large countries are added (in brackets) for comparison:
+
+    _Average Population on 1 sq. m._ (_For 1900 or 1901._)
+
+    +--------------------+---------+-------------------+---------+
+    |      Country.      | Density |      Country.     | Density |
+    |                    | of pop. |                   | of pop. |
+    +--------------------+---------+-------------------+---------+
+    | (Saxony)           |   743*  | Ceylon            |  141**  |
+    | Belgium            |   589*  | Greece            |   97    |
+    | Java               |   568** | European Turkey   |   90    |
+    | (England and Wales)|   558   | Spain             |   97    |
+    | (Bengal)           |   495** | European Russia   |   55**  |
+    | Holland            |   436   | Sweden            |   30    |
+    | United Kingdom     |   344   | United States     |   25    |
+    | Japan              |   317   | Mexico            |   18    |
+    | Italy              |   293   | Norway            |   18    |
+    | China proper       |   270** | Persia            |   15    |
+    | German Empire      |   270   | New Zealand       |    7    |
+    | Austria            |   226   | Argentina         |    5    |
+    | Switzerland        |   207   | Brazil            |    4.5  |
+    | France             |   188   | Eastern States of |         |
+    | Indian Empire      |   167** |   Australia       |    3    |
+    | Denmark            |   160** | Dominion of Canada|    1.5  |
+    | Hungary            |   154** | Siberia           |    1    |
+    | Portugal           |   146   | West Australia    |    0.2  |
+    +--------------------+---------+-------------------+---------+
+      * Almost exclusively industrial.
+     ** Almost exclusively agricultural.
+
+
+    Migration.
+
+  The movement of people from one place to another without the immediate
+  intention of returning is known as migration, and according to its
+  origin it may be classed as centrifugal (directed _from_ a particular
+  area) and centripetal (directed _towards_ a particular area).
+  Centrifugal migration is usually a matter of compulsion; it may be
+  necessitated by natural causes, such as a change of climate leading to
+  the withering of pastures or destruction of agricultural land, to
+  inundation, earthquake, pestilence or to an excess of population over
+  means of support; or to artificial causes, such as the wholesale
+  deportation of a conquered people; or to political or religious
+  persecution. In any case the people are driven out by some adverse
+  change; and when the urgency is great they may require to drive out in
+  turn weaker people who occupy a desirable territory, thus propagating
+  the wave of migration, the direction of which is guided by the forms
+  of the land into inevitable channels. Many of the great historic
+  movements of peoples were doubtless due to the gradual change of
+  geographical or climatic conditions; and the slow desiccation of
+  Central Asia has been plausibly suggested as the real cause of the
+  peopling of modern Europe and of the medieval wars of the Old World,
+  the theatres of which were critical points on the great natural lines
+  of communication between east and west.
+
+  In the case of centripetal migrations people flock to some particular
+  place where exceptionally favourable conditions have been found to
+  exist. The rushes to gold-fields and diamond-fields are typical
+  instances; the growth of towns on coal-fields and near other sources
+  of power, and the rapid settlement of such rich agricultural districts
+  as the wheat-lands of the American prairies and great plains are other
+  examples.
+
+  There is, however, a tendency for people to remain rooted to the land
+  of their birth, when not compelled or induced by powerful external
+  causes to seek a new home.
+
+
+    Political geography.
+
+  Thus arises the spirit of patriotism, a product of purely geographical
+  conditions, thereby differing from the sentiment of loyalty, which is
+  of racial origin. Where race and soil conspire to evoke both loyalty
+  and patriotism in a people, the moral qualities of a great and
+  permanent nation are secured. It is noticeable that the patriotic
+  spirit is strongest in those places where people are brought most
+  intimately into relation with the land; dwellers in the mountain or by
+  the sea, and, above all, the people of rugged coasts and mountainous
+  archipelagoes, have always been renowned for love of country, while
+  the inhabitants of fertile plains and trading communities are
+  frequently less strongly attached to their own land.
+
+  Amongst nomads the tribe is the unit of government, the political bond
+  is personal, and there is no definite territorial association of the
+  people, who may be loyal but cannot be patriotic. The idea of a
+  country arises only when a nation, either homogeneous or composed of
+  several races, establishes itself in a region the boundaries of which
+  may be defined and defended against aggression from without. Political
+  geography takes account of the partition of the earth amongst
+  organized communities, dealing with the relation of races to regions,
+  and of nations to countries, and considering the conditions of
+  territorial equilibrium and instability.
+
+
+    Boundaries.
+
+  The definition of boundaries and their delimitation is one of the most
+  important parts of political geography. Natural boundaries are always
+  the most definite and the strongest, lending themselves most readily
+  to defence against aggression. The sea is the most effective of all,
+  and an island state is recognized as the most stable. Next in
+  importance comes a mountain range, but here there is often difficulty
+  as to the definition of the actual crest-line, and mountain ranges
+  being broad regions, it may happen that a small independent state,
+  like Switzerland or Andorra, occupies the mountain valleys between two
+  or more great countries. Rivers do not form effective international
+  boundaries, although between dependent self-governing communities they
+  are convenient lines of demarcation. A desert, or a belt of country
+  left purposely without inhabitants, like the mark, marches or
+  debatable lands of the middle ages, was once a common means of
+  separating nations which nourished hereditary grievances. The
+  "buffer-state" of modern diplomacy is of the same ineffectual type. A
+  less definite though very practical boundary is that formed by the
+  meeting-line of two languages, or the districts inhabited by two
+  races. The line of fortresses protecting Austria from Italy lies in
+  some places well back from the political boundary, but just inside the
+  linguistic frontier, so as to separate the German and Italian races
+  occupying Austrian territory. Arbitrary lines, either traced from
+  point to point and marked by posts on the ground, or defined as
+  portions of meridians and parallels, are now the most common type of
+  boundaries fixed by treaty. In Europe and Asia frontiers are usually
+  strongly fortified and strictly watched in times of peace as well as
+  during war. In South America strictly defined boundaries are still the
+  exception, and the claims of neighbouring nations have very frequently
+  given rise to war, though now more commonly to arbitration.[51]
+
+
+    Forms of government.
+
+  The modes of government amongst civilized peoples have little
+  influence on political geography; some republics are as arbitrary and
+  exacting in their frontier regulations as some absolute monarchies. It
+  is, however, to be noticed that absolute monarchies are confined to
+  the east of Europe and to Asia, Japan being the only established
+  constitutional monarchy east of the Carpathians. Limited monarchies
+  are (with the exception of Japan) peculiar to Europe, and in these the
+  degree of democratic control may be said to diminish as one passes
+  eastwards from the United Kingdom. Republics, although represented in
+  Europe, are the peculiar form of government of America and are unknown
+  in Asia.
+
+  The forms of government of colonies present a series of transitional
+  types from the autocratic administration of a governor appointed by
+  the home government to complete democratic self-government. The latter
+  occurs only in the temperate possessions of the British empire, in
+  which there is no great preponderance of a coloured native population.
+  New colonial forms have been developed during the partition of Africa
+  amongst European powers, the sphere of influence being especially
+  worthy of notice. This is a vaguer form of control than a
+  protectorate, and frequently amounts merely to an agreement amongst
+  civilized powers to respect the right of one of their number to
+  exercise government within a certain area, if it should decide to do
+  so at any future time.
+
+  The central governments of all civilized countries concerned with
+  external relations are closely similar in their modes of action, but
+  the internal administration may be very varied. In this respect a
+  country is either centralized, like the United Kingdom or France, or
+  federated of distinct self-governing units like Germany (where the
+  units include kingdoms, at least three minor types of monarchies,
+  municipalities and a crown land under a nominated governor), or the
+  United States, where the units are democratic republics. The ultimate
+  cause of the predominant form of federal government may be the
+  geographical diversity of the country, as in the cantons occupying the
+  once isolated mountain valleys of Switzerland, the racial diversity of
+  the people, as in Austria-Hungary, or merely political expediency, as
+  in republics of the American type.
+
+  The minor subdivisions into provinces, counties and parishes, or
+  analogous areas, may also be related in many cases to natural features
+  or racial differences perpetuated by historical causes. The
+  territorial divisions and subdivisions often survive the conditions
+  which led to their origin; hence the study of political geography is
+  allied to history as closely as the study of physical geography is
+  allied to geology, and for the same reason.
+
+
+    Towns.
+
+  The aggregation of population in towns was at one time mainly brought
+  about by the necessity for defence, a fact indicated by the defensive
+  sites of many old towns. In later times, towns have been more often
+  founded in proximity to valuable mineral resources, and at critical
+  points or nodes on lines of communication. These are places where the
+  mode of travelling or of transport is changed, such as seaports, river
+  ports and railway termini, or natural resting-places, such as a ford,
+  the foot of a steep ascent on a road, the entrance of a valley leading
+  up from a plain into the mountains, or a crossing-place of roads or
+  railways.[52] The existence of a good natural harbour is often
+  sufficient to give origin to a town and to fix one end of a line of
+  land communication.
+
+
+    Lines of communication.
+
+  In countries of uniform surface or faint relief, roads and railways
+  may be constructed in any direction without regard to the
+  configuration. In places where the low ground is marshy, roads and
+  railways often follow the ridge-lines of hills, or, as in Finland, the
+  old glacial eskers, which run parallel to the shore. Wherever the
+  relief of the land is pronounced, roads and railways are obliged to
+  occupy the lowest ground winding along the valleys of rivers and
+  through passes in the mountains. In exceptional cases obstructions
+  which it would be impossible or too costly to turn are overcome by a
+  bridge or tunnel, the magnitude of such works increasing with the
+  growth of engineering skill and financial enterprise. Similarly the
+  obstructions offered to water communication by interruption through
+  land or shallows are overcome by cutting canals or dredging out
+  channels. The economy and success of most lines of communication
+  depend on following as far as possible existing natural lines and
+  utilizing existing natural sources of power.[53]
+
+
+    Commercial geography.
+
+  Commercial geography may be defined as the description of the earth's
+  surface with special reference to the discovery, production, transport
+  and exchange of commodities. The transport concerns land routes and
+  sea routes, the latter being the more important. While steam has been
+  said to make a ship independent of wind and tide, it is still true
+  that a long voyage even by steam must be planned so as to encounter
+  the least resistance possible from prevailing winds and permanent
+  currents, and this involves the application of oceanographical and
+  meteorological knowledge. The older navigation by utilizing the power
+  of the wind demands a very intimate knowledge of these conditions, and
+  it is probable that a revival of sailing ships may in the present
+  century vastly increase the importance of the study of maritime
+  meteorology.
+
+  The discovery and production of commodities require a knowledge of the
+  distribution of geological formations for mineral products, of the
+  natural distribution, life-conditions and cultivation or breeding of
+  plants and animals and of the labour market. Attention must also be
+  paid to the artificial restrictions of political geography, to the
+  legislative restrictions bearing on labour and trade as imposed in
+  different countries, and, above all, to the incessant fluctuations of
+  the economic conditions of supply and demand and the combinations of
+  capitalists or workers which affect the market.[54] The term "applied
+  geography" has been employed to designate commercial geography, the
+  fact being that every aspect of scientific geography may be applied to
+  practical purposes, including the purposes of trade. But apart from
+  the applied science, there is an aspect of pure geography which
+  concerns the theory of the relation of economics to the surface of the
+  earth.
+
+
+    Conclusion.
+
+  It will be seen that as each successive aspect of geographical science
+  is considered in its natural sequence the conditions become more
+  numerous, complex, variable and practically important. From the
+  underlying abstract mathematical considerations all through the
+  superimposed physical, biological, anthropological, political and
+  commercial development of the subject runs the determining control
+  exercised by crust-forms acting directly or indirectly on mobile
+  distributions; and this is the essential principle of geography.
+       (H. R. M.)
+
+
+FOOTNOTES:
+
+  [1] A concise sketch of the whole history of geographical method or
+    theory as distinguished from the history of geographical discovery
+    (see later section of this article) is only to be found in the
+    introduction to H. Wagner's _Lehrbuch der Geographie_, vol. i.
+    (Leipzig, 1900), which is in every way the most complete treatise on
+    the principles of geography.
+
+  [2] _History of Ancient Geography_ (Cambridge, 1897), p. 70.
+
+  [3] See J.L. Myres, "An Attempt to reconstruct the Maps used by
+    Herodotus," _Geographical Journal_, viii. (1896), p. 605.
+
+  [4] _Geschichte der wissenschaftlichen Erdkunde der Griechen_
+    (Leipzig, 1891), Abt. 3, p. 60.
+
+  [5] Bunbury's _History of Ancient Geography_ (2 vols., London, 1879),
+    Muller's _Geographi Graeci minores_ (2 vols., Paris, 1855, 1861) and
+    Berger's _Geschichte der wissenschaftlichen Erdkunde der Griechen_ (4
+    vols., Leipzig, 1887-1893) are standard authorities on the Greek
+    geographers.
+
+  [6] The period of the early middle ages is dealt with in Beazley's
+    _Dawn of Modern Geography_ (London; part i., 1897; part ii., 1901;
+    part iii., 1906); see also Winstedt, _Cosmos Indicopleustes_ (1910).
+
+  [7] From translator's preface to the English version by Mr Dugdale
+    (1733), entitled _A Complete System of General Geography_, revised by
+    Dr Peter Shaw (London, 1756).
+
+  [8] Printed in _Schriften zur physischen Geographie_, vol. vi. of
+    Schubert's edition of the collected works of Kant (Leipzig, 1839).
+    First published with notes by Rink in 1802.
+
+  [9] _History of Civilization_, vol. i. (1857).
+
+  [10] See H.J. Mackinder in _British Association Report_ (Ipswich),
+    1895, p. 738, for a summary of German opinion, which has been
+    expressed by many writers in a somewhat voluminous literature.
+
+  [11] H. Wagner's year-book, _Geographische Jahrbuch_, published at
+    Gotha, is the best systematic record of the progress of geography in
+    all departments; and Haack's _Geographen Kalender_, also published
+    annually at Gotha, gives complete lists of the geographical societies
+    and geographers of the world.
+
+  [12] This phrase is old, appearing in one of the earliest English
+    works on geography, William Cuningham's _Cosmographical Glasse
+    conteinyng the pleasant Principles of Cosmographie, Geographie,
+    Hydrographie or Navigation_ (London, 1559).
+
+  [13] See also S. Gunther, _Handbuch der mathematischen Geographie_
+    (Stuttgart, 1890).
+
+  [14] "On the Height of the Land and the Depth of the Ocean," _Scot.
+    Geog. Mag._ iv. (1888), p. 1. Estimates had been made previously by
+    Humboldt, De Lapparent, H. Wagner, and subsequently by Penck and
+    Heiderich, and for the oceans by Karstens.
+
+  [15] _Petermanns Mitteilungen_, xxv. (1889), p. 17.
+
+  [16] _Proc. Roy. Soc. Edin._ xvii. (1890) p. 185.
+
+  [17] _Comptes rendus Acad. Sci._ (Paris, 1890), vol. iii. p. 994.
+
+  [18] "Areal und mittlere Erhebung der Landflachen sowie der
+    Erdkruste" in Gerland's _Beitrage zur Geophysik_, ii. (1895) p. 667.
+    See also _Nature_, 54 (1896), p. 112.
+
+  [19] _Petermanns Mitteilungen_, xxxv. (1889) p. 19.
+
+  [20] The areas of the continental shelf and lowlands are
+    approximately equal, and it is an interesting circumstance that,
+    taken as a whole, the actual coast-line comes just midway on the most
+    nearly level belt of the earth's surface, excepting the ocean floor.
+    The configuration of the continental slope has been treated in detail
+    by Nansen in _Scientific Results of Norwegian North Polar
+    Expedition_, vol. iv. (1904), where full references to the literature
+    of the subject will be found.
+
+  [21] _British Association Report_ (Edinburgh, 1892), p. 699.
+
+  [22] _Das Antlitz der Erde_ (4 vols., Leipzig, 1885, 1888, 1901).
+    Translated under the editorship of E. de Margerie, with much
+    additional matter, as _La Face de la terre_, vols. i. and ii. (Paris,
+    1897, 1900), and into English by Dr Hertha Sollas as _The Face of the
+    Earth_, vols. i. and ii. (Oxford, 1904, 1906).
+
+  [23] Elie de Beaumont, _Notice sur les systemes de montagnes_ (3
+    vols., Paris, 1852).
+
+  [24] _Vestiges of the Molten Globe_ (London, 1875).
+
+  [25] See J.W. Gregory, "The Plan of the Earth and its Causes," _Geog.
+    Journal_, xiii. (1899) p. 225; Lord Avebury, _ibid._ xv. (1900) p.
+    46; Marcel Bertrand, "Deformation tetraedrique de la terre et
+    deplacement du pole," _Comptes rendus Acad. Sci._ (Paris, 1900), vol.
+    cxxx. p. 449; and A. de Lapparent, _ibid._ p. 614.
+
+  [26] See A.E.H. Love, "Gravitational Stability of the Earth," _Phil.
+    Trans._ ser. A. vol. ccvii. (1907) p. 171.
+
+  [27] _Rumpf_, in German, the language in which this distinction was
+    first made.
+
+  [28] _Lehrbuch der Geographie_ (Hanover and Leipzig, 1900), Bd. i. S.
+    245, 249.
+
+  [29] See, for example, F.G. Hahn's _Insel-Studien_ (Leipzig, 1883).
+
+  [30] See _Geographical Journal_, xxii. (1903) pp. 191-194.
+
+  [31] The most important works on the classification of land forms are
+    F. von Richthofen, _Fuhrer fur Forschungsreisende_ (Berlin, 1886); G.
+    de la Noe and E. de Margerie, _Les Formes du terrain_ (Paris, 1888);
+    and above all A. Penck, _Morphologie der Erdoberflache_ (2 vols.,
+    Stuttgart, 1894). Compare also A. de Lapparent, _Lecons de geographie
+    physique_ (2nd ed., Paris, 1898), and W.M. Davis, _Physical
+    Geography_ (Boston, 1899).
+
+  [32] "Geomorphologie als genetische Wissenschaft," in _Report of
+    Sixth International Geog. Congress_ (London, 1895), p. 735 (English
+    Abstract, p. 748).
+
+  [33] On this subject see J. Geikie, _Earth Sculpture_ (London, 1898);
+    J.E. Marr, _The Scientific Study of Scenery_ (London, 1900); Sir A.
+    Geikie, _The Scenery and Geology of Scotland_ (London, 2nd ed.,
+    1887); Lord Avebury (Sir J. Lubbock), _The Scenery of Switzerland_
+    (London, 1896) and _The Scenery of England_ (London, 1902).
+
+  [34] Some geographers distinguish a mountain from a hill by origin;
+    thus Professor Seeley says "a mountain implies elevation and a hill
+    implies denudation, but the external forms of both are often
+    identical." _Report VI. Int. Geog. Congress_ (London, 1895), p. 751.
+
+  [35] "Mountains," in _Scot. Geog. Mag._ ii. (1896) p. 145.
+
+  [36] _Fuhrer fur Forschungsreisende_, pp. 652-685.
+
+  [37] See, for a summary of river-action, A. Phillipson, _Studien uber
+    Wasserscheiden_ (Leipzig, 1886); also I.C. Russell, _River
+    Development_, (London, 1898) (published as _The Rivers of North
+    America_, New York, 1898).
+
+  [38] W.M. Davis, "The Geographical Cycle," _Geog. Journ._ xiv. (1899)
+    p. 484.
+
+  [39] A. Penck, "Potamology as a Branch of Physical Geography," _Geog.
+    Journ._ x. (1897) p. 619.
+
+  [40] See, for instance, E. Wisotzki, _Hauptfluss und Nebenfluss_
+    (Stettin, 1889). For practical studies see official reports on the
+    Mississippi, Rhine, Seine, Elbe and other great rivers.
+
+  [41] F.A. Forel, _Handbuch der Seenkunde: allgemeine Limnologie_
+    (Stuttgart, 1901); F.A. Forel, "La Limnologie, branche de la
+    geographie," _Report VI. Int. Geog. Congress_ (London, 1895), p. 593;
+    also _Le Leman_ (2 vols., Lausanne, 1892, 1894); H. Lullies, "Studien
+    uber Seen," _Jubilaumsschrift der Albertus-Universitat_ (Konigsberg,
+    1894); and G.R. Credner, "Die Reliktenseen," _Petermanns
+    Mitteilungen_, Erganzungshefte 86 and 89 (Gotha., 1887, 1888).
+
+  [42] J. Murray, "Drainage Areas of the Continents," _Scot. Geog.
+    Mag._ ii. (1886) p. 548.
+
+  [43] Wagner, _Lehrbuch der Geographie_ (1900), i. 586.
+
+  [44] For details, see A.R. Wallace, _Geographical Distribution of
+    Animals and Island Life_; A. Heilprin, _Geographical and Geological
+    Distribution of Animals_ (1887); O. Drude, _Handbuch der
+    Pflanzengeographie_; A. Engler, _Entwickelungsgeschichte der
+    Pflanzenwelt_; also Beddard, _Zoogeography_ (Cambridge, 1895); and
+    Sclater, _The Geography of Mammals_ (London, 1899).
+
+  [45] See particularly A. de Lapparent, _Traite de geologie_ (4th ed.,
+    Paris, 1900).
+
+  [46] Estimate for 1900. H. Wagner, _Lehrbuch der Geographie_, i. P.
+    658.
+
+  [47] Estimate for year not stated. A.H. Keane in _International
+    Geography_, p. 108.
+
+  [48] In _Proc. R. G. S._ xiii. (1891) p. 27.
+
+  [49] On the influence of land on people see Shaler, _Nature and Man
+    in America_ (New York and London, 1892); and Ellen C. Semple's
+    _American History and its Geographic Conditions_ (Boston, 1903).
+
+  [50] See maps of density of population in Bartholomew's great
+    large-scale atlases, _Atlas of Scotland_ and _Atlas of England_.
+
+  [51] For the history of territorial changes in Europe, see Freeman,
+    _Historical Geography of Europe_, edited by Bury (Oxford), 1903; and
+    for the official definition of existing boundaries, see Hertslet,
+    _The Map of Europe by Treaty_ (4 vols., London, 1875, 1891); _The Map
+    of Africa by Treaty_ (3 vols., London, 1896). Also Lord Curzon's
+    Oxford address on _Frontiers_ (1907).
+
+  [52] For numerous special instances of the determining causes of town
+    sites, see G.G. Chisholm, "On the Distribution of Towns and Villages
+    in England," _Geographical Journal_ (1897), ix. 76, x. 511.
+
+  [53] The whole subject of anthropogeography is treated in a masterly
+    way by F. Ratzel in his _Anthropogeographie_ (Stuttgart, vol. i. 2nd
+    ed., 1899, vol. ii. 1891), and in his _Politische Geographie_
+    (Leipzig, 1897). The special question of the reaction of man on his
+    environment is handled by G.P. Marsh in _Man and Nature, or Physical
+    Geography as modified by Human Action_ (London, 1864).
+
+  [54] For commercial geography see G.G. Chisholm, _Manual of
+    Commercial Geography_ (1890).
+
+
+
+
+GEOID (from Gr. [Greek: ge], the earth), an imaginary surface employed
+by geodesists which has the property that every element of it is
+perpendicular to the plumb-line where that line cuts it. Compared with
+the "spheroid of reference" the surface of the geoid is in general
+depressed over the oceans and raised over the great land masses. (See
+EARTH, FIGURE OF THE.)
+
+
+
+
+GEOK-TEPE, a former fortress of the Turkomans, in Russian Transcaspia,
+in the oasis of Akhal-tekke, on the Transcaspian railway, 28 m. N.W. of
+Askabad. It consisted of a walled enclosure 1-3/4 m. in circuit, the
+wall being 18 ft. high and 20 to 30 ft. thick. In December 1880 the
+place was attacked by 6000 Russians under General Skobelev, and after a
+siege of twenty-three days was carried by storm, although the defenders
+numbered 25,000. A monument and a small museum commemorate the event.
+
+
+
+
+GEOLOGY (from Gr. [Greek: ge], the earth, and [Greek: logos], science),
+the science which investigates the physical history of the earth. Its
+object is to trace the structural progress of our planet from the
+earliest beginnings of its separate existence, through its various
+stages of growth, down to the present condition of things. It seeks to
+determine the manner in which the evolution of the earth's great surface
+features has been effected. It unravels the complicated processes by
+which each continent has been built up. It follows, even into detail,
+the varied sculpture of mountain and valley, crag and ravine. Nor does
+it confine itself merely to changes in the inorganic world. Geology
+shows that the present races of plants and animals are the descendants
+of other and very different races which once peopled the earth. It
+teaches that there has been a progressive development of the
+inhabitants, as well as one of the globe on which they have dwelt; that
+each successive period in the earth's history, since the introduction of
+living things, has been marked by characteristic types of the animal and
+vegetable kingdoms; and that, however imperfectly the remains of these
+organisms have been preserved or may be deciphered, materials exist for
+a history of life upon the planet. The geographical distribution of
+existing faunas and floras is often made clear and intelligible by
+geological evidence; and in the same way light is thrown upon some of
+the remoter phases in the history of man himself. A subject so
+comprehensive as this must require a wide and varied basis of evidence.
+It is one of the characteristics of geology to gather evidence from
+sources which at first sight seem far removed from its scope, and to
+seek aid from almost every other leading branch of science. Thus, in
+dealing with the earliest conditions of the planet, the geologist must
+fully avail himself of the labours of the astronomer. Whatever is
+ascertainable by telescope, spectroscope or chemical analysis, regarding
+the constitution of other heavenly bodies, has a geological bearing. The
+experiments of the physicist, undertaken to determine conditions of
+matter and of energy, may sometimes be taken as the starting-points of
+geological investigation. The work of the chemical laboratory forms the
+foundation of a vast and increasing mass of geological inquiry. To the
+botanist, the zoologist, even to the unscientific, if observant,
+traveller by land or sea, the geologist turns for information and
+assistance.
+
+But while thus culling freely from the dominions of other sciences,
+geology claims as its peculiar territory the rocky framework of the
+globe. In the materials composing that framework, their composition and
+arrangement, the processes of their formation, the changes which they
+have undergone, and the terrestrial revolutions to which they bear
+witness, lie the main data of geological history. It is the task of the
+geologist to group these elements in such a way that they may be made to
+yield up their evidence as to the march of events in the evolution of
+the planet. He finds that they have in large measure arranged themselves
+in chronological sequence,--the oldest lying at the bottom and the
+newest at the top. Relics of an ancient sea-floor are overlain by traces
+of a vanished land-surface; these are in turn covered by the deposits of
+a former lake, above which once more appear proofs of the return of the
+sea. Among these rocky records lie the lavas and ashes of long-extinct
+volcanoes. The ripple left upon the shore, the cracks formed by the
+sun's heat upon the muddy bottom of a dried-up pool, the very imprint of
+the drops of a passing rainshower, have all been accurately preserved,
+and yield their evidence as to geographical conditions often widely
+different from those which exist where such markings are now found.
+
+But it is mainly by the remains of plants and animals imbedded in the
+rocks that the geologist is guided in unravelling the chronological
+succession of geological changes. He has found that a certain order of
+appearance characterizes these organic remains, that each great group of
+rocks is marked by its own special types of life, and that these types
+can be recognized, and the rocks in which they occur can be correlated
+even in distant countries, and where no other means of comparison would
+be possible. At one moment he has to deal with the bones of some large
+mammal scattered through a deposit of superficial gravel, at another
+time with the minute foraminifers and ostracods of an upraised
+sea-bottom. Corals and crinoids crowded and crushed into a massive
+limestone where they lived and died, ferns and terrestrial plants matted
+together into a bed of coal where they originally grew, the scattered
+shells of a submarine sand-bank, the snails and lizards which lived and
+died within a hollow-tree, the insects which have been imprisoned within
+the exuding resin of old forests, the footprints of birds and
+quadrupeds, the trails of worms left upon former shores--these, and
+innumerable other pieces of evidence, enable the geologist to realize in
+some measure what the faunas and floras of successive periods have been,
+and what geographical changes the site of every land has undergone.
+
+It is evident that to deal successfully with these varied materials, a
+considerable acquaintance with different branches of science is needful.
+Especially necessary is a tolerably wide knowledge of the processes now
+at work in changing the surface of the earth, and of at least those
+forms of plant and animal life whose remains are apt to be preserved in
+geological deposits, or which in their structure and habitat enable us
+to realize what their forerunners were. It has often been insisted that
+the present is the key to the past; and in a wide sense this assertion
+is eminently true. Only in proportion as we understand the present,
+where everything is open on all sides to the fullest investigation, can
+we expect to decipher the past, where so much is obscure, imperfectly
+preserved or not preserved at all. A study of the existing economy of
+nature ought thus to be the foundation of the geologist's training.
+
+While, however, the present condition of things is thus employed, we
+must obviously be on our guard against the danger of unconsciously
+assuming that the phase of nature's operations which we now witness has
+been the same in all past time, that geological changes have always or
+generally taken place in former ages in the manner and on the scale
+which we behold to-day, and that at the present time all the great
+geological processes, which have produced changes in the past eras of
+the earth's history, are still existent and active. As a working
+hypothesis we may suppose that the nature of geological processes has
+remained constant from the beginning; but we cannot postulate that the
+action of these processes has never varied in energy. The few centuries
+wherein man has been observing nature obviously form much too brief an
+interval by which to measure the intensity of geological action in all
+past time. For aught we can tell the present is an era of quietude and
+slow change, compared with some of the eras which have preceded it. Nor
+perhaps can we be quite sure that, when we have explored every
+geological process now in progress, we have exhausted all the causes of
+change which, even in comparatively recent times, have been at work.
+
+In dealing with the geological record, as the accessible solid part of
+the globe is called, we cannot too vividly realize that at the best it
+forms but an imperfect chronicle. Geological history cannot be compiled
+from a full and continuous series of documents. From the very nature of
+its origin the record is necessarily fragmentary, and it has been
+further mutilated and obscured by the revolutions of successive ages.
+And even where the chronicle of events is continuous, it is of very
+unequal value in different places. In one case, for example, it may
+present us with an unbroken succession of deposits many thousands of
+feet in thickness, from which, however, only a few meagre facts as to
+geological history can be gleaned. In another instance it brings before
+us, within the compass of a few yards, the evidence of a most varied and
+complicated series of changes in physical geography, as well as an
+abundant and interesting suite of organic remains. These and other
+characteristics of the geological record become more apparent and
+intelligible as we proceed in the study of the science.
+
+_Classification._--For systematic treatment the subject may be
+conveniently arranged in the following parts:--
+
+1. _The Historical Development of Geological Science._--Here a brief
+outline will be given of the gradual growth of geological conceptions
+from the days of the Greeks and Romans down to modern times, tracing the
+separate progress of the more important branches of inquiry and noting
+some of the stages which in each case have led up to the present
+condition of the science.
+
+2. _The Cosmical Aspects of Geology._--This section embraces the
+evidence supplied by astronomy and physics regarding the form and
+motions of the earth, the composition of the planets and sun, and the
+probable history of the solar system. The subjects dealt with under this
+head are chiefly treated in separate articles.
+
+3. _Geognosy._--An inquiry into the materials of the earth's substance.
+This division, which deals with the parts of the earth, its envelopes of
+air and water, its solid crust and the probable condition of its
+interior, especially treats of the more important minerals of the crust,
+and the chief rocks of which that crust is built up. Geognosy thus lays
+a foundation of knowledge regarding the nature of the materials
+constituting the mass of the globe, and prepares the way for an
+investigation of the processes by which these materials are produced and
+altered.
+
+4. _Dynamical Geology_ studies the nature and working of the various
+geological processes whereby the rocks of the earth's crust are formed
+and metamorphosed, and by which changes are effected upon the
+distribution of sea and land, and upon the forms of terrestrial
+surfaces. Such an inquiry necessitates a careful examination of the
+existing geological economy of nature, and forms a fitting introduction
+to an inquiry into the geological changes of former periods.
+
+5. _Geotectonic or Structural Geology_ has for its object the
+architecture of the earth's crust. It embraces an inquiry into the
+manner in which the various materials composing this crust have been
+arranged. It shows that some have been formed in beds or strata of
+sediment on the floor of the sea, that others have been built up by the
+slow aggregation of organic forms, that others have been poured out in a
+molten condition or in showers of loose dust from subterranean sources.
+It further reveals that, though originally laid down in almost
+horizontal beds, the rocks have subsequently been crumpled, contorted
+and dislocated, that they have been incessantly worn down, and have
+often been depressed and buried beneath later accumulations.
+
+6. _Palaeontological Geology._--This branch of the subject, starting
+from the evidence supplied by the organic forms which are found
+preserved in the crust of the earth, includes such questions as the
+relations between extinct and living types, the laws which appear to
+have governed the distribution of life in time and in space, the
+relative importance of different genera of animals in geological
+inquiry, the nature and use of the evidence from organic remains
+regarding former conditions of physical geography. Some of these
+problems belong also to zoology and botany, and are more fully discussed
+in the articles PALAEONTOLOGY and PALAEOBOTANY.
+
+7. _Stratigraphical Geology._--This section might be called geological
+history. It works out the chronological succession of the great
+formations of the earth's crust, and endeavours to trace the sequence of
+events of which they contain the record. More particularly, it
+determines the order of succession of the various plants and animals
+which in past time have peopled the earth, and thus ascertains what has
+been the grand march of life upon this planet.
+
+8. _Physiographical Geology_, proceeding from the basis of fact laid
+down by stratigraphical geology regarding former geographical changes,
+embraces an inquiry into the origin and history of the features of the
+earth's surface--continental ridges and ocean basins, plains, valleys
+and mountains. It explains the causes on which local differences of
+scenery depend, and shows under what very different circumstances, and
+at what widely separated intervals, the hills and mountains, even of a
+single country, have been produced.
+
+Most of the detail embraced in these several sections is relegated to
+separate articles, to which references are here inserted. The following
+pages thus deal mainly with the general principles and historical
+development of the science:--
+
+
+  PART I.--HISTORICAL DEVELOPMENT
+
+  _Geological Ideas among the Greeks and Romans._--Many geological
+  phenomena present themselves in so striking a form that they could
+  hardly fail to impress the imagination of the earliest and rudest
+  races of mankind. Such incidents as earthquakes and volcanic
+  eruptions, destructive storms on land and sea, disastrous floods and
+  landslips suddenly strewing valleys with ruin, must have awakened the
+  terror of those who witnessed them. Prominent features of landscape,
+  such as mountain-chains with their snows, clouds and thunderstorms,
+  dark river-chasms that seem purposely cleft open in order to give
+  passage to the torrents that rush through them, crags with their
+  impressive array of pinnacles and recesses must have appealed of old,
+  as they still do, to the awe and wonder of those who for the first
+  time behold them. Again, banks of sea-shells in far inland districts
+  would, in course of time, arrest the attention of the more intelligent
+  and reflective observers, and raise in their minds some kind of
+  surmise as to how such shells could ever have come there. These and
+  other conspicuous geological problems found their earliest solution in
+  legends and myths, wherein the more striking terrestrial features and
+  the elemental forces of nature were represented to be the
+  manifestation of the power of unseen supernatural beings.
+
+  The basin of the Mediterranean Sea was especially well adapted, from
+  its physical conditions, to be the birth-place of such fables. It is a
+  region frequently shaken by earthquakes, and contains two distinct
+  centres of volcanic activity, one in the Aegean Sea and one in Italy.
+  It is bounded on the north by a long succession of lofty snow-capped
+  mountain-ranges, whence copious rivers, often swollen by heavy rains
+  or melted snows, carry the drainage into the sea. On the south it
+  boasts the Nile, once so full of mystery; likewise wide tracts of arid
+  desert with their dreaded dust storms. The Mediterranean itself,
+  though an inland sea, is subject to gales, which, on exposed coasts,
+  raise breakers quite large enough to give a vivid impression of the
+  power of ocean waves. The countries that surround this great sheet of
+  water display in many places widely-spread deposits full of sea
+  shells, like those that still live in the neighbouring bays and gulfs.
+  Such a region was not only well fitted to supply subjects for
+  mythology, but also to furnish, on every side, materials which, in
+  their interest and suggestiveness, would appeal to the reason of
+  observant men.
+
+  It was natural, therefore, that the early philosophers of Greece
+  should have noted some of these geological features, and should have
+  sought for other explanations of them than those to be found in the
+  popular myths. The opinions entertained in antiquity on these subjects
+  may be conveniently grouped under two heads: (1) Geological processes
+  now in operation, and (2) geological changes in the past.
+
+
+    Earthquakes and volcanoes.
+
+  1. _Contemporary Processes._--The geological processes of the present
+  time are partly at work underground and partly on the surface of the
+  earth. The former, from their frequently disastrous character,
+  received much attention from Greek and Roman authors. Aristotle, in
+  his _Meteorics_, cites the speculations of several of his predecessors
+  which he rejects in favour of his own opinion to the effect that
+  earthquakes are due to the generation of wind within the earth, under
+  the influence of the warmth of the sun and the internal heat. Wind,
+  being the lightest and most rapidly moving body, is the cause of
+  motion in other bodies, and fire, united with wind, becomes flame,
+  which is endowed with great rapidity of motion. Aristotle looked upon
+  earthquakes and volcanic eruptions as closely connected with each
+  other, the discharge of hot materials to the surface being the result
+  of a severe earthquake, when finally the wind rushes out with
+  violence, and sometimes buries the surrounding country under sparks
+  and cinders, as had happened at Lipari. These crude conceptions of
+  the nature of volcanic action, and the cause of earthquakes, continued
+  to prevail for many centuries. They are repeated by Lucretius, who,
+  however, following Anaximenes, includes as one of the causes of
+  earthquakes the fall of mountainous masses of rock undermined by time,
+  and the consequent propagation of gigantic tremors far and wide
+  through the earth. Strabo, having travelled through the volcanic
+  districts of Italy, was able to recognize that Vesuvius had once been
+  an active volcano, although no eruption had taken place from it within
+  human memory. He continued to hold the belief that volcanic energy
+  arose from the movement of subterranean wind. He believed that the
+  district around the Strait of Messina, which had formerly suffered
+  from destructive earthquakes, was seldom visited by them after the
+  volcanic vents of that region had been opened, so as to provide an
+  escape for the subterranean fire, wind, water and burning masses. He
+  cites in his _Geography_ a number of examples of widespread as well as
+  local sinkings of land, and alludes also to the uprise of the
+  sea-bottom. He likewise regards some islands as having been thrown up
+  by volcanic agency, and others as torn from the mainland by such
+  convulsions as earthquakes.
+
+  The most detailed account of earthquake phenomena which has come down
+  to us from antiquity is that of Seneca in his _Quaestiones Naturales_.
+  This philosopher had been much interested in the accounts given him by
+  survivors and witnesses of the earthquake which convulsed the district
+  of Naples in February A.D. 63. He distinguished several distinct
+  movements of the ground: 1st, the up and down motion (_succussio_);
+  2nd, the oscillatory motion (_inclinatio_); and probably a third, that
+  of trembling or vibration. While admitting that some earthquakes may
+  arise from the collapse of the walls of subterranean cavities, he
+  adhered to the old idea, held by the most numerous and important
+  previous writers, that these commotions are caused mainly by the
+  movements of wind imprisoned within the earth. As to the origin of
+  volcanic outbursts he supposed that the subterranean wind in
+  struggling for an outlet, and whirling through the chasms and
+  passages, meets with great store of sulphur and other combustible
+  substances, which by mere friction are set on fire. The elder Pliny
+  reiterates the commonly accepted opinion as to the efficacy of wind
+  underground. In discussing the phenomena of earthquakes he remarks
+  that towns with many culverts and houses with cellars suffer less than
+  others, and that at Naples those houses are most shaken which stand on
+  hard ground. It thus appears that with regard to subterranean
+  geological operations, no advance was made during the time of the
+  Greeks and Romans as to the theoretical explanation of these
+  phenomena; but a considerable body of facts was collected, especially
+  as to the effects of earthquakes and the occurrence of volcanic
+  eruptions.
+
+
+    Action of rivers.
+
+  The superficial processes of geology, being much less striking than
+  those of subterranean energy, naturally attracted less attention in
+  antiquity. The operations of rivers, however, which so intimately
+  affect a human population, were watched with more or less care.
+  Herodotus, struck by the amount of alluvial silt brought down annually
+  by the Nile and spread over the flat inundated land, inferred that
+  "Egypt is the gift of the river." Aristotle, in discussing some of the
+  features of rivers, displays considerable acquaintance with the
+  various drainage-systems on the north side of the Mediterranean basin.
+  He refers to the mountains as condensers of the atmospheric moisture,
+  and shows that the largest rivers rise among the loftiest high
+  grounds. He shows how sensibly the alluvial deposits carried down to
+  the sea increase the breadth of the land, and cites some parts of the
+  shores of the Black Sea, where, in sixty years, the rivers had brought
+  down such a quantity of material that the vessels then in use required
+  to be of much smaller draught than previously, the water shallowing so
+  much that the marshy ground would, in course of time, become dry land.
+  Strabo supplies further interesting information as to the work of
+  rivers in making their alluvial plains and in pushing their deltas
+  seaward. He remarks that these deltas are prevented from advancing
+  farther outward by the ebb and flow of the tides.
+
+
+    Occurrences of fossils.
+
+  2. _Past Processes._--The abundant well-preserved marine shells
+  exposed among the upraised Tertiary and post-Tertiary deposits in the
+  countries bordering the Mediterranean are not infrequently alluded to
+  in Greek and Latin literature. Xenophanes of Colophon (614 B.C.)
+  noticed the occurrence of shells and other marine productions inland
+  among the mountains, and inferred from them that the land had risen
+  out of the sea. A similar conclusion was drawn by Xanthus the Lydian
+  (464 B.C.) from shells like scallops and cockles, which were found far
+  from the sea in Armenia and Lower Phrygia. Herodotus, Eratosthenes,
+  Strato and Strabo noted the vast quantities of fossil shells in
+  different parts of Egypt, together with beds of salt, as evidence that
+  the sea had once spread over the country. But by far the most
+  philosophical opinions on the past mutations of the earth's surface
+  are those expressed by Aristotle in the treatise already cited.
+  Reviewing the evidence of these changes, he recognized that the sea
+  now covers tracts that were once dry land, and that land will one day
+  reappear where there is now sea. These alternations are to be regarded
+  as following each other in a certain order and periodicity. But they
+  are apt to escape our notice because they require successive periods
+  of time, which, compared with our brief existence, are of enormous
+  duration, and because they are brought about so imperceptibly that we
+  fail to detect them in progress. In a celebrated passage in his
+  _Metamorphoses_, Ovid puts into the mouth of the philosopher
+  Pythagoras an account of what was probably regarded as the Pythagorean
+  view of the subject in the Augustan age. It affirms the interchange of
+  land and sea, the erosion of valleys by descending rivers, the washing
+  down of mountains into the sea, the disappearance of the rivers and
+  the submergence of land by earthquake movements, the separation of
+  some islands from, and the union of others with, the mainland, the
+  uprise of hills by volcanic action, the rise and extinction of burning
+  mountains. There was a time before Etna began to glow, and the time is
+  coming when the mountain will cease to burn.
+
+  From this brief sketch it will be seen that while the ancients had
+  accumulated a good deal of information regarding the occurrence of
+  geological changes, their interpretations of the phenomena were to a
+  considerable extent mere fanciful speculation. They had acquired only
+  a most imperfect conception of the nature and operation of the
+  geological processes; and though many writers realized that the
+  surface of the earth has not always been, and will not always remain,
+  as it is now, they had no glimpse of the vast succession of changes of
+  that surface which have been revealed by geology. They built
+  hypotheses on the slenderest basis of fact, and did not realize the
+  necessity of testing or verifying them.
+
+  _Progress of Geological Conceptions in the Middle Ages._--During the
+  centuries that succeeded the fall of the Western empire little
+  progress was made in natural science. The schoolmen in the monasteries
+  and other seminaries were content to take their science from the
+  literature of Greece and Rome. The Arabs, however, not only collected
+  and translated that literature, but in some departments made original
+  observations themselves. To one of the most illustrious of their
+  number, Avicenna, the translator of Aristotle, a treatise has been
+  ascribed, in which singularly modern ideas are expressed regarding
+  mountains, some of which are there stated to have been produced by an
+  uplifting of the ground, while others have been left prominent, owing
+  to the wearing away of the softer rocks around them. In either case,
+  it is confessed that the process would demand long tracts of time for
+  its completion.
+
+  After the revival of learning the ancient problem presented by fossil
+  shells imbedded in the rocks of the interior of many countries
+  received renewed attention. But the conditions for its solution were
+  no longer what they had been in the days of the philosophers of
+  antiquity. Men were not now free to adopt and teach any doctrine they
+  pleased on the subject. The Christian church had meanwhile arisen to
+  power all over Europe, and adjudged as heretics all who ventured to
+  impugn any of her dogmas. She taught that the land and the sea had
+  been separated on the third day of creation, before the appearance of
+  any animal life, which was not created until the fifth day. To assert
+  that the dry land is made up in great part of rocks that were formed
+  in the sea, and are crowded with the remains of animals, was plainly
+  to impugn the veracity of the Bible. Again, it had come to be the
+  orthodox belief that only somewhere about 6000 years had elapsed since
+  the time of Adam and Eve. If any thoughtful observer, impressed with
+  the overwhelming force of the evidence that the fossiliferous
+  formations of the earth's crust must have taken long periods of time
+  for their accumulation, ventured to give public expression to his
+  conviction, he ran considerable risk of being proceeded against as a
+  heretic. It was needful, therefore, to find some explanation of the
+  facts of nature, which would not run counter to the ecclesiastical
+  system of the day. Various such interpretations were proposed,
+  doubtless in an honest endeavour at reconciliation. Three of these
+  deserve special notice: (1) Many able observers and diligent
+  collectors of fossils persuaded themselves that these objects never
+  belonged to organisms of any kind, but should be regarded as mere
+  "freaks of nature," having no more connexion with any once living
+  creature than the frost patterns on a window. They were styled
+  "formed" or "figured" stones, "lapides sui generis," and were asserted
+  to be due to some inorganic imitative process within the earth or to
+  the influence of the stars. (2) Observers who could not resist the
+  evidence of their senses that the fossil shells once belonged to
+  living animals, and who, at the same time, felt the necessity of
+  accounting for the presence of marine organisms in the rocks of which
+  the dry land is largely built up, sought a way out of the difficulty
+  by invoking the Deluge of Noah. Here was a catastrophe which, they
+  said, extended over the whole globe, and by which the entire dry land
+  was submerged even up to the tops of the high hills. True, it only
+  lasted one hundred and fifty days, but so little were the facts then
+  appreciated that no difficulty seems to have been generally felt in
+  crowding the accumulation of the thousands of feet of fossiliferous
+  formations into that brief space of time. (3) Some more intelligent
+  men in Italy, recognizing that these interpretations could not be
+  upheld, fell back upon the idea that the rocks in which fossil shells
+  are imbedded might have been heaped up by repeated and vigorous
+  eruptions from volcanic centres. Certain modern eruptions in the
+  Aegean Sea and in the Bay of Naples had drawn attention to the
+  rapidity with which hills of considerable size could be piled around
+  an active crater. It was argued that if Monte Nuovo near Naples could
+  have been accumulated to a height of nearly 500 ft. in two days, there
+  seemed to be no reason against believing that, during the time of the
+  Flood, and in the course of the centuries that have elapsed since
+  that event, the whole of the fossiliferous rocks might have been
+  deposited. Unfortunately for this hypothesis it ignored the fact that
+  these rocks do not consist of volcanic materials.
+
+
+    Leonardo da Vinci; Fracastorio; Falloppio.
+
+  So long as the fundamental question remained in dispute as to the true
+  character and history of the stratified portion of the earth's crust
+  containing organic remains, geology as a science could not begin its
+  existence. The diluvialists (those who relied on the hypothesis of the
+  Flood) held the field during the 16th, 17th and a great part of the
+  18th century. They were looked on as the champions of orthodoxy; and,
+  on that account, they doubtless wielded much more influence than would
+  have been gained by them from the force of their arguments. Yet during
+  those ages there were not wanting occasional observers who did good
+  service in combating the prevalent misconceptions, and in preparing
+  the way for the ultimate triumph of truth. It was more especially in
+  Italy, where many of the more striking phenomena of geology are
+  conspicuously displayed, that the early pioneers of the science arose,
+  and that for several generations the most marked progress was made
+  towards placing the investigations of the past history of the earth
+  upon a basis of careful observation and scientific deduction. One of
+  the first of these leaders was Leonardo da Vinci (1452-1519), who,
+  besides his achievements in painting, sculpture, architecture and
+  engineering, contributed some notable observations regarding the great
+  problem of the origin of fossil shells. He ridiculed the notion that
+  these objects could have been formed by the influence of the stars,
+  and maintained that they had once belonged to living organisms, and
+  therefore that what is now land was formerly covered by the sea.
+  Girolamo Fracastorio (1483-1553) claimed that the shells could never
+  have been left by the Flood, which was a mere temporary inundation,
+  but that they proved the mountains, in which they occur, to have been
+  successively uplifted out of the sea. On the other hand, even an
+  accomplished anatomist like Gabriello Falloppio (1523-1562) found it
+  easier to believe that the bones of elephants, teeth of sharks, shells
+  and other fossils were mere earthy inorganic concretions, than that
+  the waters of Noah's Flood could ever nave reached as far as Italy.
+
+
+    Nicolas Steno.
+
+  By much the most important member of this early band of Italian
+  writers was undoubtedly Nicolas Steno (1631-1687), who, though born in
+  Copenhagen, ultimately settled in Florence. Having made a European
+  reputation as an anatomist, his attention was drawn to geological
+  problems by finding that the rocks of the north of Italy contained
+  what appeared to be sharks' teeth closely resembling those of a
+  dog-fish, of which he had published the anatomy. Cautiously at first,
+  for fear of offending orthodox opinions, but afterwards more boldly,
+  he proclaimed his conviction that those objects had once been part of
+  living animals, and that they threw light on some of the past history
+  of the earth. He published in 1669 a small tract, _De solido intra
+  solidum naturaliter contento_, in which he developed the ideas he had
+  formed of this history from an attentive study of the rocks. He showed
+  that the stratified formations of the hills and valleys consist of
+  such materials as would be laid down in the form of sediment in turbid
+  water; that where they contain marine productions this water is proved
+  to have been the sea; that diversities in their composition point to
+  commingling of currents, carrying different kinds of sediment of which
+  the heaviest would first sink to the bottom. He made original and
+  important observations on stratification, and laid down some of the
+  fundamental axioms in stratigraphy. He reasoned that as the original
+  position of strata was approximately horizontal, when they are found
+  to be steeply inclined or vertical, or bent into arches, they have
+  been disrupted by subterranean exhalations, or by the falling in of
+  the roofs of underground cavernous spaces. It is to this alteration of
+  the original position of the strata that the inequalities of the
+  earth's surface, such as mountains, are to be ascribed, though some
+  have been formed by the outburst of fire, ashes and stones from inside
+  the earth. Another effect of the dislocation has been to provide
+  fissures, which serve as outlets for springs. Steno's anatomical
+  training peculiarly fitted him for dealing authoritatively with the
+  question of the nature and origin of the fossils contained in the
+  rocks. He had no hesitation in affirming that, even if no shells had
+  ever been found living in the sea, the internal structure of these
+  fossils would demonstrate that they once formed parts of living
+  animals. And not only shells, but teeth, bones and skeletons of many
+  kinds of fishes had been quarried out of the rocks, while some of the
+  strata had skulls, horns and teeth of land-animals. Illustrating his
+  general principles by a sketch of what he supposed to have been the
+  past history of Tuscany, he added a series of diagrams which show how
+  clearly he had conceived the essential elements of stratigraphy. He
+  thought he could perceive the records of six successive phases in the
+  evolution of the framework of that country, and was inclined to
+  believe that a similar chronological sequence would be found all over
+  the world. He anticipated the objections that would be brought against
+  his views on account of the insuperable difficulty in granting the
+  length of time that would be required for all the geographical
+  vicissitudes which his interpretation required. He thought that many
+  of the fossils must be as old as the time of the general deluge, but
+  he was careful not to indulge in any speculation as to the antiquity
+  of the earth.
+
+
+    Lazzaro Moro.
+
+  To the Italian school, as especially typified in Steno, must be
+  assigned the honour of having thus begun to lay firmly and truly the
+  first foundation stones of the modern science of geology. The same
+  school included Antonio Vallisneri (1661-1730), who surpassed his
+  predecessors in his wider and more exact knowledge of the
+  fossiliferous rocks that form the backbone of the Italian peninsula,
+  which he contended were formed during a wide and prolonged submergence
+  of the region, altogether different from the brief deluge of Noah.
+  There was likewise Lazzaro Moro (1687-1740), who did good service
+  against the diluvialists, but the fundamental feature of his system of
+  nature lay in the preponderant part which, unaware of the great
+  difference between volcanic materials and ordinary sediment, he
+  assigned to volcanic action in the production of the sedimentary rocks
+  of the earth's crust. He supposed that in the beginning the globe was
+  completely surrounded with water, beneath which the solid earth lay as
+  a smooth ball. On the third day of creation, however, vast fires were
+  kindled inside the globe, whereby the smooth surface of stone was
+  broken up, and portions of it, appearing above the water, formed the
+  earliest land. From that time onward, volcanic eruptions succeeded
+  each other, not only on the emerged land, but on the sea-floor, over
+  which the ejected material spread in an ever augmenting thickness of
+  sedimentary strata. In this way Moro carried the history of the
+  stratified rocks beyond the time of the Flood back to the Creation,
+  which was supposed to have been some 1600 years earlier; and he
+  brought it down to the present day, when fresh sedimentary deposits
+  are continually accumulating. He thus incurred no censure from the
+  ecclesiastical guardians of the faith, and he succeeded in attracting
+  increased public attention to the problems of geology. The influence
+  of his teaching, however, was subsequently in great part due to the
+  Carmelite friar Generelli, who published an eloquent exposition of
+  Moro's views.
+
+  _The Cosmogonists and Theories of the Earth._--While in Italy
+  substantial progress was made in collecting information regarding the
+  fossiliferous formations of that country, and in forming conclusions
+  concerning them based upon more or less accurate observations, the
+  tendency to mere fanciful speculation, which could not be wholly
+  repressed in any country, reached a remarkable extravagance in
+  England. In proportion as materials were yet lacking from which to
+  construct a history of the evolution of our planet in accordance with
+  the teaching of the church, imagination supplied the place of
+  ascertained fact, and there appeared during the last twenty years of
+  the 18th century a group of English cosmogonists, who, by the
+  sensational character of their speculations, aroused general attention
+  both in Britain and on the continent. It may be doubted, however,
+  whether the effect of their writings was not to hinder the advance of
+  true science by diverting men from the observation of nature into
+  barren controversy over unrealities. It is not needful here to do more
+  than mention the names of Thomas Burnet, whose _Sacred Theory of the
+  Earth_ appeared in 1681, and William Whiston, whose New Theory of the
+  Earth was published in 1696. Hardly less fanciful than these writers,
+  though his practical acquaintance with rocks and fossils was
+  infinitely greater, was John Woodward, whose _Essay towards a Natural
+  History of the Earth_ dates from 1695. More important as a
+  contribution to science was the catalogue of the large collection of
+  fossils, which he had made from the rocks of England and which he
+  bequeathed to the university of Cambridge. This catalogue appeared in
+  1728-1729 with the title of _An attempt towards a Natural History of
+  the Fossils of England_.
+
+
+    Descartes.
+
+  A striking contrast to these cosmogonists is furnished by another
+  group, which arose in France and Germany, and gave to the world the
+  first rational ideas concerning the probable primeval evolution of our
+  globe. The earliest of these pioneers was the illustrious philosopher
+  Rene Descartes (1596-1650). He propounded a scheme of cosmical
+  development in which he represented the earth, like the other planets,
+  to have been originally a mass of glowing material like the sun, and
+  to have gradually cooled on the outside, while still retaining an
+  incandescent, self-luminous nucleus. Yet with this noble conception,
+  which modern science has accepted, Descartes could not shake himself
+  free from the time-honoured error in regard to the origin of volcanic
+  action. He thought that certain exhalations within the earth condense
+  into oil, which, when in violent motion, enters into the subterranean
+  cavities, where it passes into a kind of smoke. This smoke is from
+  time to time ignited by a spark of fire and, pressing violently
+  against its containing walls, gives rise to earthquakes. If the flame
+  breaks through to the surface at the top of a mountain, it may escape
+  with enormous energy, hurling forth much earth mingled with sulphur or
+  bitumen, and thus producing a volcano. The mountain might burn for a
+  long time until at last its store of fuel in the shape of sulphur or
+  bitumen would be exhausted. Not only did the philosopher refrain from
+  availing himself of the high internal temperature of the globe as the
+  source of volcanic energy, he even did not make use of it as the cause
+  of the ignition of his supposed internal fuel, but speculated on the
+  kindling of the subterranean fires by the spirits or gases setting
+  fire to the exhalations, or by the fall of masses of rock and the
+  sparks produced by their friction or percussion.
+
+
+    Leibnitz.
+
+  The ideas of Descartes regarding planetary evolution were enlarged and
+  made more definite by Wilhelm Gottfried Leibnitz (1646-1716), whose
+  teaching has largely influenced all subsequent speculation on the
+  subject. In his great tract, the _Protogaea_ (published in 1749,
+  thirty-three years after his death), he traced the probable passage of
+  our earth from an original condition of incandescent vapour into that
+  of a smooth molten globe, which, by continuous cooling, acquired an
+  external solid crust and rugose surface. He thought that the more
+  ancient rocks, such as granite and gneiss, might be portions of the
+  earliest outer crust; and that as the external solidification
+  advanced, immense subterranean cavities were left which were filled
+  with air and water. By the collapse of the roofs of these caverns,
+  valleys might be originated at the surface, while the solid
+  intervening walls would remain in place and form mountains. By the
+  disruption of the crust, enormous bodies of water were launched over
+  the surface of the earth, which swept vast quantities of sediment
+  together, and thus gave rise to sedimentary deposits. After many
+  vicissitudes of this kind, the terrestrial forces calmed down, and a
+  more stable condition of things was established.
+
+  An important feature in the cosmogony of Leibnitz is the prominent
+  place which he assigned to organic remains in the stratified rocks of
+  the crust. Ridiculing the foolish attempts to account for the presence
+  of these objects by calling them "sports of nature," he showed that
+  they are to be regarded as historical monuments; and he adduced a
+  number of instances wherein successive platforms of strata, containing
+  organic remains, bear witness to a series of advances and retreats of
+  the sea. He recognized that some of the fossils appeared to have
+  nothing like them in the living world of to-day, but some analogous
+  forms might yet be found, he thought, in still unexplored parts of the
+  earth; and even if no living representatives should ever be
+  discovered, many types of animals might have undergone transformation
+  during the great changes which had affected the surface of the earth.
+  In spite of his clear realization of the vast store of potential
+  energy residing within the highly heated interior of the earth,
+  Leibnitz continued to regard volcanic action as due to the combustion
+  of inflammable substances enclosed within the terrestrial crust, such
+  as stone-coal, naphtha and sulphur.
+
+
+    Buffon.
+
+  Appealing to a much wider public than Descartes or Leibnitz, and
+  basing his speculations on a wider acquaintance with the organic and
+  inorganic realms of nature, G.L.L. de Buffon (1707-1788) was
+  undoubtedly one of the most influential forces that in Europe guided
+  the growth of geological ideas during the 18th century. He published
+  in 1749 a _Theory of the Earth_, in which he adopted views similar to
+  those of Descartes and Leibnitz as to planetary evolution; but though
+  he realized the importance of fossils as records of former conditions
+  of the earth's surface, he accounted for them by supposing that they
+  had been deposited from a universal ocean, a large part of which had
+  subsequently been engulfed into caverns in the interior of the globe.
+  Thirty years later, after having laboured with skill and enthusiasm in
+  all branches of natural history, he published another work, his famous
+  _Epoques de la nature_ (1778), which is specially remarkable as the
+  first attempt to deal with the history of the earth in a chronological
+  manner, and to compute, on a basis of experiment, the antiquity of the
+  several stages of this history. His experiments were made with globes
+  of cast iron, and could not have yielded results of any value for his
+  purpose; but in so far as his calculations were not mere random
+  guesses but had some kind of foundation on experiment, they deserve
+  respectful recognition. He divided the history of our earth into six
+  periods of unequal duration, the whole comprising a period of some
+  70,000 or 75,000 years. He supposed that the stage of incandescence,
+  before the globe had consolidated to the centre, lasted 2936 years,
+  and that about 35,000 years elapsed before the surface had cooled
+  sufficiently to be touched, and therefore to be capable of supporting
+  living things. Terrestrial animal life, however, was not introduced
+  until 55,000 or 60,000 years after the beginning of the world or about
+  15,000 years before our time. Looking into the future, he foresaw
+  that, by continued refrigeration, our globe will eventually become
+  colder than ice, and this fair face of nature, with its manifold
+  varieties of plant and animal life, will perish after having existed
+  for 132,000 years.
+
+  Buffon's conception of the operation of the geological agents did not
+  become broader or more accurate in the interval between the appearance
+  of his two treatises. He still continued to believe in the lowering of
+  the ocean by subsidence into vast subterranean cavities, with a
+  consequent emergence of land. He still looked on volcanoes as due to
+  the burning of "pyritous and combustible stones," though he now called
+  in the co-operation of electricity. He calculated that the first
+  volcanoes could not arise until some 50,000 years after the beginning
+  of the world, by which time a sufficient extent of dense vegetation
+  had been buried in the earth to supply them with fuel. He appears to
+  have had but an imperfect acquaintance with the literature of his own
+  time. At least there can be little doubt that had he availed himself
+  of the labours of his own countryman, Jean Etienne Guettard
+  (1715-1786), of Giovanni Arduino (1714-1795) in Italy, and of Johann
+  Gottlob Lehmann (d. 1767) and George Christian Fuchsel (1722-1773) in
+  Germany, he would have been able to give to his "epochs" a more
+  definite succession of events and a greater correspondence with the
+  facts of nature.
+
+
+    James Hutton.
+
+  Among the writers of the 18th century, who formed philosophical
+  conceptions of the system of processes by which the life of our earth
+  as a habitable globe is carried on, a foremost place must be assigned
+  to James Hutton (1726-1797). Educated for the medical profession, he
+  studied at Edinburgh and at Paris, and took his doctor's degree at
+  Leiden. But having inherited a small landed property in Berwickshire,
+  he took to agriculture, and after putting his land into excellent
+  order, let his farm and betook himself to Edinburgh, there to gratify
+  the scientific tastes which he had developed early in life. He had
+  been more especially led to study minerals and rocks, and to meditate
+  on the problems which they suggest as to the constitution and history
+  of the earth. His journeys in Britain and on the continent of Europe
+  had furnished him with material for reflection; and he had gradually
+  evolved a system or theory in which all the scattered facts could be
+  arranged so as to show their mutual dependence and their place in the
+  orderly mechanism of the world. He used to discuss his views with one
+  or two of his friends, but refrained from publishing them to the world
+  until, on the foundation of the Royal Society of Edinburgh, he
+  communicated an outline of his doctrine to that learned body in 1785.
+  Some years later he expanded this first essay into a larger work in
+  two volumes, which were published in 1795 with the title of _Theory of
+  the Earth, with Proofs and Illustrations_.
+
+
+    John Playfair.
+
+  Hutton's teaching has exercised a profound influence on modern
+  geology. This influence, however, has arisen less from his own
+  writings than from the account of his doctrines given by his friend
+  John Playfair in the classic work entitled _Illustrations of the
+  Huttonian Theory_, published in 1802. Hutton wrote in so prolix and
+  obscure a style as rather to repel than attract readers. Playfair, on
+  the other hand, expressed himself in such clear and graceful language
+  as to command general attention, and to gain wide acceptance for his
+  master's views. Unlike the older cosmogonists, Hutton refrained from
+  trying to explain the origin of things, and from speculations as to
+  what might possibly have been the early history of our globe. He
+  determined from the outset to interpret the past by what can be seen
+  to be the present order of nature; and he refused to admit the
+  operation of causes which cannot be shown to be part of the actual
+  terrestrial system. Like other observers who had preceded him, he
+  recognized in the various rocks composing the dry land evidence of
+  former geographical conditions very different from those which now
+  prevail. He saw that the vast majority of rocks consist of hardened
+  sediments and must have been deposited in the sea. He could
+  distinguish among them an older or Primary series, and a younger or
+  Secondary series; and did not dispute the existence of a Tertiary
+  series claimed by Peter Simon Pallas (1741-1811). He believed that
+  these various aqueous accumulations had been consolidated by
+  subterranean heat, that the oldest and lowest rocks had suffered most
+  from this action, that into these more deep-seated masses subsequent
+  veins and larger bodies of molten matter were injected from below, and
+  thus that what was originally loose detritus eventually became changed
+  in such crystalline schists as are now found in mountain-chains. In
+  the course of these terrestrial revolutions sedimentary strata,
+  originally more or less nearly horizontal, have been pushed upward,
+  dislocated, crumpled, placed on end, and even elevated to form ranges
+  of lofty mountains. Hutton looked upon these disturbances as due to
+  the expansive power of subterranean heat; but he did not attempt to
+  sketch the mechanism of the process, and he expressly declined to
+  offer any conjecture as to how the land so elevated remains in that
+  position. He thought that the interior of our planet may "be a fluid
+  mass, melted, but unchanged by the action of heat"; and, far from
+  connecting volcanoes with the combustion of inflammable substances, as
+  had been the prevalent belief for so many centuries, he looked upon
+  them as a beneficent provision of "spiracles to the subterranean
+  furnace, in order to prevent the unnecessary elevation of land and
+  fatal effects of earthquakes."
+
+  A distinguishing feature of the Huttonian philosophy is to be seen in
+  the breadth of its conceptions regarding the geological operations
+  continually in progress on the surface of the globe. Hutton saw that
+  the land is undergoing a ceaseless process of degradation, through the
+  influence of the air, frost, rain, rivers and the sea, and that in
+  course of time, if no countervailing agency should intervene, the
+  whole of the dry land will be washed away into the sea. But he also
+  perceived that this universal erosion is not everywhere carried on at
+  the same rate; that it is specially active along the channels of
+  torrents and rivers, and that, owing to this difference these channels
+  are gradually deepened and widened, until the complicated
+  valley-system of a country is carved out. He recognized that the
+  detritus worn away from the land must be spread out over the floor of
+  the sea, so as to form there strata similar to those that compose most
+  of the dry land. As he could detect in the structure of land
+  convincing evidence that former sea floors had been elevated to form
+  the continents and islands of to-day, he could look forward to future
+  ages, when the same subterranean agency which had raised up the
+  present land would again be employed to uplift the bed of the existing
+  ocean, thus to renew the surface of our earth as a habitable globe,
+  and to start a fresh cycle of erosion and deposition.
+
+
+    Lamarck.
+
+  Though Hutton was not unaware that organic remains abound in many of
+  the stratified rocks, he left them out of consideration in the
+  elaboration of his theory. It was otherwise with one of his French
+  contemporaries, the illustrious J.B. Lamarck (1744-1829), who, after
+  having attained great eminence as a botanist, turned to zoology when
+  he was nearly fifty years of age, and before long rose to even greater
+  distinction in that department of science. His share in the
+  classification and description of the mollusca and in founding
+  invertebrate palaeontology, his theory of organic evolution and his
+  philosophical treatment of many biological questions have been tardily
+  recognized, but his contributions to geology have been less generally
+  acknowledged. When he accepted the "professorship of zoology; of
+  insects, of worms and of microscopic animals" at the Museum of Natural
+  History, Paris, in 1793, he at once entered with characteristic ardour
+  and capacity into the new field of research then opened to him. In
+  dealing with the mollusca he considered not merely the living but also
+  the extinct forms, especially the abundant, varied and well-preserved
+  genera and species furnished by the Tertiary deposits of the Paris
+  basin, of which he published descriptions and plates that proved of
+  essential service in the stratigraphical work of Cuvier and Alexandre
+  Brongniart (1770-1847). His labours among these relics of ancient seas
+  and lakes led him to ponder over the past history of the globe, and as
+  he was seldom dilatory in making known the opinions he had formed, he
+  communicated some of his conclusions to the National Institute in
+  1799. These, including a further elaboration of his views, he
+  published in 1802 in a small volume entitled Hydrogeologie.
+
+  This treatise, though it did not reach a second edition and has never
+  been reprinted, deserves an honourable place in geological literature.
+  Its object, the author states, was to present some important and novel
+  considerations, which he thought should form the basis of a true
+  theory of the earth. He entirely agreed with the doctrine of the
+  subaerial degradation of the land and the erosion of valleys by
+  running water. Not even Playfair could have stated this doctrine more
+  emphatically, and it is worthy of notice that Playfair's
+  _Illustrations of the Huttonian Theory_ appeared in the same year with
+  Lamarck's book. The French naturalist, however, carried his
+  conclusions so far as to take no account of any great movements of the
+  terrestrial crust, which might have produced or modified the main
+  physical features of the surface of the globe. He thought that all
+  mountains, except such as were thrown up by volcanic agency or local
+  accidents, have been cut out of plains, the original surfaces of which
+  are indicated by the crests and summits of these elevations.
+
+  Lamarck, in reflecting upon the wide diffusion of fossil shells and
+  the great height above the sea at which they are found, conceived the
+  extraordinary idea that the ocean basin has been scoured out by the
+  sea, and that, by an impulse communicated to the waters through the
+  influence chiefly of the moon, the sea is slowly eating away the
+  eastern margins of the continents, and throwing up detritus on their
+  western coasts, and is thus gradually shifting its basin round the
+  globe. He would not admit the operation of cataclysms; but insisted as
+  strongly as Hutton on the continuity of natural processes, and on the
+  necessity of explaining former changes of the earth's surface by
+  causes which can still be seen to be in operation. As might be
+  anticipated from his previous studies, he brought living things and
+  their remains into the forefront of his theory of the earth. He looked
+  upon fossils as one of the chief means of comprehending the
+  revolutions which the surface of the earth has undergone; and in his
+  little volume he again and again dwells on the vast antiquity to which
+  these revolutions bear witness. He acutely argues, from the condition
+  of fossil shells, that they must have lived and died where their
+  remains are now found.
+
+  In the last part of his treatise Lamarck advances some peculiar
+  opinions in physics and chemistry, which he had broached eighteen
+  years before, but which had met with no acceptance among the
+  scientific men of his time. He believed that the tendency of all
+  compound substances is to decay, and thereby to be resolved into their
+  component constituents. Yet he saw that the visible crust of the earth
+  consists almost wholly of compound bodies. He therefore set himself to
+  solve the problem thus presented. Perceiving that the biological
+  action of living organisms is constantly forming combinations of
+  matter, which would never have otherwise come into existence, he
+  proceeded to draw the extraordinary conclusion that the action of
+  plant and animal life (the _Pouvoir de la vie_) upon the inorganic
+  world is so universal and so potent, that the rocks and minerals which
+  form the outer part of the earth's crust are all, without exception,
+  the result of the operations of once living bodies. Though this
+  sweeping deduction must be allowed to detract from the value of
+  Lamarck's work, there can be no doubt that he realized, more fully
+  than any one had done before him, the efficacy of plants and animals
+  as agents of geological change.
+
+
+    Cuvier.
+
+  The last notable contributor to the cosmological literature of geology
+  was another illustrious Frenchman, the comparative anatomist Cuvier
+  (1769-1832). He was contemporary with Lamarck, but of a very different
+  type of mind. The brilliance of his speculations, and the charm with
+  which he expounded them, early gained for him a prominent place in the
+  society of Paris. He too was drawn by his zoological studies to
+  investigate fossil organic remains, and to consider the former
+  conditions of the earth's surface, of which they are memorials. It was
+  among the vertebrate organisms of the Paris basin that he found his
+  chief material, and from them that he prepared the memoirs which led
+  to him being regarded as the founder of vertebrate palaeontology. But
+  beyond their biological interest, they awakened in him a keen desire
+  to ascertain the character and sequence of the geographical
+  revolutions to which they bear witness. He approached the subject from
+  an opposite and less philosophical point of view than that of Lamarck,
+  coming to it with certain preconceived notions, which affected all
+  his subsequent writings. While Lamarck was by instinct an
+  evolutionist, who sought to trace in the history of the past the
+  operation of the same natural processes as are still at work, Cuvier,
+  on the other hand, was a catastrophist, who invoked a succession of
+  vast cataclysms to account for the interruptions in the continuity of
+  the geological record.
+
+  In a preliminary _Discourse_ prefixed to his _Recherches sur les
+  ossemens fossiles_ (1821) Cuvier gave an outline of what he conceived
+  to have been the past history of our globe, so far as he had been able
+  to comprehend it from his investigations of the Tertiary formations of
+  France. He believed that in that history evidence can be recognized of
+  the occurrence of many sudden and disastrous revolutions, which, to
+  judge from their effects on the animal life of the time, must have
+  exceeded in violence anything we can conceive at the present day, and
+  must have been brought about by other agencies than those which are
+  now in operation. Yet, in spite of these catastrophes, he saw that
+  there has been an upward progress in the animal forms inhabiting the
+  globe, until the series ended in the advent of man. He could not,
+  however, find any evidence that one species has been developed from
+  another, for in that case there should have been traces of
+  intermediate forms among the stratified formations, where he affirmed
+  that they had never been found. A prominent position in the
+  _Discourse_ is given to a strenuous argument to disprove the alleged
+  antiquity of some nations, and to show that the last great catastrophe
+  occurred not more than some 5000 or 6000 years ago. Cuvier thus linked
+  himself with those who in previous generations had contended for the
+  efficacy of the Deluge. But his researches among fossil animals had
+  given him a far wider outlook into the geological past, and had opened
+  up to him a succession of deeply interesting problems in the history
+  of life upon the earth, which, though he had not himself material for
+  their solution, he could foresee would be cleared up in the future.
+
+  _Gradual Shaping of Geology into a Distinct Branch of Science._--It
+  will be seen from the foregoing historical sketch that it was only
+  after the lapse of long centuries, and from the labours of many
+  successive generations of observers and writers, that what we now know
+  as the science of geology came to be recognized as a distinct
+  department of natural knowledge, founded upon careful and extended
+  study of the structure of the earth, and upon observation of the
+  natural processes, which are now at work in changing the earth's
+  surface. The term "geology,"[1] descriptive of this branch of the
+  investigation of nature, was not proposed until the last quarter of
+  the 18th century by Jean Andre De Luc (1727-1817) and Horace Benedict
+  De Saussure (1740-1749). But the science was then in a markedly
+  half-formed condition, theoretical speculation still in large part
+  supplying the place of deductions from a detailed examination of
+  actual fact. In 1807 a few enterprising spirits founded the Geological
+  Society of London for the special purpose of counteracting the
+  prevalent tendency and confining their intention "to investigate the
+  mineral structure of the earth." The cosmogonists and framers of
+  Theories of the Earth were succeeded by other schools of thought. The
+  Catastrophists saw in the composition of the crust of the earth
+  distinct evidence that the forces of nature were once much more
+  stupendous in their operation than they now are, and that they had
+  from time to time devastated the earth's surface; extirpating the
+  races of plants and animals, and preparing the ground for new
+  creations of organized life. Then came the Uniformitarians, who,
+  pushing the doctrines of Hutton to an extreme which he did not
+  propose, saw no evidence that the activity of the various geological
+  causes has ever seriously differed from what it is at present. They
+  were inclined to disbelieve that the stratified formations of the
+  earth's crust furnish conclusive evidence of a gradual progression,
+  from simple types of life in the oldest strata to the most highly
+  developed forms in the youngest; and saw no reason why remains of the
+  higher vertebrates should not be met with among the Palaeozoic
+  formations. Sir Charles Lyell (1797-1875) was the great leader of this
+  school. His admirably clear and philosophical presentations of
+  geological facts which, with unwearied industry, he collected from the
+  writings of observers in all parts of the world, impressed his views
+  upon the whole English-speaking world, and gave to geological science
+  a coherence and interest which largely accelerated its progress. In
+  his later years, however, he frankly accepted the views of Darwin in
+  regard to the progressive character of the geological record.
+
+  The youngest of the schools of geological thought is that of the
+  Evolutionists. Pointing to the whole body of evidence from inorganic
+  and organic nature, they maintain that the history of our planet has
+  been one of continual and unbroken development from the earliest
+  cosmical beginnings down to the present time, and that the crust of
+  the earth contains an abundant, though incomplete, record of the
+  successive stages through which the plant and animal kingdoms have
+  reached their existing organization. The publication of Darwin's
+  _Origin of Species_ in 1859, in which evolution was made the key to
+  the history of the animal and vegetable kingdoms, produced an
+  extraordinary revolution in geological opinion. The older schools of
+  thought rapidly died out, and evolution became the recognized creed of
+  geologists all over the world.
+
+
+    Werner.
+
+  _Development of Opinion regarding Igneous Rocks._--So long as the idea
+  prevailed that volcanoes are caused by the combustion of inflammable
+  substances underground, there could be no rational conception of
+  volcanic action and its products. Even so late as the middle of the
+  18th century, as above remarked, such a good observer as Lazzaro Moro
+  drew so little distinction between volcanic and other rocks that he
+  could believe the fossiliferous formations to have been mainly formed
+  of materials ejected from eruptive vents. After his time the notion
+  continued to prevail that all the rocks which form the dry land were
+  laid down under water. Even streams of lava, which were seen to flow
+  from an active crater, were regarded only as portions of sedimentary
+  or other rocks, which had been melted by the fervent heat of the
+  burning inflammable materials that had been kindled underground. In
+  spite of the speculations of Descartes and Leibnitz, it was not yet
+  generally comprehended that there exists beneath the terrestrial crust
+  a molten magma, which, from time to time, has been injected into that
+  crust, and has pierced through it, so as to escape at the surface with
+  all the energy of an active volcano. What we now recognize to be
+  memorials of these former injections and propulsions were all
+  confounded with the rocks of unquestionably aqueous origin. The last
+  great teacher by whom these antiquated doctrines were formulated into
+  a system and promulgated to the world was Abraham Gottlob Werner
+  (1749-1815), the most illustrious German mineralogist and geognost of
+  the second half of the 18th century. While still under twenty-six
+  years of age, he was appointed teacher of mining and mineralogy at the
+  Mining Academy of Freiberg in Saxony--a post which he continued to
+  fill up to the end of his life. Possessed of great enthusiasm for his
+  subject, clear, methodical and eloquent in his exposition of it, he
+  soon drew around him men from all parts of the world, who repaired to
+  study under the great oracle of what he called geognosy (Gr. [Greek:
+  ge], the earth, [Greek: gnosis], knowledge) or earth-knowledge.
+  Reviving doctrines that had been current long before his time, he
+  taught that the globe was once completely surrounded with an ocean,
+  from which the rocks of the earth's crust were deposited as chemical
+  precipitates, in a certain definite order over the whole planet. Among
+  these "universal formations" of aqueous origin were included many
+  rocks, which have long been recognized to have been once molten, and
+  to have risen from below into the upper parts of the terrestrial
+  crust. Werner, following the old tradition, looked upon volcanoes as
+  modern features in the history of the planet, which could not have
+  come into existence until a sufficient amount of vegetation had been
+  buried to furnish fuel for their maintenance. Hence he attached but
+  little importance to them, and did not include in his system of rocks
+  any division of volcanic or igneous materials. From the predominant
+  part assigned by him to the sea in the accumulation of the materials
+  of the visible part of the earth, Werner and his school were known as
+  "Neptunists."
+
+
+    Origin of basalt.
+
+  But many years before the Saxon professor began to teach, clear
+  evidence had been produced from central France that basalt, one of the
+  rocks claimed by him as a chemical precipitate and a universal
+  formation, is a lava which has been poured out in a molten state at
+  various widely separated periods of time and at many different places.
+  So far back as 1752 J.E. Guettard (1715-1786) had shown that the
+  basaltic rocks of Auvergne are true lavas, which have flowed out in
+  streams from groups of once active cones. Eleven years later the
+  observation was confirmed and greatly extended by Nicholas Desmarest
+  (1725-1815), who, during a long course of years, worked out and mapped
+  the complicated volcanic records of that interesting region, and
+  demonstrated to all who were willing impartially to examine the
+  evidence the true volcanic nature of basalt. These views found
+  acceptance from some observers, but they were vehemently opposed by
+  the followers of Werner, who, by the force of his genius, made his
+  theoretical conceptions predominate all over Europe. The controversy
+  as to the origin of basalt was waged with great vigour during the
+  later decades of the 18th century. Desmarest took no part in it. He
+  had accumulated such conclusive proof of the correctness of his
+  deductions, and had so fully expounded the clearness of the evidence
+  in their favour furnished by the region of Auvergne, that, when any
+  one came to consult him on the subject, he contented himself with
+  giving the advice to "go and see." While the debate was in progress on
+  the continent, the subject was approached from a new and independent
+  point of view by Hutton in Scotland. This illustrious philosopher, as
+  already stated, realized the importance of the internal heat of the
+  globe in consolidating the sedimentary rocks, and believed that molten
+  material from the earth's interior has been protruded from below into
+  the overlying crust. Some of the material thus injected could be
+  recognized, he thought, in granite and in the various dark massive
+  rocks which, known in Scotland under the name of "whinstone," were
+  afterwards called "Trap," and are now grouped under various names,
+  such as basalt, dolerite and diorite. So important a share did Hutton
+  thus assign to the internal heat in the geological evolution of the
+  planet, that he and those who adopted the same opinions were styled
+  "Plutonists," or, especially where they concerned themselves with the
+  volcanic origin of basalt, "Vulcanists." The geological world was thus
+  divided into two hostile camps, that of the Neptunists or Wernerians,
+  and that of the Plutonists, Vulcanists or Huttonians.
+
+  After many years of futile controversy the first serious weakening of
+  the position of the dominant Neptunist school arose from the defection
+  of some of the most prominent of Werner's pupils. In particular Jean
+  Francois D'Aubuisson de Voisins (1769-1819), who had written a
+  treatise on the aqueous origin of the basalts of Saxony, went
+  afterwards to Auvergne, where he was speedily a convert to the views
+  expounded by Desmarest as to the volcanic nature of basalt. Having
+  thus to relinquish one of the fundamental articles of the Freiberg
+  faith, he was subsequently led to modify his adherence to others
+  until, as he himself confessed, his views came almost wholly to agree
+  with those of Hutton. Not less complete, and even more important, was
+  the conversion of the great Leopold von Buch (1774-1853). He, too, was
+  trained by Werner himself, and proved to be the most illustrious pupil
+  of the Saxon professor. Full of admiration for the Neptunism in which
+  he had been reared, he, in his earliest separate work, maintained the
+  aqueous origin of basalt, and contrasted the wide field opened up to
+  the spirit of observation by his master's teaching with the narrower
+  outlook offered by "the volcanic theory." But a little further
+  acquaintance with the facts of nature led Von Buch also to abandon his
+  earlier prepossessions. It was a personal visit to the volcanic region
+  of Auvergne that first opened his eyes, and led him to recant what he
+  had believed and written about basalt. But the abandonment of so
+  essential a portion of the Wernerian creed prepared the way for
+  further relinquishments. When a few years later he went to Norway and
+  found to his astonishment that granite, which he had been taught to
+  regard as the oldest chemical precipitate from the universal ocean,
+  could there be seen to have broken through and metamorphosed
+  fossiliferous limestones, and to have sent veins into them, his faith
+  in Werner's order of the succession of the rocks in the earth's crust
+  received a further momentous shock. While one after another of the
+  Freiberg doctrines crumbled away before him, he was now able to
+  interrogate nature on a wider field than the narrow limits of Saxony,
+  and he was thus gradually led to embrace the tenets of the opposite
+  school. His commanding position, as the most accomplished geologist on
+  the continent, gave great importance to his recantation of the
+  Neptunist creed. His defection indeed was the severest blow that this
+  creed had yet sustained. It may be said to have rung the knell of
+  Wernerianism, which thereafter rapidly declined in influence, while
+  Plutonism came steadily to the front, where it has ever since
+  remained.
+
+  Although Desmarest had traced in Auvergne a long succession of
+  volcanic eruptions, of which the oldest went back to a remote period
+  of time, and although he had shown that this succession, coupled with
+  the records of contemporaneous denudation, might be used in defining
+  epochs of geological history, it was not until many years after his
+  day that volcanic action came to be recognized as a normal part of the
+  mechanism of our globe, which had been in operation from the remotest
+  past, and which had left numerous records among the rocks of the
+  terrestrial crust. During the progress of the controversy between the
+  two great opposing factions in the later portion of the 18th and the
+  first three decades of the 19th century, those who espoused the
+  Vulcanist cause were intent on proving that certain rocks, which are
+  intercalated among the stratified formations and which were claimed by
+  the Neptunists as obviously formed by water, are nevertheless of truly
+  igneous origin. These observers fixed their eyes on the evidence that
+  the material of such rocks, instead of having been deposited from
+  aqueous solution, had once been actually molten, and had in that
+  condition been thrust between the strata, had enveloped portions of
+  them, and had indurated or otherwise altered them. They spoke of these
+  masses as "unerupted lavas"; and undoubtedly in innumerable instances
+  they were right. But their zeal to establish an intrusive origin led
+  them to overlook the proofs that some intercalated sheets of igneous
+  material had not been injected into the strata, but had been poured
+  out at the surface as truly volcanic discharges, and therefore
+  belonged to the ancient periods represented by the strata between
+  which they are interposed. It may readily be supposed that any proofs
+  of the contemporaneous intercalation of such sheets would be eagerly
+  seized upon by the Neptunists in favour of their aqueous theory. The
+  influence of the ancient belief that "burning mountains" could only
+  rise from the combustion of subterranean inflammable materials
+  extended even into the ranks of the Vulcanists, so far at least as to
+  lead to a general acquiescence in the assumption that volcanoes
+  appeared to belong to a late phase in the history of the planet. It
+  was not until after considerable progress had been made in determining
+  the palaeontological distinctions and order of succession of the
+  stratified formations of the earth's crust that it became possible to
+  trace among these formations a succession of volcanic episodes which
+  were contemporaneous with them. In no part of the world has an ampler
+  record of such episodes been preserved than in the British Isles. It
+  was natural, therefore, that the subject should there receive most
+  attention. As far back as 1820 Ami Boue (1794-1881) showed that the
+  Old Red Sandstone of Scotland includes a great series of volcanic
+  rocks, and that other rocks of volcanic origin are associated with the
+  Carboniferous formations. H.T. de la Beche (1796-1855) afterwards
+  traced proofs of contemporaneous eruptions among the Devonian rocks of
+  the south-west of England. Adam Sedgwick (1785-1873) showed, first in
+  the Lake District, and afterwards in North Wales, the presence of
+  abundant volcanic sheets among the oldest divisions of the Palaeozoic
+  series; while Roderick Impey Murchison (1792-1871) made similar
+  discoveries among the Lower Silurian rocks. From the time of these
+  pioneers the volcanic history of the country has been worked out by
+  many observers until it is now known with a fulness as yet unattained
+  in any other region.
+
+  _Growth of Opinion regarding Earthquakes._--We have seen how crude
+  were the conceptions of the ancients regarding the causes of volcanic
+  action, and that they connected volcanoes and earthquakes as results
+  of the commotion of wind imprisoned within subterranean caverns and
+  passages. One of the earliest treatises, in which the phenomena of
+  terrestrial movements were discussed in the spirit of modern science,
+  was the posthumous collection of papers by Robert Hooke (1635-1703),
+  entitled _Lectures and Discourses of Earthquakes and Subterranean
+  Eruptions_, where the probable agency of earthquakes in upheaving and
+  depressing land is fully considered, but without any definite
+  pronouncement as to the author's conception of its origin. Hooke still
+  associated earthquakes with volcanic action, and connected both with
+  what he called "the general congregation of sulphurous subterraneous
+  vapours." He conceived that some kind of "fermentation" takes place
+  within the earth, and that the materials which catch fire and give
+  rise to eruptions or earthquakes are analogous to those that
+  constitute gunpowder. The first essay wherein earthquakes are treated
+  from the modern point of view as the results of a shock that sends
+  waves through the crust of the earth was written by the Rev. John
+  Michell, and communicated to the Royal Society in the year 1760. Still
+  under the old misconception that volcanoes are due to the combustion
+  of inflammable materials, which he thought might be set on fire by the
+  spontaneous combustion of pyritous strata, he supposed that, by the
+  sudden access of large bodies of water to these subterranean fires,
+  vapour is produced in such quantity and with such force as to give
+  rise to the shock. From the centre of origin of this shock waves, he
+  thought, are propagated through the earth, which are largest at the
+  start and gradually diminish as they travel outwards. By drawing lines
+  at different places in the direction of the track of these waves, he
+  believed that the place of common intersection of these lines would be
+  nearly the centre of the disturbance. In this way he showed that the
+  great Lisbon earthquake of 1755 had its focus under the Atlantic,
+  somewhere between the latitudes of Lisbon and Oporto, and he estimated
+  that the depth at which it originated could not be much less than 1
+  m., and probably did not exceed 3 m. Michell, however, misconceived
+  the character of the waves which he described, seeing that he believed
+  them to be due to the actual propagation of the vapour itself
+  underneath the surface of the earth. A century had almost passed after
+  the date of his essay before modern scientific methods of observation
+  and the use of recording instruments began to be applied to the study
+  of earthquake phenomena. In 1846 Robert Mallet (1810-1881) published
+  an important paper "On the Dynamics of Earthquakes" in the
+  _Transactions of the Royal Irish Academy_. From that time onward he
+  continued to devote his energies to the investigation, studying the
+  effects of the Calabrian earthquake of 1857, experimenting on the
+  transmission of waves of shock through various materials, caused by
+  exploding charges of gunpowder, and collecting all the information to
+  be obtained on the subject. His writings, and especially his work in
+  two volumes on _The First Principles of Observational Seismology_,
+  must be regarded as having laid the foundations of this branch of
+  modern geology (see EARTHQUAKE; SEISMOMETER).
+
+  _History of the Evolution of Stratigraphical Geology._--Men had long
+  been familiar with the evidence that the present dry land once lay
+  under the sea, before they began to realize that the rocks, of which
+  the land consists, contain a record of many alternations of land and
+  sea, and relics of a long succession of plants and animals from early
+  and simple types up to the manifold and complex forms of to-day. In
+  countries where coal-mining had been prosecuted for generations, it
+  had been recognized that the rocks consist of strata superposed on
+  each other in a definite order, which was found to extend over the
+  whole of a district. As far back as 1719 John Strachey drew attention
+  to this fact in a communication published in the _Philosophical
+  Transactions_. John Michell (1760), in the paper on earthquakes
+  already cited, showed that he had acquired a clear understanding of
+  the order of succession among stratified formations, and perceived
+  that to disturbances of the terrestrial crust must be ascribed the
+  fact that the lower or older and more inclined strata form the
+  mountains, while the younger and more horizontal strata are spread
+  over the plains.
+
+  In Italy G. Arduino (1713-1795) classified the rocks in the north of
+  the peninsula as Primitive, Secondary, Tertiary and Volcanic. A
+  similar threefold order was announced for the Harz and Erzgebirge by
+  J.G. Lehmann in 1756. He recognized in that region an ancient series
+  of rocks in inclined or vertical strata, which rise to the tops of the
+  hills and descend to an unknown depth into the interior. These masses,
+  he thought, were contemporaneous with the making of the world. Next
+  came the Flotzgebirge, consisting of younger sediments, disposed in
+  flat or gently inclined sheets which overlie the first and more
+  disturbed series, and are full of petrified remains of plants and
+  animals. Lastly he included the mountains which have from time to time
+  been formed by local accidents. Still more advanced were the
+  conceptions of G.C. Fuchsel, who in the year 1762 published in Latin
+  _A History of the Earth and the Sea, based on a History of the
+  Mountains of Thuringia_; and in 1773, in German, a _Sketch of the most
+  Ancient History of the Earth and Man_. In these works he described the
+  stratigraphical relations and general characters of the various
+  geological formations in his little principality; and taking them as
+  indicative of a general order of succession, he traced what he
+  believed to have been a series of revolutions through which the earth
+  has passed. In interpreting this geological history, he laid great
+  stress on the evidence of the fossils contained in the rocks. He
+  recognized that the various formations differ from each other in their
+  enclosed organic remains, and that from these differences the
+  existence of former sea-bottoms and land surfaces can be determined.
+
+  The labours of these pioneers paved the way for the advent of Werner.
+  Though the system evolved by this teacher claimed to discard theory
+  and to be established on a basis of observed facts, it rested on a
+  succession of hypotheses, for which no better foundation could be
+  shown than the belief of their author in their validity. Starting from
+  the extremely limited stratigraphical range displayed in the
+  geological structure of Saxony, he took it as a type for the rest of
+  the globe, persuading himself and impressing upon his followers that
+  the rocks of that small kingdom were to be taken as examples of his
+  "universal formations." The oldest portion of the series, classed by
+  him as "Primitive," consisted of rocks which he maintained had been
+  deposited from chemical solution. Yet they included granite, gneiss,
+  basalt, porphyry and serpentine, which, even in his own day, were by
+  many observers correctly regarded as of igneous origin. A later group
+  of rocks, to which he gave the name of "Transition," comprised, in his
+  belief, partly chemical, partly mechanical sediments, and contained
+  the earliest fossil organic remains. A third group, for which he
+  reserved Lehmann's name "Flotz," was made up chiefly of mechanical
+  detritus, while youngest of all came the "Alluvial" series of loams,
+  clays, sands, gravels and peat. It was by the gradual subsidence of
+  the ocean that, as he believed, the general mass of the dry land
+  emerged, the first-formed rocks being left standing up, sometimes on
+  end, to form the mountains, while those of later date, less steeply
+  inclined, occupied successively lower levels down to the flat alluvial
+  accumulations of the plains. Neither Werner, nor any of his followers,
+  ventured to account for what became of the water as the sea-level
+  subsided, though, in despite of their antipathy to anything like
+  speculation, they could not help suggesting, as an answer to the
+  cogent arguments of their opponents, that "one of the celestial bodies
+  which sometimes approach near to the earth may have been able to
+  withdraw a portion of our atmosphere and of our ocean." Nor was any
+  attempt made to explain the extraordinary nature of the supposed
+  chemical precipitates of the universal ocean. The progress of inquiry
+  even in Werner's lifetime disproved some of the fundamental portions
+  of his system. Many of the chemical precipitates were shown to be
+  masses that had been erupted in a molten state from below. His order
+  of succession was found not to hold good; and though he tried to
+  readjust his sequence and to introduce into it modifications to suit
+  new facts, its inherent artificiality led to its speedy decline after
+  his death. It must be conceded, however, that the stress which he laid
+  upon the fact that the rocks of the earth's crust were deposited in a
+  definite order had an important influence in directing attention to
+  this subject, and in preparing the way for a more natural system,
+  based not on mere mineralogical characters, but having regard to the
+  organic remains, which were now being gathered in ever-increasing
+  numbers and variety from stratified formations of many different ages
+  and from all parts of the globe.
+
+  It was in France and in England that the foundations of stratigraphy,
+  based upon a knowledge of organic remains, were first successfully
+  laid. Abbe J.L. Giraud-Soulavie (1752-1813), in his _Histoire
+  naturelle de la France meridionale_, which appeared in seven volumes,
+  subdivided the limestones of Vivarais into five ages, each marked by a
+  distinct assemblage of shells. In the lowest strata, representing the
+  first age, none of the fossils were believed by him to have any living
+  representatives, and he called these rocks "Primordial." In the next
+  group a mingling of living with extinct forms was observable. The
+  third age was marked by the presence of shells of still existing
+  species. The strata of the fourth series were characterized by
+  carbonaceous shales or slates, containing remains of primordial
+  vegetation, and perhaps equivalents of the first three calcareous
+  series. The fifth age was marked by recent deposits containing remains
+  of terrestrial vegetation and of land animals. It is remarkable that
+  these sagacious conclusions should have been formed and published at a
+  time when the geologists of the Continent were engaged in the
+  controversy about the origin of basalt, or in disputes about the
+  character and stratigraphical position of the supposed universal
+  formations, and when the interest and importance of fossil organic
+  remains still remained unrecognized by the vast majority of the
+  combatants.
+
+  The rocks of the Paris basin display so clearly an orderly
+  arrangement, and are so distinguished for the variety and perfect
+  preservation of their enclosed organic remains, that they could not
+  fail to attract the early notice of observers. J. E. Guettard, G.F.
+  Rouelle (1703-1770), N. Desmarest, A.L. Lavoisier (1743-1794) and
+  others made observations in this interesting district. But it was
+  reserved for Cuvier (1769-1832) and A. Brongniart (1770-1847) to work
+  out the detailed succession of the Tertiary formations, and to show
+  how each of these is characterized by its own peculiar assemblage of
+  organic remains. The later progress of investigation has slightly
+  corrected and greatly amplified the tabular arrangement established by
+  these authors in 1808, but the broad outlines of the Tertiary
+  stratigraphy of the Paris basin remain still as Cuvier and Brongniart
+  left them. The most important subsequent change in the classification
+  of the Tertiary formations was made by Sir Charles Lyell, who,
+  conceiving in 1828 the idea of a classification of these rocks by
+  reference to their relative proportions of living and extinct species
+  of shells, established, in collaboration with G.P. Deshayes, the now
+  universally accepted divisions Eocene, Miocene and Pliocene.
+
+  Long before Cuvier and Brongniart published an account of their
+  researches, another observer had been at work among the Secondary
+  formations of the west of England, and had independently discovered
+  that the component members of these formations were each distinguished
+  by a peculiar group of organic remains; and that this distinction
+  could be used to discriminate them over all the region through which
+  he had traced them. The remarkable man who arrived at this
+  far-reaching generalization was William Smith (1769-1839), a land
+  surveyor who, in the prosecution of his professional business, found
+  opportunities of traversing a great part of England, and of putting
+  his deductions to the test. As the result of these journeys he
+  accumulated materials enough to enable him to produce a geological map
+  of the country, on which the distribution and succession of the rocks
+  were for the first time delineated. Smith's labours laid the
+  foundation of stratigraphical geology in England and he was styled
+  even in his lifetime the "Father of English geology." From his day
+  onward the significance of fossil organic remains gained rapidly
+  increasing recognition. Thus in England the outlines traced by him
+  among the Secondary and Tertiary formations were admirably filled in
+  by Thomas Webster (1773-1844); while the Cretaceous series was worked
+  out in still greater detail in the classic memoirs of William Henry
+  Fitton (1780-1861).
+
+  There was one stratigraphical domain, however, into which William
+  Smith did not enter. He traced his sequence of rocks down into the
+  Coal Measures, but contented himself with only a vague reference to
+  what lay underneath that formation. Though some of these underlying
+  rocks had in various countries yielded abundant fossils, they had
+  generally suffered so much from terrestrial disturbances, and their
+  order of succession was consequently often so much obscured throughout
+  western Europe, that they remained but little known for many years
+  after the stratigraphy of the Secondary and Tertiary series had been
+  established. At last in 1831 Murchison began to attack this _terra
+  incognita_ on the borders of South Wales, working into it from the Old
+  Red Sandstone, the stratigraphical position of which was well known.
+  In a few years he succeeded in demonstrating the existence of a
+  succession of formations, each distinguished by its own peculiar
+  assemblage of organic remains which were distinct from those in any of
+  the overlying strata. To these formations he gave the name of Silurian
+  (q.v.). From the key which his researches supplied, it was possible to
+  recognize in other countries the same order of formations and the same
+  sequence of fossils, so that, in the course of a few years,
+  representatives of the Silurian system were found far and wide over
+  the globe. While Murchison was thus engaged, Sedgwick devoted himself
+  to the more difficult task of unravelling the complicated structure of
+  North Wales. He eventually made out the order of the several
+  formations there, with their vast intercalations of volcanic material.
+  He named them the Cambrian system (q.v.), and found them to contain
+  fossils, which, however, lay for some time unexamined by him. He at
+  first believed, as Murchison also did, that his rocks were all older
+  than any part of the Silurian series. It was eventually discovered
+  that a portion of them was equivalent to the lower part of that
+  series. The oldest of Sedgwick's groups, containing distinctive
+  fossils, retain the name Cambrian, and are of high interest, as they
+  enclose the remains of the earliest faunas which are yet well known.
+  Sedgwick and Murchison rendered yet another signal service to
+  stratigraphical geology by establishing, in 1839, on a basis of
+  palaeontological evidence supplied by W. Lonsdale, the independence of
+  the Devonian system (q.v.).
+
+  For many years the rocks below the oldest fossiliferous deposits
+  received comparatively little attention. They were vaguely described
+  as the "crystalline schists" and were often referred to as parts of
+  the primeval crust in which no chronology was to be looked for. W.E.
+  Logan (1798-1875) led the way, in Canada, by establishing there
+  several vast series of rocks, partly of crystalline schists and
+  gneisses (Laurentian) and partly of slates and conglomerates
+  (Huronian). Later observers, both in Canada and the United States,
+  have greatly increased our knowledge of these rocks, and have shown
+  their structure to be much more complex than was at first supposed
+  (see ARCHEAN SYSTEM).
+
+  During the latter half of the 19th century the most important
+  development of stratigraphical geology was the detailed working out
+  and application of the principle of zonal classification to the
+  fossiliferous formations--that is, the determination of the sequence
+  and distribution of organic remains in these formations, and the
+  arrangement of the strata into zones, each of which is distinguished
+  by a peculiar assemblage of fossil species (see under Part VI.). The
+  zones are usually named after one especially characteristic species.
+  This system of classification was begun in Germany with reference to
+  the members of the Jurassic system (q.v.) by A. Oppel (1856-1858) and
+  F.A. von Quenstedt (1858), and it has since been extended through the
+  other Mesozoic formations. It has even been found to be applicable to
+  the Palaeozoic rocks, which are now subdivided into palaeontological
+  zones. In the Silurian system, for example, the graptolites have been
+  shown by C. Lapworth to furnish a useful basis for zonal subdivisions.
+  The lowest fossiliferous horizon in the Cambrian rocks of Europe and
+  North America is known as the _Olenellus_ zone, from the prominence in
+  it of that genus of trilobite.
+
+  Another conspicuous feature in the progress of stratigraphy during the
+  second half of the 19th century was displayed by the rise and rapid
+  development of what is known as Glacial geology. The various deposits
+  of "drift" spread over northern Europe, and the boulders scattered
+  across the surface of the plains had long attracted notice, and had
+  even found a place in popular legend and superstition. When men began
+  to examine them with a view to ascertain their origin, they were
+  naturally regarded as evidences of the Noachian deluge. The first
+  observer who drew attention to the smoothed and striated surfaces of
+  rock that underlie the Drifts was Hutton's friend, Sir James Hall, who
+  studied them in the lowlands of Scotland and referred them to the
+  action of great debacles of water, which, in the course of some
+  ancient terrestrial convulsion, had been launched across the face of
+  the country. Playfair, however, pointed out that the most potent
+  geological agents for the transportation of large blocks of stone are
+  the glaciers. But no one was then bold enough to connect the travelled
+  boulders with glaciers on the plains of Germany and of Britain. Yet
+  the transporting agency of ice was invoked in explanation of their
+  diffusion. It came to be the prevalent belief among the geologists of
+  the first half of the 19th century, that the fall of temperature,
+  indicated by the gradual increase in the number of northern species of
+  shells in the English Crag deposits, reached its climax during the
+  time of the Drift, and that much of the north and centre of Europe was
+  then submerged beneath a sea, across which floating icebergs and floes
+  transported the materials of the Drift and dropped the scattered
+  boulders. As the phenomena are well developed around the Alps, it was
+  necessary to suppose that the submergence involved the lowlands of the
+  Continent up to the foot of that mountain chain--a geographical change
+  so stupendous as to demand much more evidence than was adduced in its
+  support. At last Louis Agassiz (1807-1873), who had varied his
+  palaeontological studies at Neuchatel by excursions into the Alps, was
+  so much struck by the proofs of the former far greater extension of
+  the Swiss glaciers, that he pursued the investigation and satisfied
+  himself that the ice had formerly extended from the Alpine valleys
+  right across the great plain of Switzerland, and had transported huge
+  boulders from the central mountains to the flanks of the Jura. In the
+  year 1840 he visited Britain and soon found evidence of similar
+  conditions there. He showed that it was not by submergence in a sea
+  cumbered with floating ice, but by the former presence of vast
+  glaciers or sheets of ice that the Drift and erratic blocks had been
+  distributed. The idea thus propounded by him did not at once command
+  complete approval, though traces of ancient glaciers in Scotland and
+  Wales were soon detected by native geologists, particularly by W.
+  Buckland, Lyell, J.D. Forbes and Charles Maclaren. Robert Chambers
+  (1802-1871) did good service in gathering additional evidence from
+  Scotland and Norway in favour of Agassiz's views, which steadily
+  gained adherents until, after some quarter of a century, they were
+  adopted by the great majority of geologists in Britain, and
+  subsequently in other countries. Since that time the literature of
+  geology has been swollen by a vast number of contributions in which
+  the history of the Glacial period, and its records both in the Old and
+  New World, have been fully discussed.
+
+  _Rise and Progress of Palaeontological Geology._--As this branch of
+  the science deals with the evidence furnished by fossil organic
+  remains as to former geographical conditions, it early attracted
+  observers who, in the superficial beds of marine shells found at some
+  distance from the coast, saw proofs of the former submergence of the
+  land under the sea. But the occurrence of fossils embedded in the
+  heart of the solid rocks of the mountains offered much greater
+  difficulties of explanation, and further progress was consequently
+  slow. Especially baneful was the belief that these objects were mere
+  sports of nature, and had no connexion with any once living organisms.
+  So long as the true organic origin of the fossil plants and animals
+  contained in the rocks was in dispute, it was hardly possible that
+  much advance could be made in their systematic study, or in the
+  geological deductions to be drawn from them. One good result of the
+  controversy, however, was to be seen in the large collections of these
+  "formed stones" that were gathered together in the cabinets and
+  museums of the 17th and 18th centuries. The accumulation and
+  comparison of these objects naturally led to the production of
+  treatises in which they were described and not unfrequently
+  illustrated by good engravings. Switzerland was more particularly
+  noted for the number and merit of its works of this kind, such as
+  that of K.N. Lang (_Historia lapidum figuratorum Helvetiae_, 1708) and
+  those of Johann Jacob Scheuchzer (1672-1733). In England, also,
+  illustrated treatises were published both by men who looked on fossils
+  as mere freaks of nature, and by those who regarded them as proofs of
+  Noah's flood. Of the former type were the works of Martin Lister
+  (1638-1712) and Robert Plot (_Natural History of Oxfordshire_, 1677).
+  The Celtic scholar Edward Llwyd (1660-1709) wrote a Latin treatise
+  containing good plates of a thousand fossils in the Ashmolean Museum,
+  Oxford, and J. Woodward, in 1728-1729, published his _Natural History
+  of the Fossils of England_, already mentioned, wherein he described
+  his own extensive collection, which he bequeathed to the University of
+  Cambridge, where it is still carefully preserved. The most voluminous
+  and important of all these works, however, appeared at a later date at
+  Nuremberg. It was begun by G.W. Knorr (1705-1761), who himself
+  engraved for it a series of plates, which for beauty and accuracy have
+  seldom been surpassed. After his death the work was continued by
+  J.E.I. Walch (1725-1778), and ultimately consisted of four massive
+  folio volumes and nearly 300 plates under the title of _Lapides
+  diluvii universalis testes_. Although the authors supposed their
+  fossils to be relics of Noah's flood, their work must be acknowledged
+  to mark a distinct onward stage in the palaeontological department of
+  geology.
+
+  It was in France that palaeontological geology began to be cultivated
+  in a scientific spirit. The potter Bernard Palissy, as far back as
+  1580, had dwelt on the importance of fossil shells as monuments of
+  revolutions of the earth's surface; but the observer who first
+  undertook the detailed study of the subject was Jean Etienne Guettard,
+  who began in 1751 to publish his descriptions of fossils in the form
+  of memoirs presented to the Academy of Sciences of Paris. To him they
+  were not only of deep interest as monuments of former types of
+  existence, but they had an especial value as records of the changes
+  which the country had undergone from sea to land and from land to sea.
+  More especially noteworthy was a monograph by him which appeared in
+  1765 bearing the title "On the accidents that have befallen Fossil
+  Shells compared with those which are found to happen to shells now
+  living in the Sea." In this treatise he showed that the fossils have
+  been encrusted with barnacles and serpulae, have been bored into by
+  other organisms, and have often been rounded or broken before final
+  entombment; and he inferred that these fossils must have lived and
+  died on the sea-floor under similar conditions to those which obtain
+  on the sea-floor to-day. His argument was the most triumphant that had
+  ever been brought against the doctrine of _lusus naturae_, and that of
+  the efficacy of Noah's flood--doctrines which still held their ground
+  in Guettard's day. When Soulavie, Cuvier and Brongniart in France, and
+  William Smith in England, showed that the rock formations of the
+  earth's crust could be arranged in chronological order, and could be
+  recognized far and wide by means of their enclosed organic remains,
+  the vast significance of these remains in geological research was
+  speedily realized, and palaeontological geology at once entered on a
+  new and enlarged phase of development. But apart from their value as
+  chronological monuments, and as witnesses of former conditions of
+  geography, fossils presented in themselves a wide field of
+  investigation as types of life that had formerly existed, but had now
+  passed away. It was in France that this subject first took definite
+  shape as an important branch of science. The mollusca of the Tertiary
+  deposits of the Paris basin became, in the hands of Lamarck, the basis
+  on which invertebrate palaeontology was founded. The same series of
+  strata furnished to Cuvier the remains of extinct land animals, of
+  which, by critical study of their fragmentary bones and skeletons, he
+  worked out restorations that may be looked on as the starting-point of
+  vertebrate palaeontology. These brilliant researches, rousing
+  widespread interest in such studies, showed how great a flood of light
+  could be thrown on the past history of the earth and its inhabitants.
+  But the full significance of these extinct types of life could not be
+  understood so long as the doctrine of the immutability of species, so
+  strenuously upheld by Cuvier, maintained its sway among naturalists.
+  Lamarck, as far back as the year 1800, had begun to propound his
+  theory of evolution and the transformation of species; but his views,
+  strongly opposed by Cuvier and the great body of naturalists of the
+  day, fell into neglect. Not until after the publication in 1859 of the
+  _Origin of Species_ by Charles Darwin were the barriers of old
+  prejudice in this matter finally broken down. The possibility of
+  tracing the ancestry of living forms back into the remotest ages was
+  then perceived; the time-honoured fiction that the stratified
+  formations record a series of catastrophes and re-creations was
+  finally dissipated; and the earth's crust was seen to contain a noble,
+  though imperfect, record of the grand evolution of organic types of
+  which our planet has been the theatre.
+
+  _Development of Petrographical Geology._--Theophrastus, the favourite
+  pupil of Aristotle, wrote a treatise _On Stones_, which has come down
+  to our own day, and may be regarded as the earliest work on
+  petrography. At a subsequent period Pliny, in his _Natural History_,
+  collected all that was known in his day regarding the occurrence and
+  uses of minerals and rocks. But neither of these works is of great
+  scientific importance, though containing much interesting information.
+  Minerals from their beauty and value attracted notice before much
+  attention was paid to rocks, and their study gave rise to the science
+  of mineralogy long before geology came into existence. When rocks
+  began to be more particularly scrutinized, it was chiefly from the
+  side of their usefulness for building and other economic purposes. The
+  occurrence of marine shells in many of them had early attracted
+  attention to them. But their varieties of composition and origin did
+  not become the subject of serious study until after Linnaeus and J.G.
+  Wallerius in the 18th century had made a beginning. The first
+  important contribution to this department of the science was that of
+  Werner, who in 1786 published a classification and description of
+  rocks in which he arranged them in two divisions, simple and compound,
+  and further distinguished them by various external characters and by
+  their relative age. The publication of this scheme may be said to mark
+  the beginning of scientific petrography. Werner's system, however, had
+  the serious defect that the chronological order in which he grouped
+  the rocks, and the hypothesis by which he accounted for them as
+  chemical precipitates from the original ocean, were both alike
+  contrary to nature. It was hardly possible indeed that much progress
+  could be made in this branch of geology until chemistry and mineralogy
+  had made greater advances; and especially until it was possible to
+  ascertain the intimate chemical and mineralogical composition, and the
+  minute structure of rocks. The study, however, continued to be pursued
+  in Germany, where the influence of Werner's enthusiasm still led men
+  to enter the petrographical rather than the palaeontological domain.
+  The resources of modern chemistry were pressed into the service, and
+  analyses were made and multiplied to such a degree that it seemed as
+  if the ultimate chemical constitution of every type of rock had now
+  been thoroughly revealed. The condition of the science in the middle
+  of the 19th century was well shown by J.L.A. Roth, who in 1861
+  collected about 1000 trustworthy analyses which up to that time had
+  been made. But though the chemical elements of the rocks had been
+  fairly well determined, the manner in which they were combined in the
+  compound rocks could for the most part be only more or less plausibly
+  conjectured. As far back as 1831 an account was published of a process
+  devised by William Nicol of Edinburgh, whereby sections of fossil wood
+  could be cut, mounted on glass, and reduced to such a degree of
+  transparency as to be easily examined under a microscope. Henry Sorby,
+  of Sheffield, having seen Nicol's preparations, perceived how
+  admirably adapted the process was for the study of the minute
+  structure and composition of rocks. In 1858 he published in the
+  _Quarterly Journal of the Geological Society_ a paper "On the
+  Microscopical Structure of Crystals." This essay led to a complete
+  revolution of petrographical methods and gave a vast impetus to the
+  study of rocks. Petrology entered upon a new and wider field of
+  investigation. Not only were the mineralogical constituents of the
+  rocks detected, but minute structures were revealed which shed new
+  light on the origin and history of these mineral masses, and opened up
+  new paths in theoretical geology. In the hands of H. Vogelsang, F.
+  Zirkel, H. Rosenbusch, and a host of other workers in all civilized
+  countries, the literature of this department of the science has grown
+  to a remarkable extent. Armed with the powerful aid of modern optical
+  instruments, geologists are now able with far more prospect of success
+  to resume the experiments begun a century before by de Saussure and
+  Hall. G.A. Daubree, C. Friedel, E. Sarasin, F. Fouque and A. Michel
+  Levy in France, C. Doelter y Cisterich and E. Hussak of Gratz, J.
+  Morozewicz of Warsaw and others, have greatly advanced our knowledge
+  by their synthetical analyses, and there is every reason to hope that
+  further advances will be made in this field of research.
+
+  _Rise of Physiographical Geology._--Until stratigraphical geology had
+  advanced so far as to show of what a vast succession of rocks the
+  crust of the earth is built up, by what a long and complicated series
+  of revolutions these rocks have come to assume their present
+  positions, and how enormous has been the lapse of time which all these
+  changes represent, it was not possible to make a scientific study of
+  the surface features of our globe. From ancient times it had been
+  known that many parts of the land had once been under the sea; but
+  down even to the beginning of the 19th century the vaguest conceptions
+  continued to prevail as to the operations concerned in the submergence
+  and elevation of land, and as to the processes whereby the present
+  outlines of terrestrial topography were determined. We have seen, for
+  instance, that according to the teaching of Werner the oldest rocks
+  were first precipitated from solution in the universal ocean to form
+  the mountains, that the vertical position of their strata was
+  original, that as the waters subsided successive formations were
+  deposited and laid bare, and that finally the superfluous portion of
+  the ocean was whisked away into space by some unexplained co-operation
+  of another planetary body. Desmarest, in his investigation of the
+  volcanic history of Auvergne, was the first observer to perceive by
+  what a long process of sculpture the present configuration of the land
+  has been brought about. He showed conclusively that the valleys have
+  been carved out by the streams that flow in them, and that while they
+  have sunk deeper and deeper into the framework of the land, the spaces
+  of ground between them have been left as intervening ridges and hills.
+  De Saussure learnt a similar lesson from his studies of the Alps, and
+  Hutton and Playfair made it a cardinal feature in their theory of the
+  earth. Nevertheless the idea encountered so much opposition that it
+  made but little way until after the middle of the 19th century.
+  Geologists preferred to believe in convulsions of nature, whereby
+  valleys were opened and mountains were upheaved. That the main
+  features of the land, such as the great mountain-chains, had been
+  produced by gigantic plication of the terrestrial crust was now
+  generally admitted, and also that minor fractures and folds had
+  probably initiated many of the valleys. But those who realized most
+  vividly the momentous results achieved by ages of subaerial denudation
+  perceived that, as Hutton showed, even without the aid of underground
+  agency, the mere flow of water in streams across a mass of land must
+  in course of time carve out just such a system of valleys as may
+  anywhere be seen. It was J.B. Jukes who, in 1862, first revived the
+  Huttonian doctrine, and showed how completely it explained the
+  drainage-lines in the south of Ireland. Other writers followed in
+  quick succession until, in a few years, the doctrine came to be widely
+  recognized as one of the established principles of modern geology.
+  Much help was derived from the admirable illustrations of
+  land-sculpture and river-erosion supplied from the Western Territories
+  and States of the American Union.
+
+  Another branch of physiographical geology which could only come into
+  existence after most of the other departments of the science had made
+  large progress, deals with the evolution of the framework of each
+  country and of the several continents and oceans of the globe. It is
+  now possible, with more or less confidence, to trace backward the
+  history of every terrestrial area, to see how sea and land have there
+  succeeded each other, how rivers and lakes have come and gone, how the
+  crust of the earth has been ridged up at widely separated intervals,
+  each movement determining some line of mountains or plains, how the
+  boundaries of the oceans have shifted again and again in the past, and
+  thus how, after so prolonged a series of revolutions, the present
+  topography of each country, and of the globe as a whole, has been
+  produced. In the prosecution of this subject maps have been
+  constructed to show what is conjectured to have been the distribution
+  of sea and land during the various geological periods in different
+  parts of the world, and thus to indicate the successive stages through
+  which the architecture of the land has been gradually evolved. The
+  most noteworthy contribution to this department of the science is the
+  _Antlitz der Erde_ of Professor Suess of Vienna. This important and
+  suggestive work has been translated into French and English.
+
+
+PART II.--COSMICAL ASPECTS
+
+Before geology had attained to the position of an inductive science, it
+was customary to begin investigations into the history of the earth by
+propounding or adopting some more or less fanciful hypothesis in
+explanation of the origin of our planet, or even of the universe. Such
+preliminary notions were looked upon as essential to a right
+understanding of the manner in which the materials of the globe had been
+put together. One of the distinguishing features of Hutton's Theory of
+the Earth consisted in his protest that it is no part of the province of
+geology to discuss the origin of things. He taught that in the materials
+from which geological evidence is to be compiled there can be found "no
+traces of a beginning, no prospect of an end." In England, mainly to the
+influence of the school which he founded, and to the subsequent rise of
+the Geological Society of London, which resolved to collect facts
+instead of fighting over hypotheses, is due the disappearance of the
+crude and unscientific cosmologies by which the writings of the earlier
+geologists were distinguished.
+
+But there can now be little doubt that in the reaction against those
+visionary and often grotesque speculations, geologists were carried too
+far in an opposite direction. In allowing themselves to believe that
+geology had nothing to do with questions of cosmogony, they gradually
+grew up in the conviction that such questions could never be other than
+mere speculation, interesting or amusing as a theme for the employment
+of the fancy, but hardly coming within the domain of sober and inductive
+science. Nor would they soon have been awakened out of this belief by
+anything in their own science. It is still true that in the data with
+which they are accustomed to deal, as comprising the sum of geological
+evidence, there can be found no trace of a beginning, though the
+evidence furnished by the terrestrial crust shows a general evolution of
+organic forms from some starting-point which cannot be seen. The oldest
+rocks which have been discovered on any part of the globe have probably
+been derived from other rocks older than themselves. Geology by itself
+has not yet revealed, and is little likely ever to reveal, a trace of
+the first solid crust of our globe. If, then, geological history is to
+be compiled from direct evidence furnished by the rocks of the earth, it
+cannot begin at the beginning of things, but must be content to date
+its first chapter from the earliest period of which any record has been
+preserved among the rocks.
+
+Nevertheless, though geology in its usual restricted sense has been, and
+must ever be, unable to reveal the earliest history of our planet, it no
+longer ignores, as mere speculation, what is attempted in this subject
+by its sister sciences. Astronomy, physics and chemistry have in late
+years all contributed to cast light on the earlier stages of the earth's
+existence, previous to the beginning of what is commonly regarded as
+geological history. But whatever extends our knowledge of the former
+conditions of our globe may be legitimately claimed as part of the
+domain of geology. If this branch of inquiry, therefore, is to continue
+worthy of its name as the science of the earth, it must take cognizance
+of these recent contributions from other sciences. It must no longer be
+content to begin its annals with the records of the oldest rocks, but
+must endeavour to grope its way through the ages which preceded the
+formation of any rocks. Thanks to the results achieved with the
+telescope, the spectroscope and the chemical laboratory, the story of
+these earliest ages of our earth is every year becoming more definite
+and intelligible.
+
+Up to the present time no definite light has been thrown by physics on
+the origin and earliest condition of our globe. The famous nebular
+theory (q.v.) of Kant and Laplace sketched the supposed evolution of the
+solar system from a gaseous nebula, slowly rotating round a more
+condensed central portion of its mass, which eventually became the sun.
+As a consequence of increased rapidity of rotation resulting from
+cooling and contraction, the nebula acquired a more and more lenticular
+form, until at last it threw off from its equatorial protuberance a ring
+of matter. Subsequently the same process was repeated, and other similar
+rings successively separated from the parent mass. Each ring went
+through a corresponding series of changes until it ultimately became a
+planet, with or without one or more attendant satellites. The intimate
+relationship of our earth to the sun and the other planets was, in this
+way, shown. But there are some serious physical difficulties in the way
+of the acceptance of the nebular hypothesis. Another explanation is
+given by the meteoritic hypothesis, according to which, out of the
+swarms of meteorites with which the regions of space are crowded, the
+sun and planets have been formed by gradual accretion.
+
+According to these theoretical views we should expect to find a general
+uniformity of composition in the constituent matter of the solar system.
+For many years the only available evidence on this point was derived
+from the meteorites (q.v.) which so constantly fall from outer space
+upon the surface of the earth. These bodies were found to consist of
+elements, all of which had been recognized as entering into the
+constitution of the earth. But the discoveries of spectroscopic research
+have made known a far more widely serviceable method of investigation,
+which can be applied even to the luminous stars and nebulae that lie far
+beyond the bounds of the solar system. By this method information has
+been obtained regarding the constitution of the sun, and many of our
+terrestrial metals, such as iron, nickel and magnesium, have been
+ascertained to exist in the form of incandescent vapour in the solar
+atmosphere. The present condition of the sun probably represents one of
+the phases through which stars and planets pass in their progress
+towards becoming cool and dark bodies in space. If our globe was at
+first, like its parent sun, an incandescent mass of probably gaseous
+matter, occupying much more space than it now fills, we can conceive
+that it has ever since been cooling and contracting until it has reached
+its present form and dimensions, and that it still retains a high
+internal temperature. Its oblately spheroidal form is such as would be
+assumed by a rotating mass of matter in the transition from a vaporous
+and self-luminous or liquid condition to one of cool and dark solidity.
+But it has been claimed that even a solid spherical globe might develop,
+under the influence of protracted rotation, such a shape as the earth at
+present possesses.
+
+The observed increase of temperature downwards in our planet has
+hitherto been generally accepted as a relic and proof of an original
+high temperature and mobility of substance. Recently, however, the
+validity of this proof has been challenged on the ground that the
+ascertained amount of radium in the rocks of the outer crust is more
+than sufficient to account for the observed downward increase of
+temperature. Too little, however, is known of the history and properties
+of what is called radium to afford a satisfactory ground on which to
+discard what has been, and still remains, the prevalent belief on this
+subject.
+
+An important epoch in the geological history of the earth was marked by
+the separation of the moon from its mass (see TIDE). Whether the
+severance arose from the rupture of a surrounding ring or the gradual
+condensation of matter in such a ring, or from the ejection of a single
+mass of matter from the rapidly rotating planet, it has been shown that
+our satellite was only a few thousand miles from the earth's surface,
+since when it has retreated to its present distance of 240,000 m. Hence
+the influence of the moon's attraction, and all the geological effects
+to which it gives rise, attained their maximum far back in the
+development of the globe, and have been slowly diminishing throughout
+geological history.
+
+The sun by virtue of its vast size has not yet passed out of the
+condition of glowing gas, and still continues to radiate heat beyond the
+farthest planet of the solar system. The earth, however, being so small
+a body in comparison, would cool down much more quickly. Underneath its
+hot atmosphere a crust would conceivably begin to form over its molten
+surface, though the interior might still possess a high temperature and,
+owing to the feeble conducting power of rocks, would remain intensely
+hot for a protracted series of ages.
+
+Full information regarding the form and size of the earth, and its
+relations to the other planetary members of the solar system, will be
+found in the articles PLANET and SOLAR SYSTEM. For the purposes of
+geological inquiry the reader will bear in mind that the equatorial
+diameter of our globe is estimated to be about 7925 m., and the polar
+diameter about 7899 m.; the difference between these two sums
+representing the amount of flattening at the poles (about 26-1/2 m.).
+The planet has been compared in shape to an orange, but it resembles an
+orange which has been somewhat squeezed, for its equatorial
+circumference is not a regular circle but an ellipse, of which the major
+axis lies in long. 8 deg. 15' W.--on a meridian which cuts the
+north-west corner of America, passing through Portugal and Ireland, and
+the north-east corner of Asia in the opposite hemisphere.
+
+The rotation of the earth on its axis exerts an important influence on
+the movements of the atmosphere, and thereby affects the geological
+operations connected with these movements. The influence of rotation is
+most marked in the great aerial circulation between the poles and the
+equator. Currents of air, which set out in a meridional direction from
+high latitudes towards the equator, come from regions where the velocity
+due to rotation is small to where it is greater, and they consequently
+fall behind. Thus, in the northern hemisphere a north wind, as it moves
+away from its northern source of origin, is gradually deflected more and
+more towards the west and becomes a north-east current; while in the
+opposite hemisphere a wind making from high southern latitudes towards
+the equator becomes, from the same cause, a south-east current. Where,
+on the other hand, the air moves from the equatorial to the polar
+regions its higher velocity of rotation carries it eastward, so that on
+the south side of the equator it becomes a north-west current and on the
+north side a south-west current. It is to this cause that the easting
+and westing of the great atmospheric currents are to be attributed, as
+is familiarly exemplified in the trade winds.
+
+The atmospheric circulation thus deflected influences the circulation of
+the ocean. The winds which persistently blow from the north-east on the
+north side of the equator, and from the south-east on the south side,
+drive the superficial waters onwards, and give rise to converging
+oceanic currents which unite to form the great westerly equatorial
+current.
+
+A more direct effect of terrestrial rotation has been claimed in the
+case of rivers which flow in a meridional direction. It has been
+asserted that those, which in the northern hemisphere flow from north to
+south, like the Volga, by continually passing into regions where the
+velocity of rotation is increasingly greater, are thrown more against
+their western than their eastern banks, while those whose general course
+is in an opposite direction, like the Irtisch and Yenesei, press more
+upon their eastern sides. There cannot be any doubt that the tendency of
+the streams must be in the directions indicated. But when the
+comparatively slow current and constantly meandering course of most
+rivers are taken into consideration, it may be doubted whether the
+influence of rotation is of much practical account so far as
+river-erosion is concerned.
+
+One of the cosmical relations of our planet which has been more
+especially prominent in geological speculations relates to the position
+of the earth's axis of rotation. Abundant evidence has now been obtained
+to prove that at a comparatively late geological period a rich flora,
+resembling that of warm climates at the present day, existed in high
+latitudes even within less than 9 deg. of the north pole, where, with an
+extremely low temperature and darkness lasting for half of the year, no
+such vegetation could possibly now exist. It has accordingly been
+maintained by many geologists that the axis of rotation must have
+shifted, and that when the remarkable Arctic assemblage of fossil plants
+lived the region of their growth must have lain in latitudes much nearer
+to the equator of the time.
+
+The possibility of any serious displacement of the rotational axis since
+a very early period in the earth's history has been strenuously denied
+by astronomers, and their arguments have been generally, but somewhat
+reluctantly, accepted by geologists, who find themselves confronted with
+a problem which has hitherto seemed insoluble. That the axis is not
+rigidly stable, however, has been postulated by some physicists, and has
+now been demonstrated by actual observation and measurement. It is
+admitted that by the movement of large bodies of water the air over the
+surface of the globe, and more particularly by the accumulation of vast
+masses of snow and ice in different regions, the position of the axis
+might be to some extent shifted; more serious effects might follow from
+widespread upheavals or depressions of the surface of the lithosphere.
+On the assumption of the extreme rigidity of the earth's interior,
+however, the general result of mathematical calculation is to negative
+the supposition that in any of these ways within the period represented
+by what is known as the "geological record," that is, since the time of
+the oldest known sedimentary formations, the rotational axis has ever
+been so seriously displaced as to account for such stupendous geological
+events as the spread of a luxuriant vegetation far up into polar
+latitudes. If, however, the inside of the globe possesses a great
+plasticity than has been allowed, the shifting of the axis might not be
+impossible, even to such an extent as would satisfy the geological
+requirements. This question is one on which the last word has not been
+said, and regarding which judgment must remain in suspense.
+
+In recent years fresh information bearing on the minor devagations of
+the pole has been obtained from a series of several thousand careful
+observations made in Europe and North America. It has thus been
+ascertained that the pole wanders with a curiously irregular but
+somewhat spiral movement, within an amplitude of between 40 and 50 ft.,
+and completes its erratic circuit in about 428 days. It was not supposed
+that its movement had any geological interest, but Dr John Milne has
+recently pointed out that the times of sharpest curvature in the path of
+the pole coincide with the occurrence of large earthquakes, and has
+suggested that, although it can hardly be assumed that this coincidence
+shows any direct connexion between earthquake frequency and changes in
+the position of the earth's axis, both effects may not improbably arise
+from the same redistribution of surface material by ocean currents and
+meteorological causes.
+
+If for any reason the earth's centre of gravity were sensibly displaced,
+momentous geological changes would necessarily ensue. That the centre of
+gravity does not coincide with the centre of figure of the globe, but
+lies to the south of it, has long been known. This greater aggregation
+of dense material in the southern hemisphere probably dates from the
+early ages of the earth's consolidation, and it is difficult to believe
+that any readjustment of the distribution of this material in the
+earth's interior is now possible. But certain rearrangements of the
+hydrosphere on the surface of the globe may, from time to time, cause a
+shifting of the centre of gravity, which will affect the level of the
+ocean. The accumulation of enormous masses of ice around the pole will
+give rise to such a displacement, and will thus increase the body of
+oceanic water in the glaciated hemisphere. Various calculations have
+been made of the effect of the transference of the ice-cap from one pole
+to the other, a revolution which may possibly have occurred more than
+once in the past history of the globe. James Croll estimated that if the
+mass of ice in the southern hemisphere be assumed to be 1000 ft. thick
+down to lat. 60 deg., its removal to the opposite hemisphere would raise
+the level of the sea 80 ft. at the north pole, while the Rev. Osmond
+Fisher made the rise as much as 409 ft. The melting of the ice would
+still further raise the sea-level by the addition of so large a volume
+of water to the ocean. To what extent superficial changes of this kind
+have operated in geological history remains an unsolved problem, but
+their probable occurrence in the past has to be recognized as one of the
+factors that must be considered in tracing the revolutions of the
+earth's surface.
+
+_The Age of the Earth._--Intimately connected with the relations of our
+globe to the sun and the other members of the solar system is the
+question of the planet's antiquity--a subject of great geological
+importance, regarding which much discussion has taken place since the
+middle of the 19th century. Though an account of this discussion
+necessarily involves allusion to departments of geology which are more
+appropriately referred to in later parts of this article, it may perhaps
+be most conveniently included here.
+
+Geologists were for many years in the habit of believing that no limit
+could be assigned to the antiquity of the planet, and that they were at
+liberty to make unlimited drafts on the ages of the past. In 1862 and
+subsequent years, however, Lord Kelvin (then Sir William Thomson)
+pointed out that these demands were opposed to known physical facts, and
+that the amount of time required for geological history was not only
+limited, but must have been comprised within a comparatively narrow
+compass. His argument rested on three kinds of evidence: (1) the
+internal heat and rate of cooling of the earth; (2) the tidal
+retardation of the earth's rotation; and (3) the origin and age of the
+sun's heat.
+
+1. Applying Fourier's theory of thermal conductivity, Lord Kelvin
+contended that in the known rate of increase of temperature downward and
+beneath the surface, and the rate of loss of heat from the earth, we
+have a limit to the antiquity of the planet. He showed, from the data
+available at the time, that the superficial consolidation of the globe
+could not have occurred less than 20 million years ago, or the
+underground heat would have been greater than it is; nor more than 400
+million years ago, otherwise the underground temperature would have
+shown no sensible increase downwards. He admitted that very wide limits
+were necessary. In subsequently discussing the subject, he inclined
+rather towards the lower than the higher antiquity, but concluded that
+the limit, from a consideration of all the evidence, must be placed
+within some such period of past time as 100 millions of years.
+
+2. The argument from tidal retardation proceeds on the admitted fact
+that, owing to the friction of the tide-wave, the rotation of the earth
+is retarded, and is, therefore, much slower now than it must have been
+at one time. Lord Kelvin affirmed that had the globe become solid some
+10,000 million years ago, or indeed any high antiquity beyond 100
+million years, the centrifugal force due to the more rapid rotation must
+have given the planet a very much greater polar flattening than it
+actually possesses. He admitted, however, that, though 100 million years
+ago that force must have been about 3% greater than now, yet "nothing
+we know regarding the figure of the earth, and the disposition of land
+and water, would justify us in saying that a body consolidated when
+there was more centrifugal force by 3% than now, might not now be in all
+respects like the earth, so far as we know it at present."
+
+3. The third argument, based upon the age of the sun's heat, is
+confessedly less to be relied on than the two previous ones. It proceeds
+upon calculations as to the amount of heat which would be available by
+the falling together of masses from space, which gave rise by their
+impact to our sun. The vagueness of the data on which this argument
+rests may be inferred from the fact that in one passage P.G. Tait placed
+the limit of time during which the sun has been illuminating the earth
+as, "on the very highest computation, not more than about 15 or 20
+millions of years"; while, in another sentence of the same volume, he
+admitted that, "by calculations in which there is no possibility of
+large error, this hypothesis [of the origin of the sun's heat by the
+falling together of masses of matter] is thoroughly competent to explain
+100 millions of years' solar radiation at the present rate, perhaps
+more." In more recently reviewing his argument, Lord Kelvin expressed
+himself in favour of more strictly limiting geological time than he had
+at first been disposed to do. He insists that the time "was more than 20
+and less than 40 millions of years and probably much nearer 20 than 40."
+Geologists appear to have reluctantly brought themselves to believe that
+perhaps, after all, 100 millions of years might suffice for the
+evolution of geological history. But when the time was cut down to 15 or
+20 millions they protested that such a restricted period was
+insufficient for that evolution, and though they did not offer any
+effective criticism of the arguments of the physicists they felt
+convinced that there must be some flaw in the premises on which these
+arguments were based.
+
+By degrees, however, there have arisen among the physicists themselves
+grave doubts as to the validity of the physical evidence on which the
+limitation of the earth's age has been founded, and at the same time
+greater appreciation has been shown of the signification and strength of
+the geological proofs of the high antiquity of our planet. In an address
+from the chair of the Mathematical Section of the British Association in
+1886, Professor (afterwards Sir) George Darwin reviewed the controversy,
+and pronounced the following deliberate judgment in regard to it: "In
+considering these three arguments I have adduced some reasons against
+the validity of the first [tidal friction], and have endeavoured to show
+that there are elements of uncertainty surrounding the second [secular
+cooling of the earth]; nevertheless, they undoubtedly constitute a
+contribution of the first importance to physical geology. Whilst, then,
+we may protest against the precision with which Professor Tait seeks to
+deduce results from them, we are fully justified in following Sir
+William Thomson, who says that 'the existing state of things on the
+earth, life on the earth--all geological history showing continuity of
+life--must be limited within some such period of past time as 100
+million years'." Lord Kelvin has never dealt with the geological and
+palaeontological objections against the limitation of geological time to
+a few millions of years. But Professor Darwin, in the address just
+cited, uttered the memorable warning: "At present our knowledge of a
+definite limit to geological time has so little precision that we should
+do wrong summarily to reject theories which appear to demand longer
+periods of time than those which now appear allowable." In his
+presidential address to the British Association at Cape Town in 1905 he
+returned to the subject, remarking that the argument derived from the
+increase of underground temperature "seems to be entirely destroyed" by
+the discovery of the properties of radium. He thinks that "it does not
+seem extravagant to suppose that 500 to 1000 million years may have
+elapsed since the birth of the moon." He has "always believed that the
+geologists were more nearly correct than the physicists, notwithstanding
+the fact that appearances were so strongly against them," and he
+concludes thus: "It appears, then, that the physical argument is not
+susceptible of a greater degree of certainty than that of the
+geologists, and the scale of geological time remains in great measure
+unknown" (see also Tide, chap. viii.).
+
+In an address to the mathematical section of the American Association
+for the Advancement of Science in 1889, the vice-president of the
+section, R.S. Woodward, thus expressed himself with regard to the
+physical arguments brought forward by Lord Kelvin and Professor Tait in
+limitation of geological time: "Having been at some pains to look into
+this matter, I feel bound to state that, although the hypothesis appears
+to be the best which can be formulated at present, the odds are against
+its correctness. Its weak links are the unverified assumptions of an
+initial uniform temperature and a constant diffusivity. Very likely
+these are approximations, but of what order we cannot decide.
+Furthermore, if we accept the hypothesis, the odds appear to be against
+the present attainment of trustworthy numerical results, since the data
+for calculation, obtained mostly from observations on continental areas,
+are far too meagre to give satisfactory average values for the entire
+mass of the earth."
+
+Still more emphatic is the protest made from the physical side by
+Professor John Perry. He has attacked each of the three lines of
+argument of Lord Kelvin, and has impugned the validity of the
+conclusions drawn from them. The argument from tidal retardation he
+dismisses as fallacious, following in this contention the previous
+criticism of the Rev. Maxwell Close and Sir George Darwin. In dealing
+with the argument based on the secular cooling of the earth, he holds it
+to be perfectly allowable to assume a much higher conductivity for the
+interior of the globe, and that such a reasonable assumption would
+enable us greatly to increase our estimate of the earth's antiquity. As
+for the third argument, from the age of the sun's heat, he points out
+that the sun may have been repeatedly fed by a supply of meteorites from
+outside, while the earth may have been protected from radiation, and
+been able to retain much of its heat by being enveloped in a dense
+atmosphere. Remarking that "almost anything is possible as to the
+present internal state of the earth," he concludes thus: "To sum up, we
+can find no published record of any lower maximum age of life on the
+earth, as calculated by physicists, than 400 millions of years. From the
+three physical arguments Lord Kelvin's higher limits are 1000, 400 and
+500 million years. I have shown that we have reasons for believing that
+the age, from all these, may be very considerably underestimated. It is
+to be observed that if we exclude everything but the arguments from mere
+physics, the _probable_ age of life on the earth is much less than any
+of the above estimates; but if the palaeontologists have good reasons
+for demanding much greater times, I see nothing from the physicists'
+point of view which denies them four times the greatest of these
+estimates."
+
+A fresh line of argument against Lord Kelvin's limitation of the
+antiquity of our globe has recently been started by the remarkable
+discoveries in radio-activity. From the ascertained properties of radium
+it appears to be possible that our estimates of solar heat, as derived
+from the theory of gravitation, may have to be augmented ten or twenty
+times; that stores of radium and similar bodies within the earth may
+have indefinitely deferred the establishment of the present temperature
+gradient from the surface inward; that consequently the earth may have
+remained for long ages at a temperature not greatly different from that
+which it now possesses, and hence that the times during which our globe
+has supported animal and vegetable life may be very much longer than
+that allowed in the estimates previously made by physicists from other
+data (see RADIOACTIVITY).
+
+The arguments from the geological side against the physical contention
+that would limit the age of our globe to some 10 or 20 millions of years
+are mainly based on the observed rates of geological and biological
+changes at the present time upon land and sea, and on the nature,
+physical history and organic contents of the stratified crust of the
+earth. Unfortunately, actual numerical data are not obtainable in many
+departments of geological activity, and even where they can be procured
+they do not yet rest on a sufficiently wide collection of accurate and
+co-ordinated observations. But in some branches of dynamical geology,
+material exists for, at least, a preliminary computation of the rate of
+change. This is more especially the case in respect of the wide domain
+of denudation. The observational records of the action of the sea, of
+springs, rivers and glaciers are becoming gradually fuller and more
+trustworthy. A method of making use of these records for estimating the
+rate of denudation of the land has been devised. Taking the Mississippi
+as a general type of river action, it has been shown that the amount of
+material conveyed by this stream into the sea in one year is equivalent
+to the lowering of the general surface of the drainage basin of the
+river by 1/6000 of a foot. This would amount to one foot in 6000 years
+and 1000 ft. in 6 million years. So that at the present rate of waste in
+the Mississippi basin a whole continent might be worn away in a few
+millions of years.
+
+It is evident that as deposition and denudation are simultaneous
+processes, the ascertainment of the rate at which solid material is
+removed from the surface of the land supplies some necessary information
+for estimating the rate at which new sedimentary formations are being
+accumulated on the floor of the sea, and for a computation of the length
+of time that would be required at the present rate of change for the
+deposition of all the stratified rocks that enter into the composition
+of the crust of our globe. If the thickness of these rocks be assumed to
+be 100,000 ft., and if we could suppose them to have been laid down over
+as wide an area as that of the drainage basins from the waste of which
+they were derived, then at the present rate of denudation their
+accumulation would require some 600 millions of years. But, as Dr A.R.
+Wallace has justly pointed out, the tract of sea-floor over which the
+material derived from the waste of the terrestrial surface is laid down
+is at present much less than that from which this material is worn away.
+We have no means, however, of determining what may have been the ratio
+between the two areas in past time. Certainly ancient marine sedimentary
+rocks cover at the present day a much more extensive area than that in
+which they are now being elaborated. If we take the ratio postulated by
+Dr Wallace--1 to 19--the 100,000 ft. of sedimentary strata would require
+31 millions of years for their accumulation. It is quite possible,
+however, that this ratio may be much too high. There are reasons for
+believing that the proportion of coast-line to land area has been
+diminishing during geological time; in other words, that in early times
+the land was more insular and is now more continental. So that the 31
+millions of years may be much less than the period that would be
+required, even on the supposition of continuous uninterrupted denudation
+and sedimentation, during the whole of the time represented by the
+stratified formations.
+
+But no one who has made himself familiar with the actual composition of
+these formations and the detailed structure of the terrestrial crust can
+fail to recognize how vague, imperfect and misleading are the data on
+which such computations are founded. It requires no prolonged
+acquaintance with the earth's crust to impress upon the mind that one
+all-important element is omitted, and indeed can hardly be allowed for
+from want of sufficiently precise data, but the neglect of which must
+needs seriously impair the value of all numerical calculations made
+without it. The assumption that the stratified formations can be treated
+as if they consisted of a continuous unbroken sequence of sediments,
+indicating a vast and uninterrupted process of waste and deposition, is
+one that is belied on every hand by the actual structure of these
+formations. It can only give us a minimum of the time required; for,
+instead of an unbroken series, the sedimentary formations are full of
+"unconformabilities"--gaps in the sequence of the chronological
+records--as if whole chapters and groups of chapters had been torn out
+of a historical work. It can often be shown that these breaks of
+continuity must have been of vast duration, and actually exceeded in
+chronological importance thick groups of strata lying below and above
+them (see Part VI.). Moreover, even among the uninterrupted strata,
+where no such unconformabilities exist, but where the sediments follow
+each other in apparently uninterrupted sequence, and might be thought to
+have been deposited continuously at the same general rate, and without
+the intervention of any pause, it can be demonstrated that sometimes an
+inch or two of sediment might, on certain horizons, represent the
+deposit of an enormously longer period than a hundred or a thousand
+times the same amount of sediment on other horizons. A prolonged study
+of these questions leads to a profound conviction that in many parts of
+the geological record the time represented by sedimentary deposits may
+be vastly less than the time which is not so represented.
+
+It has often been objected that the present rate of geological change
+ought not to be taken as a measure of the rate in past time, because the
+total sum of terrestrial energy has been steadily diminishing, and
+geological processes must consequently have been more vigorous in former
+ages than they are now. Geologists do not pretend to assert that there
+has been no variation or diminution in the activities of the various
+processes which they have to study. What they do insist on is that the
+present rate of change is the only one which we can watch and measure,
+and which will thus supply a statistical basis for any computations on
+the subject. But it has been dogmatically affirmed that because
+terrestrial energy has been diminishing therefore all kinds of
+geological work must have been more vigorously and more rapidly carried
+on in former times than now; that there were far more abundant and more
+stupendous volcanoes, more frequent and more destructive earthquakes,
+more gigantic upheavals and subsidences, more powerful oceanic waves and
+tides, more violent atmospheric disturbances with heavier rainfall and
+more active denudation.
+
+It is easy to make these assertions, and they look plausible; but, after
+all, they rest on nothing stronger than assumption. They can be tested
+by an appeal to the crust of the earth, in which the geological history
+of our planet has been so fully recorded. Had such portentous
+manifestations of geological activity ever been the normal condition of
+things since the beginning of that history, there ought to be a record
+of them in the rocks. But no evidence for them has been found there,
+though it has been diligently sought for in all quarters of the globe.
+We may confidently assert that while geological changes may quite
+possibly have taken place on a gigantic scale in the earliest ages of
+the earth's existence, of which no geological record remains, there is
+no proof that they have ever done so since the time when the very oldest
+of the stratified formations were deposited. There is no need to
+maintain that they have always been conducted precisely on the same
+scale as now, or to deny that they may have gradually become less
+vigorous as the general sum of terrestrial energy has diminished. But we
+may unhesitatingly affirm that no actual evidence of any such
+progressive diminution of activity has been adduced from the geological
+record in the crust of the earth: that, on the contrary, no appearances
+have been detected there which necessarily demand the assumption of
+those more powerful operations postulated by physicists, or which are
+not satisfactorily explicable by reference to the existing scale of
+nature's processes.
+
+That this conclusion is warranted even with regard to the innate energy
+of the globe itself will be seen if we institute a comparison between
+the more ancient and the more recent manifestations of that energy.
+Take, for example, the proofs of gigantic plication, fracture and
+displacement within the terrestrial crust. These, as they have affected
+the most ancient rocks of Europe, have been worked out in great detail
+in the north-west of Scotland. But they are not essentially different
+from or on a greater scale than those which have been proved to have
+affected the Alps, and to have involved strata of so recent a date as
+the older Tertiary formations. On the contrary, it may be doubted
+whether any denuded core of an ancient mountain-chain reveals traces of
+such stupendous disturbances of the crust as those which have given rise
+to the younger mountain-chains of the globe. It may, indeed, quite well
+have been the rule that instead of diminishing in intensity of effect,
+the consequences of terrestrial contraction have increased in magnitude,
+the augmenting thickness of the crust offering greater resistance to
+the stresses, and giving rise to vaster plications, faults,
+thrust-planes and metamorphism, as this growing resistance had to be
+overcome.
+
+The assertion that volcanic action must have been more violent and more
+persistent in ancient times than it is now has assuredly no geological
+evidence in its support. It is quite true that there are vastly more
+remains of former volcanoes scattered over the surface of the globe than
+there are active craters now, and that traces of copious eruptions of
+volcanic material can be followed back into some of the oldest parts of
+the geological record. But we have no proof that ever at any one time in
+geological history there have been more or larger or more vigorous
+volcanoes than those of recent periods. It may be said that the absence
+of such proof ought not to invalidate the assertion until a far wider
+area of the earth's surface has been geologically studied. But most
+assuredly, as far as geological investigation has yet gone, there is an
+overwhelming body of evidence to show that from the earliest epochs in
+geological history, as registered in the stratified rocks, volcanic
+action has manifested itself very much as it does now, but on a less
+rather than on a greater scale. Nowhere can this subject be more
+exhaustively studied than in the British Isles, where a remarkably
+complete series of volcanic eruptions has been chronicled ranging from
+the earliest Palaeozoic down to older Tertiary time. The result of a
+prolonged study of British volcanic geology has demonstrated that, even
+to minute points of detail, there has been a singular uniformity in the
+phenomena from beginning to end. The oldest lavas and ashes differ in no
+essential respect from the youngest. Nor have they been erupted more
+copiously or more frequently. Many successive volcanic periods have
+followed each other after prolonged intervals of repose, each displaying
+the same general sequence of phenomena and similar evidence of gradual
+diminution and extinction. The youngest, instead of being the feeblest,
+were the most extensive outbursts in the whole of this prolonged series.
+
+If now we turn for evidence of the alleged greater activity of all the
+epigene or superficial forces, and especially for proofs of more rapid
+denudation and deposition on the earth's surface, we search for it in
+vain among the stratified formations of the terrestrial crust. Had the
+oldest of these rocks been accumulated in a time of great atmospheric
+perturbation, of torrential rains, colossal tides and violent storms, we
+might surely expect to find among the sediments some proof of such
+disturbed meteorological and geographical conditions. We should look, on
+the one hand, for tumultuous accumulations of coarse unworn detritus,
+rapidly swept by rains, floods and waves from land to sea, and on the
+other hand, for an absence of any evidence of the tranquil and
+continuous deposit of such fine laminated silt as could only settle in
+quiet water. But an appeal to the geological record is made in vain for
+any such proofs. The oldest sediments, like the youngest, reveal the
+operation only of such agents and such rates of activity as are still to
+be witnessed in the accumulation of the same kind of deposits. If, for
+instance, we search the most ancient thick sedimentary formation in
+Britain--the Torridon Sandstone of north-west Scotland, which is older
+than the oldest fossiliferous deposits--we meet with nothing which might
+not be found in any Palaeozoic, Mesozoic or Cainozoic group of similar
+sediments. We see an accumulation, at least 8000 or 10,000 ft. thick, of
+consolidated sand, gravel and mud, such as may be gathering now on the
+floor of any large mountain-girdled lake. The conglomerates of this
+ancient series are not pell-mell heaps of angular detritus, violently
+swept away from the land and huddled promiscuously on the sea-floor.
+They are, in general, built up of pebbles that have been worn smooth,
+rounded and polished by prolonged attrition in running water, and they
+follow each other on successive platforms with intervening layers of
+finer sediment. The sandstones are composed of well water-worn sand,
+some of which has been laid down so tranquilly that its component grains
+have been separated out in layers according to their specific gravity,
+in such manner that they now present dark laminae in which particles of
+magnetic iron, zircon and other heavy minerals have been sifted out
+together, just as iron-sand may be seen gathered into thin sheets on
+sandy beaches at the present day. Again, the same series of primeval
+sediments includes intercalations of fine silt, which has been deposited
+as regularly and intermittently there as it has been among the most
+recent formations. These bands of shale have been diligently searched
+for fossils, as yet without success; but they may eventually disclose
+organic remains older than any hitherto found in Europe.
+
+We now come to the consideration of the palaeontological evidence as to
+the value of geological time. Here the conclusions derived from a study
+of the structure of the sedimentary formations are vastly strengthened
+and extended. In the first place, the organization of the most ancient
+plants and animals furnishes no indication that they had to contend with
+any greater violence of storm, flood, wave or ocean-current than is
+familiar to their modern descendants. The oldest trees, shrubs, ferns
+and club-mosses display no special structures that suggest a difference
+in the general conditions of their environment. The most ancient
+crinoids, sponges, crustaceans, arachnids and molluscs were as
+delicately constructed as those of to-day, and their remains are often
+found in such perfect preservation as to show that neither during their
+lifetime nor after their death were they subject to any greater violence
+of the elements than their living representatives now experience. Of
+much more cogency, however, is the evidence supplied by the grand upward
+succession of organic forms, from the most ancient stratified rocks up
+to the present day. No biologist now doubts for a moment that this
+marvellous succession is the result of a gradual process of evolution
+from lower to higher types of organization. There may be differences of
+opinion as to the causes which have governed this process and the order
+of the steps through which it has advanced, but no one who is conversant
+with the facts will now venture to deny that it has taken place, and
+that, on any possible explanation of its progress, it must have demanded
+an enormous lapse of time. In the Cambrian or oldest fossiliferous
+formations there is already a large and varied fauna, in which the
+leading groups of invertebrate life are represented. On no tenable
+hypothesis can these be regarded as the first organisms that came into
+being on our planet. They must have had a long ancestry, and as Darwin
+first maintained, the time required for their evolution may have been
+"as long as, or probably far longer than, the whole interval from the
+Silurian [Cambrian] age to the present day." The records of these
+earliest eras of organic development have unfortunately not survived the
+geological revolutions of the past; at least, they have not yet been
+recovered. But it cannot be doubted that they once existed and
+registered their testimony to the prodigious lapse of time prior to the
+deposition of the most ancient fossiliferous formations which have
+escaped destruction.
+
+The impressive character of the evidence furnished by the sequence of
+organic forms throughout the great series of fossiliferous strata can
+hardly be fully realized without a detailed and careful study of the
+subject. Professor E.B. Poulton, in an address to the zoological section
+of the British Association at the Liverpool Meeting in 1896, showed how
+overwhelming are the demands which this evidence makes for long periods
+of time, and how impossible it is of comprehension unless these demands
+be conceded. The history of life upon the earth, though it will probably
+always be surrounded with great and even insuperable difficulties,
+becomes broadly comprehensible in its general progress when sufficient
+time is granted for the evolution which it records; but it remains
+unintelligible on any other conditions.
+
+Taken then as a whole, the body of evidence, geological and
+palaeontological, in favour of the high antiquity of our globe is so
+great, so manifold, and based on such an ever-increasing breadth of
+observation and reflection, that it may be confidently appealed to in
+answer to the physical arguments which would seek to limit that
+antiquity to ten or twenty millions of years. In the present state of
+science it is out of our power to state positively what must be the
+lowest limit of the age of the earth. But we cannot assume it to be much
+less, and it may possibly have been much more, than the 100 millions of
+years which Lord Kelvin was at one time willing to concede.[2]
+
+
+PART III.--GEOGNOSY. THE INVESTIGATION OF THE NATURE AND COMPOSITION OF
+THE MATERIALS OF WHICH THE EARTH CONSISTS
+
+This division of the science is devoted to a description of the parts of
+the earth--of the atmosphere and ocean that surround the planet, and
+more especially of the solid materials that underlie these envelopes and
+extend downwards to an unknown distance into the interior. These various
+constituents of the globe are here considered as forms of matter capable
+of being analysed, and arranged according to their composition and the
+place they take in the general composition of the globe.
+
+Viewed in the simplest way the earth may be regarded as made up of three
+distinct parts, each of which ever since an early period of planetary
+history has been the theatre of important geological operations. (1) An
+envelope of air, termed the _atmosphere_, which surrounds the whole
+globe; (2) A lower and less extensive envelope of water, known as the
+_hydrosphere_ (Gr. [Greek: hydor], water) which, constituting the oceans
+and seas, covers nearly three-fourths of the underlying solid surface of
+the planet; (3) A globe, called the _lithosphere_ (Gr. [Greek: lithos],
+stone), the external part of which, consisting of solid stone, forms the
+_crust_, while underneath, and forming the vast mass of the interior,
+lies the _nucleus_, regarding the true constitution of which we are
+still ignorant.
+
+1. _The Atmosphere._--The general characters of the atmosphere are
+described in separate articles (see especially ATMOSPHERE; METEOROLOGY).
+Only its relations to geology have here to be considered. As this
+gaseous envelope encircles the whole globe it is the most universally
+present and active of all the agents of geological change. Its efficacy
+in this respect arises partly from its composition, and the chemical
+reactions which it effects upon the surface of the land, partly from its
+great variations in temperature and moisture, and partly from its
+movements.
+
+  Many speculations have been made regarding the chemical composition of
+  the atmosphere during former geological periods. There can indeed be
+  little doubt that it must originally have differed greatly from its
+  present condition. If the whole mass of the planet originally existed
+  in a gaseous state, there would be practically no atmosphere. The
+  present outer envelope of air may be considered to be the surviving
+  relic of this condition, after all the other constituents have been
+  incorporated into the hydrosphere and lithosphere. The oxygen, which
+  now forms fully a half of the outer crust of the earth, was doubtless
+  originally, whether free or in combination, part of the atmosphere.
+  So, too, the vast beds of coal found all over the world, in geological
+  formations of many different ages, represent so much carbonic acid
+  once present in the air. The chlorides and other salts in the sea may
+  likewise partly represent materials carried down out of the atmosphere
+  in the primitive condensation of the aqueous vapour, though they have
+  been continually increased ever since by contributions from the
+  drainage of the land. It has often been suggested that, during the
+  Carboniferous period, the atmosphere must have been warmer and more
+  charged with aqueous vapour and carbon dioxide than at the present
+  day, to admit of so luxuriant a flora as that from which the
+  coal-seams were formed. There seems, however, to be at present no
+  method of arriving at any certainty on this subject. Lastly, the
+  amount of carbonic acid absorbed in the weathering of rocks at the
+  surface, and the consequent production of carbonates, represents an
+  enormous abstraction of this gas.
+
+  As at present constituted, the atmosphere is regarded as a mechanical
+  mixture of nearly four volumes of nitrogen and one of oxygen, together
+  with an average of 3.5 parts of carbon dioxide in every 10,000 parts
+  of air, and minute quantities of various other gases and solid
+  particles. Of the vapours contained in it by far the most important is
+  that of water which, although always present, varies greatly in amount
+  according to variations in temperature. By condensation the water
+  vapour appears in visible form as dew, mist, cloud, rain, hail, snow
+  and ice, and in these forms includes and carries down some of the
+  other vapours, gases and solid particles present in the air. The
+  circulation of water from the atmosphere to the land, from the land to
+  the sea, and again from the sea to the land, forms the great
+  geological process whereby the habitable condition of the planet is
+  maintained and the surface of the land is sculptured (Part IV.).
+
+2. _The Hydrosphere._--The water envelope covers nearly three-fourths of
+the surface of the earth, and forms the various oceans and seas which,
+though for convenience of reference distinguished by separate names, are
+all linked together in one great body. The physical characters of this
+vast envelope are discussed in separate articles (see OCEAN and
+OCEANOGRAPHY). Viewed from the geological standpoint, the features of
+the sea that specially deserve attention are first the composition of
+its waters, and secondly its movements.
+
+  Sea-water is distinguished from that of ordinary lakes and rivers by
+  its greater specific gravity and its saline taste. Its average density
+  is about 1.026, but it varies even within the same ocean, being least
+  where large quantities of fresh water are added from rain or melting
+  snow and ice, and greatest where evaporation is most active. That
+  sea-water is heavier than fresh arises from the greater proportion of
+  salts which it contains in solution. These salts constitute about
+  three and a half parts in every hundred of water. They consist mainly
+  of chlorides of sodium and magnesium, the sulphates of magnesium,
+  calcium and potassium, with minuter quantities of magnesium bromide
+  and calcium carbonate. Still smaller proportions of other substances
+  have been detected, gold for example having been found in the
+  proportion of 1 part in 15,180,000.
+
+  That many of the salts have existed in the sea from the time of its
+  first condensation out of the primeval atmosphere appears to be
+  probable. It is manifest, however, that, whatever may have been the
+  original composition of the oceans, they have for a vast section of
+  geological time been constantly receiving mineral matter in solution
+  from the land. Every spring, brook and river removes various salts
+  from the rocks over which it moves, and these substances, thus
+  dissolved, eventually find their way into the sea. Consequently
+  sea-water ought to contain more or less traceable proportions of every
+  substance which the terrestrial waters can remove from the land, in
+  short, of probably every element present in the outer shell of the
+  globe, for there seems to be no constituent of this earth which may
+  not, under certain circumstances, be held in solution in water.
+  Moreover, unless there be some counteracting process to remove these
+  mineral ingredients, the ocean water ought to be growing, insensibly
+  perhaps, but still assuredly, saltier, for the supply of saline matter
+  from the land is incessant.
+
+  To the geologist the presence of mineral solutions in sea-water is a
+  fact of much importance, for it explains the origin of a considerable
+  part of the stratified rocks of the earth's crust. By evaporation the
+  water has given rise to deposits of rock-salt, gypsum and other
+  materials. The lime contained in solution, whether as sulphate or
+  carbonate, has been extracted by many tribes of marine animals, which
+  have thus built up out of their remains vast masses of solid
+  limestone, of which many mountain-chains largely consist.
+
+  Another important geological feature of the sea is to be seen in the
+  fact that its basins form the great receptacles for the detritus worn
+  away from the land. Besides the limestones, the visible parts of the
+  terrestrial crust are, in large measure, composed of sedimentary rocks
+  which were originally laid down on the sea-bottom. Moreover, by its
+  various movements, the sea occupies a prominent place among the
+  epigene or superficial agents which produce geological changes on the
+  surface of the globe.
+
+3. _The Lithosphere._--Beneath the gaseous and liquid envelopes lies the
+solid part of the planet, which is conveniently regarded as consisting
+of two parts,--(a) the crust, and (b) the interior or nucleus.
+
+
+  The crust.
+
+It was for a long time a prevalent belief that the interior of the globe
+is a molten mass round which an outer shell has gradually formed through
+cooling. Hence the term "crust" was applied to this external solid
+envelope, which was variously computed to be 10, 20, or more miles in
+thickness. The portion of this crust accessible to human observation was
+seen to afford abundant evidence of vast plications and corrugations of
+its substance, which were regarded as only explicable on the supposition
+of a thin solid collapsible shell floating on a denser liquid interior.
+When, however, physical arguments were adduced to show the great
+rigidity of the earth as a whole, the idea of a thin crust enclosing a
+molten nucleus was reluctantly abandoned by geologists, who found the
+problem of the earth's interior to be incapable of solution by any
+evidence which their science could produce. They continued, however, to
+use the term "crust" as a convenient word to denote the cool outer layer
+of the earth's mass, the structure and history of which form the main
+subjects of geological investigation. More recently, however, various
+lines of research have concurred in suggesting that, whatever may be the
+condition of the interior, its substance must differ greatly from that
+of the outer shell, and that there may be more reason than appeared for
+the retention of the name of crust. Observations on earthquake motion by
+Dr John Milne and others, show that the rate and character of the waves
+transmitted through the interior of the earth differ in a marked degree
+from those propagated along the crust. This difference indicates that
+rocky material, such as we know at the surface, may extend inwards for
+some 30 m., below which the earth's interior rapidly becomes fairly
+homogeneous and possesses a high rigidity. From measurements of the
+force of gravity in India by Colonel S.G. Burrard, it has been inferred
+that the variations in density of the outer parts of the earth do not
+descend farther than 30 or 40 m., which might be assumed to be the limit
+of the thickness of the crust. Recent researches in regard to the
+radio-active substances present in rocks suggest that the crust is not
+more than 50 m. thick, and that the interior differs from it in
+possessing little or no radio-active material.
+
+
+  The interior.
+
+Though we cannot hope ever to have direct acquaintance with more than
+the mere outside skin of our planet, we may be led to infer the
+irregular distribution of materials within the crust from the present
+distribution of land and water, and the observed differences in the
+amount of deflection of the plumb-line near the sea and near
+mountain-chains. The fact that the southern hemisphere is almost wholly
+covered with water appears explicable only on the assumption of an
+excess of density in the mass of that portion of the planet. The
+existence of such a vast sheet of water as that of the Pacific Ocean is
+to be accounted for, as Archdeacon J.H. Pratt pointed out, by the
+presence of "some excess of matter in the solid parts of the earth
+between the Pacific Ocean and the earth's centre, which retains the
+water in its place, otherwise the ocean would flow away to the other
+parts of the earth." A deflection of the plumb-line towards the sea,
+which has in a number of cases been observed, indicates that "the
+density of the crust beneath the mountains must be less than that below
+the plains, and still less than that below the ocean-bed." Apart
+therefore from the depression of the earth's surface in which the oceans
+lie, we must regard the internal density, whether of crust or nucleus,
+to be somewhat irregularly arranged, there being an excess of heavy
+materials in the water hemisphere, and beneath the ocean-beds, as
+compared with the continental masses.
+
+In our ignorance regarding the chemical constitution of the nucleus of
+our planet, an argument has sometimes been based upon the known fact
+that the specific gravity of the globe as a whole is about double that
+of the crust. This has been held by some writers to prove that the
+interior must consist of much heavier material and is therefore probably
+metallic. But the effect of pressure ought to make the density of the
+nucleus much higher, even if the interior consisted of matter no heavier
+than the crust. That the total density of the planet does not greatly
+exceed its observed amount seems only explicable on the supposition that
+some antagonistic force counteracts the effects of pressure. The only
+force we can suppose capable of so acting is heat. But comparatively
+little is yet known regarding the compression of gases, liquids and
+solids under such vast pressures as must exist within the nucleus.
+
+That the interior of the earth possesses a high temperature is inferred
+from the evidence of various sources. (1) Volcanoes, which are openings
+that constantly, or intermittently, give out hot vapours and molten lava
+from reservoirs beneath the crust. Besides active volcanoes, it is known
+that former eruptive vents have been abundantly and widely distributed
+over the globe from the earliest geological periods down to our own day.
+(2) Hot springs are found in many parts of the globe, with temperatures
+varying up to the boiling point of water. (3) From mines, tunnels and
+deep borings into the earth it has been ascertained that in all quarters
+of the globe below the superficial zone of invariable temperature, there
+is a progressive increase of heat towards the interior. The rate of this
+increase varies, being influenced, among other causes, by the varying
+conductivity of the rocks. But the average appears to be about 1 deg.
+Fahr. for every 50 or 60 ft. of descent, as far down as observations
+have extended. Though the increase may not advance in the same
+proportion at great depths, the inference has been confidently drawn
+that the temperature of the nucleus must be exceedingly high.
+
+The probable condition of the earth's interior has been a fruitful
+source of speculation ever since geology came into existence; but no
+general agreement has been arrived at on the subject. Three chief
+hypotheses have been propounded: (1) that the nucleus is a molten mass
+enclosed within a solid shell; (2) that, save in local vesicular spaces
+which may be filled with molten or gaseous material, the globe is solid
+and rigid to the centre; (3) that the great body of the nucleus consists
+of incandescent vapours and gases, especially vaporous iron, which under
+the gigantic pressure within the earth are so compressed as to confer
+practical rigidity on the globe as a whole, and that outside this main
+part of the nucleus the gases pass into a shell of molten magma, which,
+in turn, shades off outwards into the comparatively thin, cool
+solidified crust. Recent seismological observations have led to the
+inference that the outer crust, some 30 to 45 m. thick, must rapidly
+merge into a fairly homogeneous nucleus which, whatever be its
+constitution, transmits undulatory movements through its substance with
+uniform velocity and is believed to possess a high rigidity.
+
+The origin of the earth's high internal temperature has been variously
+accounted for. Most usually it has been assumed to be the residue of the
+original "tracts of fluent heat" out of which the planet shaped itself
+into a globe. According to another supposition the effects of the
+gradual gravitational compression of the earth's mass have been the main
+source of the high temperature. Recent researches in radio-activity, to
+which reference has already been made, have indicated another possible
+source of the internal heat in the presence of radium in the rocks of
+the crust. This substance has been detected in all igneous rocks,
+especially among the granites, in quantity sufficient, according to the
+Hon. R.J. Strutt, to account for the observed temperature-gradient in
+the crust, and to indicate that this crust cannot be more than 45 m.
+thick, otherwise the outflow of heat would be greater than the amount
+actually ascertained. Inside this external crust containing radio-active
+substances, it is supposed, as already stated, that the nucleus consists
+of some totally different matter containing little or no radium.
+
+  _Constitution of the Earth's Crust._--As the crust of the earth
+  contains the "geological record," or stony chronicle from which
+  geology interprets the history of our globe, it forms the main subject
+  of study to the geologist. The materials of which this crust consists
+  are known as minerals and rocks. From many chemical analyses, which
+  have been made of these materials, the general chemical constitution
+  of, at least, the accessible portion of the crust has been
+  satisfactorily ascertained. This information becomes of much
+  importance in speculations regarding the early history of the globe.
+  Of the elements known to the chemist the great majority form but a
+  small proportion of the composition of the crust, which is mainly
+  built up of about twenty of them. Of these by far the most important
+  are the non-metallic elements oxygen and silicon. The former forms
+  about 47% and the latter rather more than 28% of the original crust,
+  so that these two elements make up about three-fourths of the whole.
+  Next after them come the metals aluminium (8.16%), iron (4.64),
+  calcium (3.50), magnesium (2.62), sodium (2.63), and potassium (2.35).
+  The other twelve elements included in the twenty vary in amount from a
+  proportion of 0.41% in the case of titanium, to not more than 0.01% of
+  chlorine, fluorine, chromium, nickel and lithium. The other fifty or
+  more elements exist in such minute proportions in the crust that,
+  probably, not one of them amounts to as much as 0.01%, though they
+  include the useful metals, except iron. Taking the crust, and the
+  external envelopes of the ocean and the air, we thus perceive that
+  these outer parts of our planet consist of more than three-fourths of
+  non-metals and less than one-fourth of metals.
+
+  The combinations of the elements which are of most importance in the
+  constitution of the terrestrial crust consist of oxides. From the mean
+  of a large number of analyses of the rocks of the lower or primitive
+  portion of the crust, it has been ascertained that silica (SiO2) forms
+  almost 60% and alumina (Al2O3) upwards of 15% of the whole. The other
+  combinations in order of importance are lime (CaO) 4.90%, magnesia
+  (MgO) 4.36, soda (Na2O) 3.55, ferrous oxide (FeO) 3.52, potash (K2O)
+  2.80, ferric oxide (Fe2O3) 2.63, water (H2O) 1.52, titanium oxide
+  (TiO2) 0.60, phosphoric acid (P2O5) 0.22; the other combinations of
+  elements thus form less than 1% of the crust.
+
+  These different combinations of the elements enter into further
+  combinations with each other so as to produce the wide assortment of
+  simple minerals (see MINERALOGY). Thus, silica and alumina are
+  combined to form the aluminous silicates, which enter so largely into
+  the composition of the crust of the earth. The silicates of magnesia,
+  potash and soda constitute other important families of minerals. A
+  mass of material composed of one, but more usually of more than one
+  mineral, is known as a _rock_. Under this term geologists are
+  accustomed to class not only solid stone, such as granite and
+  limestone, but also less coherent materials such as clay, peat and
+  even loose sand. The accessible portion of the earth's crust consists
+  of various kinds of rocks, which differ from each other in structure,
+  composition and origin, and are therefore susceptible of diverse
+  classifications according to the point of view from which they are
+  considered. The details of this subject will be found in the article
+  PETROLOGY.
+
+  _Classification of Rocks._--Various systems of classification of rocks
+  have been proposed, but none of them is wholly satisfactory. The most
+  useful arrangement for most purposes of the geologist is one based on
+  the broad differences between them in regard to their mode of origin.
+  From this point of view they may be ranged in three divisions:
+
+  1. In the first place, a large number of rocks may be described as
+  original or underived, for it is not possible to trace them back to
+  any earlier source. They belong to the primitive constitution of the
+  planet, and, as they have all come up from below through the crust,
+  they serve to show the nature of the material which lies immediately
+  below the outer parts of that crust. They include the numerous
+  varieties of lava, which have been poured out in a molten state from
+  volcanic vents, also a great series of other rocks which, though they
+  may never have been erupted to the surface, have been forced upward in
+  a melted condition into the other rocks of the crust and have
+  solidified there. From their mode of origin this great class of rocks
+  has been called "igneous" or "eruptive." As they generally show no
+  definite internal structure save such as may result from joints, they
+  have been termed "massive" or "unstratified," to distinguish them from
+  those of the second division which are strongly marked out by the
+  presence of a stratified structure. The igneous rocks present a
+  considerable range of composition. For the most part they consist
+  mainly of aluminous silicates, some of them being highly acid
+  compounds with 75% or more of silica. But they also include highly
+  basic varieties wherein the proportion of silica sinks to 40%, and
+  where magnesia greatly predominates over alumina. The textures of
+  igneous rocks likewise comprise a wide series of varieties. On the one
+  hand, some are completely vitreous, like obsidian, which is a natural
+  glass. From this extreme every gradation may be traced through gradual
+  increase of the products of devitrification, until the mass may become
+  completely crystalline. Again, some crystalline igneous rocks are so
+  fine in grain as not to show their component crystals save under the
+  microscope, while in others the texture is so coarse as to present the
+  component minerals in separate crystals an inch or more in length.
+  These differences indicate that, at first, the materials of the rock
+  may have been as completely molten as artificial glass, and that the
+  crystalline condition has been subsequently developed by cooling, and
+  the separation of the chemical constituents into definite crystalline
+  minerals. Many of the characters of igneous rocks have been reproduced
+  experimentally by fusing together their minerals, or the constituents
+  of their minerals, in the proper proportion. But it has not yet been
+  found possible to imitate the structure of such rocks as granite.
+  Doubtless these rocks consolidated with extreme slowness at great
+  depths below the surface, under vast pressures and probably in the
+  presence of water or water-vapour--conditions which cannot be
+  adequately imitated in a laboratory.
+
+  Though the igneous rocks occupy extensive areas in some countries,
+  they nevertheless cover a much smaller part of the whole surface of
+  the land than is taken up by the second division or stratified rocks.
+  But they increase in quantity downwards and probably extend
+  continuously round the globe below the other rocks. This important
+  series brings before us the relations of the molten magma within the
+  earth to the overlying crust and to the outer surface. On the one
+  hand, it includes the oldest and most deep-seated extravasations of
+  that magma, which have been brought to light by ruptures and upheavals
+  of the crust and prolonged denudation. On the other, it presents to
+  our study the varied outpourings of molten and fragmentary materials
+  in the discharges of modern and ancient volcanoes. Between these two
+  extremes of position and age, we find that the crust has been, as it
+  were, riddled with injections of the magma from below. These features
+  will be further noticed in Part V. of this article.
+
+  2. The "sedimentary" or "stratified rocks" form by much the larger
+  part of the dry land of the globe, and they are prolonged to an
+  unknown distance from the shores under the bed of the sea. They
+  include those masses of mineral matter which, unlike the igneous
+  rocks, can be traced back to a definite origin on the surface of the
+  earth. Three distinct types may be recognized among them: (a) By far
+  the largest proportion of them consists of different kinds of sediment
+  derived from the disintegration of pre-existing rocks. In this
+  "fragmental" group are placed all the varieties of shingle, gravel,
+  sand, clay and mud, whether these materials remain in a loose
+  incoherent condition, or have been compacted into solid stone. (b)
+  Another group consists of materials that have been deposited by
+  chemical precipitation from solution in water. The white sinter laid
+  down by calcareous springs is a familiar example on a small scale.
+  Beds of rock-salt, gypsum and dolomite have, in some regions, been
+  accumulated to a thickness of many thousand feet, by successive
+  precipitations of the salt contained in the water of inland seas. (c)
+  An abundant and highly important series of sedimentary formations has
+  been formed from the remains of plants and animals. Such accumulations
+  may arise either from the transport and deposit of these remains, as
+  in the case of sheets of drift-wood, and banks of drifted sea-shells,
+  or from the growth and decay of the organisms on the spot, as happens
+  in peat bogs and in coral-reefs.
+
+  As the sedimentary rocks have for the most part been laid down under
+  water, and more especially on the sea-floor, they are often spoken of
+  as "aqueous," in contradistinction to the igneous rocks. Some of them,
+  however, are accumulated by the drifting action of wind upon loose
+  materials, and are known as "aeolian" formations. Familiar instances
+  of such wind-formed deposits are the sand-dunes along many parts of
+  the sea coast. Much more extensive in area are the sands of the great
+  deserts in the arid regions of the globe.
+
+  It is from the sedimentary rocks that the main portion of geological
+  history is derived. They have been deposited one over another in
+  successive strata from a remote period in the development of the globe
+  down to the present time. From this arrangement they have been termed
+  "stratified," in contrast to the unstratified or igneous series. They
+  have preserved memorials of the geographical revolutions which the
+  surface of the earth has undergone; and above all, in the abundant
+  fossils which they have enclosed, they furnish a momentous record of
+  the various tribes of plants and animals which have successively
+  flourished on land and sea. Their investigation is thus the most
+  important task which devolves upon the geologist.
+
+  3. In the third place comes a series of rocks which are not now in
+  their original condition, but have undergone such alteration as to
+  have acquired new characters that more or less conceal their first
+  structures. Some of them can be readily recognized as altered igneous
+  masses; others are as manifestly of sedimentary origin; while of many
+  it is difficult to decide what may have been their pristine character.
+  To this series the term "metamorphic" has been applied. Its members
+  are specially distinguished by a prevailing fissile, or schistose,
+  structure which they did not at first possess, and which differs from
+  anything found in unaltered igneous or sedimentary rocks. This
+  fissility is combined with a more or less pronounced crystalline
+  structure. These changes are believed to be the result of movements
+  within the crust of the earth, whereby the most solid rocks were
+  crushed and sheared, while, at the same time, under the influence of a
+  high temperature and the presence of water, they underwent internal
+  chemical reactions, which led to a rearrangement and recomposition of
+  their mineral constituents and the production of a crystalline
+  structure (see METAMORPHISM).
+
+  Among the less altered metamorphic rocks of sedimentary origin, the
+  successive laminae of deposit of the original sediment can be easily
+  observed; but they are also traversed by a new set of divisional
+  planes, along which they split across the original bedding. Together
+  with this superinduced cleavage there have been developed in them
+  minute hairs, scales and rudimentary crystals. Further stages of
+  alteration are marked by the increase of micaceous scales, garnets and
+  other minerals, especially along the planes of cleavage, until the
+  whole rock becomes crystalline, and displays its chief component
+  minerals in successive discontinuous folia which merge into each
+  other, and are often crumpled and puckered. Massive igneous rocks can
+  be observed to have undergone intense crushing and cleavage, and to
+  have ultimately assumed a crystalline foliated character. Rocks which
+  present this aspect are known as schists (q.v.). They range from the
+  finest silky slates, or phyllites, up to the coarsest gneisses, which
+  in hand-specimens can hardly be distinguished from granites. There is
+  indeed every reason to believe that such gneisses were probably
+  originally true granites, and that their foliation and
+  recrystallization have been the result of metamorphism.
+
+  The schists are more especially to be found in the heart of
+  mountain-chains, and in regions where the lowest and oldest parts of
+  the earth's crust have, in the course of geological revolutions, been
+  exposed to the light of day. They have been claimed by some writers
+  to be part of the original or primitive surface of our globe that
+  first consolidated on the molten nucleus. But the progress of
+  investigation all over the world has shown that this supposition
+  cannot be sustained. The oldest known rocks present none of the
+  characters of molten material that has cooled and hardened in the air,
+  like the various forms of recent lava. On the contrary, they possess
+  many of the features characteristic of bodies of eruptive material
+  that have been injected into the crust at some depth underground, and
+  are now visible at the surface, owing to the removal by denudation of
+  the rocks under which they consolidated. In their less foliated
+  portions they can be recognized as true eruptive rocks. In many places
+  gneisses that possess a thoroughly typical foliation have been found
+  to pierce ancient sedimentary formations as intrusive bosses and
+  veins.
+
+
+PART IV.--DYNAMICAL GEOLOGY
+
+This section of the science includes the investigation of those
+processes of change which are at present in progress upon the earth,
+whereby modifications are made on the structure and composition of the
+crust, on the relations between the interior and the surface, as shown
+by volcanoes, earthquakes and other terrestrial disturbances, on the
+distribution of oceans and continents, on the outlines of the land, on
+the form and depth of the sea-bottom, on climate, and on the races of
+plants and animals by which the earth is tenanted. It brings before us,
+in short, the whole range of activities which it is the province of
+geology to study, and leads us to precise notions regarding their
+relations to each other and the results which they achieve. A knowledge
+of this branch of the subject is thus the essential groundwork of a true
+and fruitful acquaintance with the principles of geology, seeing that it
+necessitates a study of the present order of nature, and thus provides a
+key for the interpretation of the past.
+
+The whole range of operations included within the scope of inquiry in
+this branch of the science may be regarded as a vast cycle of change,
+into which we may break at any point, and round which we may travel,
+only to find ourselves brought back to our starting-point. It is a
+matter of comparatively small moment at what part of the cycle we begin
+our inquiries. We shall always find that the changes we see in action
+have resulted from some that preceded, and give place to others which
+follow them.
+
+At an early time in the earth's history, anterior to any of the periods
+of which a record remains in the visible rocks, the chief sources of
+geological action probably lay within the earth itself. If, as is
+generally supposed, the planet still retained a great store of its
+initial heat, it was doubtless the theatre of great chemical changes,
+giving rise, perhaps, to manifestations of volcanic energy somewhat like
+those which have so marvellously roughened the surface of the moon. As
+the outer layers of the globe cooled, and the disturbances due to
+internal heat and chemical action became less marked, the conditions
+would arise in which the materials for geological history were
+accumulated. The influence of the sun, which must always have operated,
+would then stand out more clearly, giving rise to that wide circle of
+superficial changes wherein variations of temperature and the
+circulation of air and water over the surface of the earth come into
+play.
+
+In the pursuit of his inquiries into the past history and into the
+present _regime_ of the earth, the geologist must needs keep his mind
+ever open to the reception of evidence for kinds and especially for
+degrees of action which he had not before imagined. Human experience has
+been too short to allow him to assume that all the causes and modes of
+geological change have been definitively ascertained. On the earth
+itself there may remain for future discovery evidence of former
+operations by heat, magnetism, chemical change or otherwise, which may
+explain many of the phenomena with which geology has to deal. Of the
+influences, so many and profound, which the sun exerts upon our planet,
+we can as yet only perceive a little. Nor can we tell what other
+cosmical influences may have lent their aid in the evolution of
+geological changes.
+
+Much useful information regarding many geological processes has been
+obtained from experimental research in laboratories and elsewhere, and
+much more may be confidently looked for from future extensions of this
+method of inquiry. The early experiments of Sir James Hall, already
+noticed, formed the starting-point for numerous subsequent researches,
+which have elucidated many points in the origin and history of rocks. It
+is true that we cannot hope to imitate those operations of nature which
+demand enormous pressures and excessively high temperatures combined
+with a long lapse of time. But experience has shown that in regard to a
+large number of processes, it is possible to imitate nature's working
+with sufficient accuracy to enable us to understand them, and so to
+modify and control the results as to obtain a satisfactory solution of
+some geological problems.
+
+In the present state of our knowledge, all the geological energy upon
+and within the earth must ultimately be traced back to the primeval
+energy of the parent nebula or sun. There is, however, a certain
+propriety and convenience in distinguishing between that part of it
+which is due to the survival of some of the original energy of the
+planet and that part which arises from the present supply of energy
+received day by day from the sun. In the former case we have to deal
+with the interior of the earth, and its reaction upon the surface; in
+the latter, we deal with the surface of the earth and to some extent
+with its reaction on the interior. This distinction allows of a broad
+treatment of the subject under two divisions:
+
+I. Hypogene or Plutonic Action: The changes within the earth caused by
+internal heat, mechanical movement and chemical rearrangements.
+
+II. Epigene or Surface Action: The changes produced on the superficial
+parts of the earth, chiefly by the circulation of air and water set in
+motion by the sun's heat.
+
+
+_DIVISION I.--HYPOGENE OR PLUTONIC ACTION_
+
+In the discussion of this branch of the subject we must carry in our
+minds the conception of a globe still possessing a high internal
+temperature, radiating heat into space and consequently contracting in
+bulk. Portions of molten rocks from inside are from time to time poured
+out at the surface. Sudden shocks are generated by which destructive
+earthquakes are propagated through the diameter of the globe as well as
+to and along its surface. Wide geographical areas are pushed up or sink
+down. In the midst of these movements remarkable changes are produced
+upon the rocks of the crust; they are plicated, fractured, crushed,
+rendered crystalline and even fused.
+
+
+  (A) _Volcanoes and Volcanic Action._
+
+  This subject is discussed in the article VOLCANO, and only a general
+  view of its main features will be given here. Under the term volcanic
+  action (vulcanism, vulcanicity) are embraced all the phenomena
+  connected with the expulsion of heated materials from the interior of
+  the earth to the surface. A volcano may be defined as a conical hill
+  or mountain, built up wholly or mainly of materials which have been
+  ejected from below, and which have accumulated around the central vent
+  of eruption. As a rule its truncated summit presents a cup-shaped
+  cavity, termed the crater, at the bottom of which is the opening of
+  the main funnel or pipe whereby communication is maintained with the
+  heated interior. From time to time, however, in large volcanoes rents
+  are formed on the sides of the cone, whence steam and other hot
+  vapours and also streams of molten lava are poured forth. On such
+  rents smaller or parasitic cones are often formed, which imitate the
+  operations of the parent cone and, after repeated eruptions, may rise
+  to hills hundreds of feet in height. In course of centuries the result
+  of the constant outpouring of volcanic materials may be to build up a
+  large mountain like Etna, which towers above the sea to a height of
+  10,840 feet, and has some 200 minor cones along its flanks.
+
+  But all volcanic eruptions do not proceed from central orifices. In
+  Iceland it has been observed that, from fissures opened in the ground
+  and extending for long distances, molten material has issued in such
+  abundance as to be spread over the surrounding country for many miles,
+  while along the lines of fissure small cones or hillocks of
+  fragmentary material have accumulated round more active parts of the
+  rent. There is reason to believe that in the geological past this
+  fissure-type of eruption has repeatedly been developed, as well as the
+  more common form of central cones like Vesuvius or Etna.
+
+  In the operations of existing volcanoes only the superficial
+  manifestations of volcanic action are observable. But when the rocks
+  of the earth's crust are studied, they are found to enclose the relics
+  of former volcanic eruptions. The roots of ancient volcanoes have thus
+  been laid bare by geological revolutions; and some of the
+  subterranean phases of volcanic action are thereby revealed which are
+  wholly concealed in an active volcano. Hence to obtain as complete a
+  conception as possible of the nature and history of volcanic action,
+  regard must be had, not merely to modern volcanoes, but to the records
+  of ancient eruptions which have been preserved within the crust.
+
+  The substances discharged from volcanic vents consist of--(1) Gases
+  and vapours: which, dissolved in the molten magma of the interior,
+  take the chief share in volcanic activity. They include in greatest
+  abundance water-gas, which condenses into the clouds of steam so
+  conspicuous in volcanic eruptions. Hydrochloric acid and sulphuretted
+  hydrogen are likewise plentiful, together with many other substances
+  which, sublimed by the high internal temperature, take a solid form on
+  cooling at the surface. (2) Molten rock or lava: which ranges from the
+  extremely acid type of the obsidians and rhyolites with 70% or more of
+  silica, to the more basic and heavy varieties such as basalts and
+  leucite-lavas with much iron, and sometimes no more than 45% of
+  silica. The specific gravity of lavas varies between 2.37 and 3.22,
+  and the texture ranges from nearly pure glass, like obsidian, to a
+  coarse granitoid compound, as in some rhyolites. (3) Fragmentary
+  materials, which are sometimes discharged in enormous quantity and
+  dispersed over a wide extent of country, the finer particles being
+  transported by upper air-currents for hundreds of miles. These
+  materials arise either from the explosion of lava by the sudden
+  expansion of the dissolved vapours and gases, as the molten rock rises
+  to the surface, or from the breaking up and expulsion of portions of
+  the walls of the vent, or of the lava, which happens to have
+  solidified within these walls. They vary from the finest impalpable
+  dust and ashes, through increasing stages of coarseness up to huge
+  "bombs" torn from the upper surface of the molten rock in the vent,
+  and large blocks of already solidified lava, or of non-volcanic rock
+  detached from the sides of the pipe up which the eruptions take place.
+
+  Nothing is yet known as to the determining cause of any particular
+  volcanic eruption. Some vents, like that of Stromboli, in the
+  Mediterranean, are continually active, and have been so ever since man
+  has observed them. Others again have been only intermittently in
+  eruption, with intervals of centuries between their periods of
+  activity. We are equally in the dark as to what has determined the
+  sites on which volcanic action has manifested itself. There is reason,
+  indeed, to believe that extensive fractures of the terrestrial crust
+  have often provided passages up which the vapours, imprisoned in the
+  internal magma, have been able to make their way, accompanied by other
+  products. Where chains of volcanoes rise along definite lines, like
+  those of Sumatra, Java, and many other tracts both in the Old and the
+  New World, there appears to be little doubt that their linear
+  distribution should be attributed to this cause. But where a volcano
+  has appeared by itself, in a region previously exempt from volcanic
+  action, the existence of a contributing fissure cannot be so
+  confidently presumed. The study of certain ancient volcanoes, the
+  roots of which have been exposed by long denudation, has shown an
+  absence of any visible trace of their having availed themselves of
+  fractures in the crust. The inference has been drawn that volcanic
+  energy is capable of itself drilling an orifice through the crust,
+  probably at some weaker part, and ejecting its products at the
+  surface. The source of this energy is to be sought in the enormous
+  expansive force of the vapours and gases dissolved in the magma. They
+  are kept in solution by the enormous pressure within the earth; but as
+  the lava approaches the surface and this pressure is relieved these
+  dissolved vapours and gases rush out with explosive violence, blowing
+  the upper part of the lava column into dust, and allowing portions of
+  the liquid mass below to rise and escape, either from the crater or
+  from some fissure which the vigour of explosion has opened on the side
+  of the cone. So gigantic is the energy of these pent-up vapours, that,
+  after a long period of volcanic quiescence, they sometimes burst forth
+  with such violence as to blow off the whole of the upper part or even
+  one side of a large cone. The history of Vesuvius, and the great
+  eruptions of Krakatoa in 1883 and of Bandaizan in 1888 furnish
+  memorable examples of great volcanic convulsions. It has been observed
+  that such stupendous discharges of aeriform and fragmentary matter may
+  be attended with the emission of little or no lava. On the other hand,
+  some of the largest outflows of lava have been accompanied by
+  comparatively little fragmentary material. Thus, the great lava-floods
+  of Iceland in 1783 spread for 40 m. away from their parent fissure,
+  which was marked only by a line of little cones of slag.
+
+  The temperature of lava as it issues from underground has been
+  measured more or less satisfactorily, and affords an indication of
+  that existing within the earth. At Vesuvius it has been ascertained to
+  be more than 2000 deg. Fahr. At first the molten rock glows with a
+  white light, which rapidly reddens, and disappears under the rugged
+  brown and black crust that forms on the surface. Underneath this badly
+  conducting crust, the lava cools so slowly that columns of steam have
+  been noticed rising from its surface more than 80 years after its
+  eruption.
+
+  Considerable alteration in the topography of volcanic regions may be
+  produced by successive eruptions. The fragmentary materials are
+  sometimes discharged in such abundance as to cover the ground for many
+  miles around with a deposit of loose ashes, cinders and slag. Such a
+  deposit accumulating to a depth of many feet may completely bury
+  valleys and water-courses, and thus greatly affect the drainage. The
+  coarsest materials accumulate nearest to the vent that emits them. The
+  finer dust is not infrequently hurled forth with such an impetus as to
+  be carried for thousands of feet into the tracks of upper
+  air-currents, whereby it may be borne for hundreds of miles away from
+  the vent so as ultimately to fall to the ground in countries far
+  removed from any active volcano. Outflows of lava, from their greater
+  solidity and durability, produce still more serious and lasting
+  changes in the external features of the ground over which they flow.
+  As they naturally seek the lowest levels, they find their way into the
+  channels of streams. If they keep along the channels, they seal them
+  up under a mass of compact stone which the running water, if not
+  wholly diverted elsewhere, will take many long centuries to cut
+  through. If, on the other hand, the lava crosses a stream, it forms a
+  massive dam, above which the water is ponded back so as to form a
+  lake.
+
+  As the result of prolonged activity a volcanic cone is gradually built
+  up by successive outflows of lava and showers of dust and stones.
+  These materials are arranged in beds, or sheets, inclined outwards
+  from the central vent. On surrounding level ground the alternating
+  beds are flat. In course of time, deep gullies are cut on the outer
+  slopes of the cone by rain, and by the heavy showers that arise from
+  the condensation of the copious discharges of steam during eruptions.
+  Along the sides of these ravines instructive sections may be studied
+  of the volcanic strata. The larger rivers of some volcanic regions
+  have likewise eroded vast gorges in the more horizontal lavas and
+  ashes of the flatter country, and have thus laid bare stupendous
+  cliffs, along which the successive volcanic sheets can be seen piled
+  above each other for many hundred feet. On a small scale, some of
+  these features are well displayed among the rivers that drain the
+  volcanic tracts of central France; on a great scale, they are
+  presented in the course of the Snake river, and other streams that
+  traverse the great volcanic country of western North America. Similar
+  volcanic scenery has been produced in western Europe by the action of
+  denudation in dissecting the flat Tertiary lavas of Scotland, the
+  Faeroe Isles and Iceland.
+
+  Of special interest to the geologist are those volcanoes which have
+  taken their rise on the sea-bottom; for the volcanic intercalations
+  among the stratified formations of the earth's crust are almost
+  entirely of submarine origin. Many active volcanoes situated on
+  islands have begun their eruptions below sea-level. Both Vesuvius and
+  Etna sprang up on the floor of the Mediterranean sea, and have
+  gradually built up their cones into conspicuous parts of the dry land.
+  Examples of a similar history are to be found among the volcanic
+  islands of the Pacific Ocean. In some of these cases a movement of
+  elevation has carried the submarine lavas, tuffs and agglomerates
+  above sea-level, and has furnished opportunities of comparing these
+  materials with those of recent subaerial origin, and also with the
+  ancient records of submarine eruptions which have been preserved among
+  the stratified formations. From the evidence thus supplied, it can be
+  shown that the materials ejected from modern submarine volcanic vents
+  closely resemble those accumulated by subaerial volcanoes; that the
+  dust, ashes and stones become intermingled or interstratified with
+  coral-mud, or other non-volcanic deposit of the sea-bottom, that
+  vesicular lavas may be intercalated among them as on land, and that
+  between the successive sheets of volcanic origin, layers of limestone
+  may be laid down which are composed chiefly, or wholly, of the remains
+  of calcareous marine organisms.
+
+  Though active volcanoes are widely distributed over the globe, and are
+  especially abundant around the vast basin of the Pacific Ocean, they
+  afford an incomplete picture of the extent to which volcanic action
+  has displayed itself on the surface of our planet. When the rocks of
+  the land are attentively studied they disclose proofs of that action
+  in many districts where there is now no outward sign of it. Not only
+  so, but they reveal that volcanoes have been in eruption in some of
+  these districts during many different periods of the past, back to the
+  beginnings of geological history. The British Islands furnish a
+  remarkable example of such a series of ancient eruptions. From the
+  Cambrian period all through Palaeozoic times there rose at intervals
+  in that country a succession of volcanic centres from some of which
+  thousands of feet of lavas and tuffs were discharged. Again in older
+  Tertiary times the same region witnessed a stupendous outpouring of
+  basalt, the surviving relics of which are more than 3000 ft. thick,
+  and cover many hundreds of square miles. Similar evidence is supplied
+  in other countries both in the Old and the New world. Hence it is
+  proved that, in the geological past, volcanic action has been vigorous
+  at long intervals on the same sites during a vast series of ages,
+  though no active vents are to be seen there now. The volcanoes now
+  active form but a small proportion of the total number which has
+  appeared on the surface of the earth.
+
+  With regard to the cause of volcanic action much has been speculated,
+  but little can be confidently affirmed. That water in the form of
+  occluded gas plays the chief part in forcing the lava column up a
+  volcanic chimney, and in the violent explosions that accompany the
+  rise of the molten material, is generally admitted. But opinions
+  differ as to the source of this water. According to some
+  investigators, it should be regarded as in large measure of meteoric
+  origin, derived from the descent of rain into the earth, and its
+  absorption by the molten magma in the interior. Others, contending
+  that the supply so furnished, even if it could reach and be dissolved
+  in the magma, would yet be insufficient to furnish the prodigious
+  quantity of aqueous vapour discharged during an eruption, maintain
+  that the water belongs to the magma itself. They point to the admitted
+  fact that many substances, particularly metals in a state of fusion,
+  can absorb large quantities of vapours and gases without chemical
+  combination, and on cooling discharge them with eruptive phenomena
+  somewhat like those of volcanoes. This question must be regarded as
+  one of the still unsolved problems of geology.
+
+
+  (B) _Movements of the Earth's Crust._
+
+  Among the hypogene forces in geological dynamics an important place
+  must be assigned to movements of the terrestrial crust. Though the
+  expression "the solid earth" has become proverbial, it appears
+  singularly inappropriate in the light of the results obtained in
+  recent years by the use of delicate instruments of observation. With
+  the facilities supplied by these instruments (see SEISMOMETER), it has
+  been ascertained that the ground beneath our feet is subject to
+  continual slight tremors, and feeble pulsations of longer duration,
+  some of which may be due to daily or seasonal variations of
+  temperature, atmospheric pressure or other meteorological causes. The
+  establishment of self-recording seismometers all over the world has
+  led to the detection of many otherwise imperceptible shocks, over and
+  above the appreciable earth-waves propagated from earthquake centres
+  of disturbance. Moreover, it has been ascertained that some parts of
+  the surface of the land are slowly rising, while others are falling
+  with reference to the sea-level. From time to time the surface suffers
+  calamitous devastation from earthquakes, when portions of the crust
+  under great strain suddenly give way. Lastly, at intervals, probably
+  separated from each other by vast periods of time, the terrestrial
+  crust undergoes intense plication and fracture, and is consequently
+  ridged up into mountain-chains. No event of this kind has been
+  witnessed since man began to record his experiences. But from the
+  structure of mountains, as laid open by prolonged denudation, it is
+  possible to form a vivid conception of the nature and effects of these
+  most stupendous of all geological revolutions.
+
+  In considering this department of geological inquiry it will be
+  convenient to treat it under the following heads: (1) Slow depression
+  and upheaval; (2) Earthquakes; (3) Mountain-making; (4) Metamorphism
+  of rocks.
+
+  1. _Slow Depression and Upheaval._--On the west side of Japan the land
+  is believed to be sinking below the sea, for fields are replaced by
+  beaches of sand or shingle, while the depth of the sea off shore has
+  perceptibly increased. A subsidence of the south of Sweden has taken
+  place in comparatively recent times, for streets and foundations of
+  houses at successive levels are found below high-water mark. The west
+  coast of Greenland over an extent of more than 600 m. is sinking, and
+  old settlements are now submerged. Proofs of submergence of land are
+  furnished by "submerged forests," and beds of terrestrial peat now
+  lying at various depths below the level of the sea, of which many
+  examples have been collected along the shores of the British Isles,
+  Holland and France. Interesting evidence that the west of Europe now
+  stands at a lower level than it did at a late geological period is
+  supplied in the charts of the North Sea and Atlantic, which show that
+  the valleys of the land are prolonged under the sea. These valleys
+  have been eroded out of the rocks by the streams which flow in them,
+  and the depth of their submerged portions below the sea level affords
+  an indication of the extent of the subsidence.
+
+  The uprise of land has been detected in various parts of the world.
+  One of the most celebrated instances is that of the shores of the Gulf
+  of Bothnia, where, at Stockholm, the elevation, between the years 1774
+  and 1875, appears to have been 48 centimetres (18-1/2 in.) in a
+  century. But on the west side of Sweden, fronting the Skager Rak, the
+  coast, between the years 1820 and 1870, rose 30 centimetres, which is
+  at the rate of 60 centimetres, or nearly 2 ft. in a century. In the
+  region of the Great Lakes in the interior of Canada and the United
+  States it has been ascertained that the land is undergoing a slow tilt
+  towards the south-west, of which the mean rate appears to be rather
+  less than 6 in. in a century. If this rate of change should continue
+  the waters of Lake Michigan, owing to the progress of the tilt, will,
+  in some 500 or 600 years, submerge the city of Chicago, and eventually
+  the drainage of the lakes will be diverted into the basin of the
+  Mississippi. Proof of recent emergence of land is supplied by what are
+  called "raised beaches" or "strand-lines," that is, lines of former
+  shores marked by sheets of littoral deposits, or platforms cut by
+  shore-waves in rock and flanked by old sea-cliffs and lines of
+  sea-worn caves. Admirable examples of these features are to be seen
+  along the west coast of Europe from the south of England to the north
+  of Norway. These lines of old shores become fainter in proportion to
+  their antiquity. In Britain they occur at various heights, the
+  platforms at 25, 50 and 100 ft. being well marked.
+
+  The cause of these slow upward and downward movements of the crust of
+  the earth is still imperfectly understood. Upheaval might conceivably
+  be produced by an ascent of the internal magma, and the consequent
+  expansion of the overlying crust by heat; while depression might
+  follow any subsidence of the magma, or its displacement to another
+  district. If, as is generally believed, the globe is still
+  contracting, the shrinkage of the surface may cause both these
+  movements. Subsidence will be in excess, but between subsiding tracts
+  lateral thrust may suffice to push upward intervening more solid and
+  stable ground; but no solution of the problem yet proposed is wholly
+  satisfactory.
+
+  2. _Earthquakes._--As this subject is discussed in a separate article
+  it will be sufficient here to take note of its more important
+  geological bearings. It was for many centuries taken for granted that
+  earthquakes and volcanoes are due to a common cause. We have seen that
+  in classical antiquity they were looked on as the results of the
+  movements of wind imprisoned within the earth. Long after this notion
+  was discarded, and a more scientific appreciation of volcanic action
+  was reached, it was still thought that earthquakes should be regarded
+  as manifestations of the same source of energy as that which displays
+  itself in volcanic eruptions. It is true that earthquakes are frequent
+  in districts of active volcanoes, and they may undoubtedly be often
+  due there to the explosions of the magma, or to the rupture of rocks
+  caused by its ascent towards the surface. But such shocks are
+  comparatively local in their range and feeble in their effects. There
+  is now a general agreement that between the great world-shaking
+  earthquakes and volcanic phenomena, no immediate and intimate
+  relationship can be traced, though they may be connected in ways which
+  are not yet perceived. Some of the more recent great earthquakes on
+  land have proved that the waves of shock are produced by the sudden
+  rupture or collapse of rocks under great strain, either along lines of
+  previous fracture or of new rents in the terrestrial crust; and that
+  such ruptures may occur at a remote distance from any volcano. Thus
+  the recent disastrous San Francisco earthquake has been recognized to
+  have resulted from a slipping of ground along the line of an old
+  fault, which has been traced for a long distance in California
+  generally parallel to the coast. The position of this fault at the
+  surface has long been clearly followed by its characteristic
+  topography. After the earthquake these superficial features were found
+  to have been removed by the same cause that had originated them. For
+  some 300 m. on the track of this old fault-line a renewed slipping was
+  seen to have taken place along one or both sides, and the ground at
+  the surface was ruptured as well as displaced horizontally. Obviously,
+  the jar occasioned by the sudden and simultaneous subsidence of a
+  portion of the earth's crust several hundred miles long, must be far
+  more serious than could be produced by an earthquake radiating from a
+  single local volcanic focus.
+
+  From their disastrous effects on buildings and human lives, an
+  exaggerated importance has been imputed to earthquakes as agents of
+  geological change. Experience shows that even after a severe shock
+  which may have destroyed numerous towns and villages, together with
+  thousands of their inhabitants, the face of the country has suffered
+  scarcely any perceptible change, and that, in the course of a year or
+  two, when the ruined houses and prostrate trees have been cleared
+  away, little or no obvious trace of the catastrophe may remain. Among
+  the more enduring records of a great earthquake may be enumerated (a)
+  landslips, which lay bare hillsides, and sometimes pond back the
+  drainage of valleys so as to give rise to lakes; (b) alterations of
+  the topography, as in fissuring of the ground, or in the production of
+  inequalities whereby the drainage is affected; new valleys and new
+  lakes may thus be formed, while previously existing lakes may be
+  emptied; (c) permanent changes of level, either in an upward or
+  downward direction.
+
+  3. _Mountain-making._--This subject may be referred to here for the
+  striking evidence which it supplies of the importance of movements of
+  the earth's crust among geological processes. The structure of a great
+  mountain-chain such as the Alps proves that the crust of the earth has
+  been intensely plicated, crumpled and fractured. Vast piles of
+  sedimentary strata have been folded to such an extent as to occupy now
+  only half of their original horizontal extent. This compression in the
+  case of the Alps has been computed to amount to as much as 120,000
+  metres or 74 English miles, so that two points on the opposite sides
+  of that chain have been brought by so much nearer to each other than
+  they were originally before the movements. Besides such intense
+  plication, extensive rupturing of the crust has taken place in the
+  same range of mountains. Not only have the most ancient rocks been
+  squeezed up into the central axis of the chain, but huge slices of
+  them have been torn away from the main body, and thrust forward for
+  many miles, so as now actually to form the summits of mountains, which
+  are almost entirely composed of much younger formations. If these
+  colossal disturbances occurred rapidly, they would give rise to
+  cataclysms of inconceivable magnitude over the surface of the globe.
+  No record has been discovered of such accompanying devastation. But
+  whether sudden and violent, or prolonged and gradual, such stupendous
+  upturnings of the crust did undoubtedly take place, as is clearly
+  revealed in innumerable natural sections, which have been laid open by
+  the denudation of the crests and sides of the mountains.
+
+  4. _Metamorphism of Rocks_ (see METAMORPHISM).--During the movements
+  to which the crust of the earth has been subject, not only have the
+  rocks been folded and fractured, but they have likewise, in many
+  regions, acquired new internal structures, and have thus undergone a
+  process of "regional metamorphism." This rearrangement of their
+  substance has been governed by conditions which are probably not yet
+  all recognized, but among them we should doubtless include a high
+  temperature, intense pressure, mechanical movement resulting in
+  crushing, shearing and foliation, and the presence of water in their
+  pores. It is among igneous rocks that the progressive stages of
+  metamorphism can be most easily traced. Their definite original
+  structure and mineral composition afford a starting-point from which
+  the investigation may be begun and pursued. Where an igneous rock has
+  been invaded by metamorphic changes, it may be observed to have been
+  first broken down into separate lenticles, the cores of which may
+  still retain, with little or no alteration, the original
+  characteristic minerals and crystalline structure of the rock. Between
+  these lenticles, the intervening portions have been crushed down into
+  a powder or paste, which seems to have been squeezed round and past
+  them, and shows a laminated arrangement that resembles the
+  flow-structure in lavas. As the degree of metamorphism increases, the
+  lenticles diminish in size, and the intervening crushed and foliated
+  matrix increases in amount, until at last it may form the entire mass
+  of the rock. While the original minerals are thus broken down, new
+  varieties make their appearance. Of these, among the earliest to
+  present themselves are usually the micas, that impart their
+  characteristic silvery sheen to the surfaces of the folia along which
+  they spread. Younger felspars, as well as mica, are developed, and
+  there arise also sillimanite, garnet, andalusite and many others. The
+  texture becomes more coarsely crystalline, and the segregation of the
+  constituent minerals more definite along the lines of foliation. From
+  the finest silky phyllites a graduation may be traced through
+  successively coarser mica-schists, until we reach the almost granitic
+  texture of the coarsest gneisses.
+
+  Regional metamorphism has arisen in the heart of mountain-chains, and
+  in any other district where the deformation of the crust has been
+  sufficiently intense. There is another type of alteration termed
+  "contact-metamorphism," which is developed around masses of igneous
+  rock, especially where these have been intruded in large bosses among
+  stratified formations. It is particularly displayed around masses of
+  granite, where sandstones are found altered into quartzite, shales and
+  grits into schistose compounds, and where sometimes fossils are still
+  recognizable among the metamorphic minerals.
+
+
+_DIVISION II.--EPIGENE OR SUPERFICIAL ACTION_
+
+It is on the surface of the globe, and by the operation of agents
+working there, that at present the chief amount of visible geological
+change is effected. In considering this branch of inquiry, we are not
+involved in a preliminary difficulty regarding the very nature of the
+agencies as is the case in the investigation of plutonic action. On the
+contrary, the surface agents are carrying on their work under our very
+eyes. We can watch it in all its stages, measure its progress, and mark
+in many ways how accurately it represents similar changes which, for
+long ages previously, must have been effected by the same means. But in
+the systematic treatment of this subject we encounter a difficulty of
+another kind. We discover that while the operations to be discussed are
+numerous and readily observable, they are so interwoven into one great
+network that any separation of them under different subdivisions is sure
+to be more or less artificial and to convey an erroneous impression.
+While, therefore, under the unavoidable necessity of making use of such
+a classification of subjects, we must always bear in mind that it is
+employed merely for convenience, and that in nature superficial
+geological action must be continually viewed as a whole, since the work
+of each agent has constant reference to that of the others, and is not
+properly intelligible unless that connexion be kept in view.
+
+The movements of the air; the evaporation from land and sea; the fall of
+rain, hail and snow; the flow of rivers and glaciers; the tides,
+currents and waves of the ocean; the growth and decay of organized
+existence, alike on land and in the depths of the sea;--in short, the
+whole circle of movement, which is continually in progress upon the
+surface of our planet, are the subjects now to be examined. It is
+desirable to adopt some general term to embrace the whole of this range
+of inquiry. For this end the word epigene (Gr. [Greek: epi], upon) has
+been suggested as a convenient term, and antithetical to hypogene (Gr.
+[Greek: hypo], under), or subterranean action.
+
+A simple arrangement of this part of Geological Dynamics is in three
+sections:
+
+A. _Air._--The influence of the atmosphere in destroying and forming
+rocks.
+
+B. _Water._--The geological functions of the circulation of water
+through the air and between sea and land, and the action of the sea.
+
+C. _Life._--The part taken by plants and animals in preserving,
+destroying or reproducing geological formations.
+
+The words destructive, reproductive and conservative, employed in
+describing the operations of the epigene agents, do not necessarily
+imply that anything useful to man is destroyed, reproduced or preserved.
+On the contrary, the destructive action of the atmosphere may turn
+barren rock into rich soil, while its reproductive effects sometimes
+turn rich land into barren desert. Again, the conservative influence of
+vegetation has sometimes for centuries retained as barren morass what
+might otherwise have become rich meadow or luxuriant woodland. The
+terms, therefore, are used in a strictly geological sense, to denote the
+removal and re-deposition of material, and its agency in preserving what
+lies beneath it.
+
+
+  (A) _The Air._
+
+  As a geological agent, the air brings about changes partly by its
+  component gases and partly by its movements. Its destructive action is
+  both chemical and mechanical. The chemical changes are probably
+  mainly, if not entirely, due to the moisture of the air, and
+  particularly to the gases, vapours and organic matter which the
+  moisture contains. Dry air seems to have little or no appreciable
+  influence in promoting these reactions. As the changes in question are
+  similar to those much more abundantly brought about by rain they are
+  described in the following section under the division on rain.
+
+  Among the more recognizable mechanical changes effected in the
+  atmosphere, one of considerable importance is to be seen in the result
+  of great and rapid changes of temperature. Heat expands rocks, while
+  cold contracts them. In countries with a great annual range of
+  temperature, considerable difficulty is sometimes experienced in
+  selecting building materials liable to be little affected by the
+  alternate expansion and contraction, which prevents the joints of
+  masonry from remaining close and tight. In dry tropical climates,
+  where the days are intensely hot and the nights extremely cold, the
+  rapid nocturnal contraction produces a strain so great as to rival
+  frost in its influence upon the surface of exposed rocks,
+  disintegrating them into sand, or causing them to crack or peel off in
+  skins or irregular pieces. Dr Livingstone found in Africa (12 deg. S.
+  lat., 34 deg. E. long.) that surfaces of rock which during the day
+  were heated up to 137 deg. Fahr., cooled so rapidly by radiation at
+  night that, unable to sustain the strain of contraction, they split
+  and threw off sharp angular fragments from a few ounces to 100 or 200
+  [lb] in weight. In temperate regions this action, though much less
+  pronounced, still makes itself felt. In these climates, however, and
+  still more in high latitudes, somewhat similar results are brought
+  about by frost.
+
+  By its motion in wind the air drives loose sand over rocks, and in
+  course of time abrades and smoothes them. "Desert polish" is the name
+  given to the characteristic lustrous surface thus imparted. Holes are
+  said to be drilled in window glass at Cape Cod by the same agency.
+  Cavities are now and then hollowed out of rocks by the gyration in
+  them of little fragments of stone or grains of sand kept in motion by
+  the wind. Hurricanes form important geological agents upon land in
+  uprooting trees, and thus sometimes impeding the drainage of a country
+  and giving rise to the formation of peat mosses.
+
+  The reproductive action of the air arises partly from the effect of
+  the chemical and mechanical disintegration involved in the process of
+  "weathering," and partly from the transporting power of wind and of
+  aerial currents. The layer of soil, which covers so much of the
+  surface of the land, is the result of the decay of the underlying
+  rocks, mingled with mineral matter blown over the ground by wind, or
+  washed thither by rain, and with the mouldering remains of plants and
+  animals. The extent to which fine dust may be transported over the
+  surface of the land can hardly be realized in countries clothed with a
+  covering of vegetation, though even there, in dry weather during
+  spring, clouds of dust may often be seen blown away by wind from bare
+  ploughed fields. Intercepted by the leaves of plants and washed down
+  to their roots by rain, this dust goes to increase the soil below. In
+  arid climates, where dust clouds are dense and frequent, enormous
+  quantities of fine mineral particles are thus borne along and
+  accumulated. The remarkable deposit of "Loess," which is sometimes
+  more than 1500 ft. thick and covers extensive areas in China and other
+  countries, is regarded as due to the drifting of dust by wind. Again
+  the dunes of sand so abundant along the inner side of sandy
+  sea-beaches in many different parts of the world are attributable to
+  the same action.
+
+
+  (B) _Water._
+
+  In treating of the epigene action of water in geological processes it
+  will be convenient to deal first with its operations in traversing the
+  land, and then with those which it performs in the sea. The
+  circulation of water from land to sea and again from sea to land
+  constitutes the fundamental cause of most of the daily changes by
+  which the surface of the land is affected.
+
+  1. _Rain._--Rain effects two kinds of changes upon the surface of the
+  land. It acts _chemically_ upon soils and stones, and sinking under
+  ground continues a great series of similar reactions there. It acts
+  _mechanically_, by washing away loose materials, and thus powerfully
+  affecting the contours of the land. Its chemical action depends mainly
+  upon the nature and proportion of the substances which, in descending
+  to the earth, it abstracts from the atmosphere. Rain always absorbs a
+  little air, which, in addition to its nitrogen and oxygen, contains
+  carbonic acid, and in minute proportions, sodium chloride, sulphuric
+  acid and other ingredients, especially inorganic dust, organic
+  particles and living germs. Probably the most generally efficient of
+  these constituents are oxygen, carbonic acid and organic matter. Armed
+  with these reagents, rain effects a chemical decomposition of the
+  rocks on which it falls, and through which it sinks underground. The
+  principal changes thus produced are as follows: (a) Oxidation.--Owing
+  to the prominence of oxygen in rain-water, and its readiness to unite
+  with any substance which can contain more of it, a thin oxidized
+  pellicle is formed on the surface of many rocks on which rain falls,
+  and this oxidized layer if not at once washed off, sinks deeper until
+  a crust is formed over the stone. A familiar illustration of this
+  action is afforded by the rust, or oxide, which forms on iron when
+  exposed to moisture, though this iron may be kept long bright if
+  allowed to remain screened from moist air and rain. (b)
+  Deoxidation.--Organic matter having an affinity for more oxygen
+  decomposes peroxides by depriving them of some part of their share of
+  that element and reducing them to protoxides. These changes are
+  especially noticeable among the iron oxides so abundantly diffused
+  among rocks. Hence rain-water, in sinking through soil and obtaining
+  such organic matter, becomes thereby a reducing agent. (c)
+  Solution.--This may take place either by the simple action of the
+  water, as in the solution of rock-salt, or by the influence of the
+  carbonic acid present in the rain. (d) Formation of Carbonates.--A
+  familiar example of the action of carbonic acid in rain is to be seen
+  in the corrosion of exposed marble slabs. The carbonic acid dissolves
+  some of the lime, which, as a bicarbonate, is held in solution in the
+  carbonated water, but is deposited again when the water loses its
+  carbonic acid or evaporates. It is not merely carbonates, however,
+  which are liable to this kind of destruction. Even silicates of lime,
+  potash and soda, combinations existing abundantly as constituents of
+  rocks, are attacked; their silica is liberated, and their alkalis or
+  alkaline earths, becoming carbonates, are removed in solution. (e)
+  Hydration.--Some minerals, containing little or no water, and
+  therefore called anhydrous, when exposed to the action of the
+  atmosphere, absorb water, or become hydrous, and are then usually more
+  prone to further change. Hence the rocks of which they form part
+  become disintegrated.
+
+  Besides the reactions here enumerated, a considerable amount of decay
+  may be observed as the result of the presence of sulphuric and nitric
+  acid in the air, especially in that of large towns and manufacturing
+  districts, where much coal is consumed. Metallic surfaces, as well as
+  various kinds of stone, are there corroded, while the mortar of walls
+  may often be observed to be slowly swelling out and dropping off,
+  owing to the conversion of the lime into sulphate. Great injury is
+  likewise done from a similar cause to marble monuments in exposed
+  graveyards.
+
+  The general result of the disintegrating action of the air and of
+  rain, including also that of plants and animals, to be noticed in the
+  sequel, is denoted by the term "weathering." The amount of decay
+  depends partly on conditions of climate, especially the range of
+  temperature, the abundance of moisture, height above the sea and
+  exposure to prevalent winds. Many rocks liable to be saturated with
+  rain and rapidly dried under a warm sun are apt to disintegrate at the
+  surface with comparative rapidity. The nature and progress of the
+  weathering are mainly governed by the composition and texture of the
+  rocks exposed to it. Rocks composed of particles liable to little
+  chemical change from the influence of moisture are best fitted to
+  resist weathering, provided they possess sufficient cohesion to
+  withstand the mechanical processes of disintegration. Siliceous
+  sandstones are excellent examples of this permanence. Consisting
+  wholly or mainly of the durable mineral quartz, they are sometimes
+  able so to withstand decay that buildings made of them still retain,
+  after the lapse of centuries, the chisel-marks of the builders. Some
+  rocks, which yield with comparative rapidity to the chemical attacks
+  of moisture, may show little or no mark of disintegration on their
+  surface. This is particularly the case with certain calcareous rocks.
+  Limestone when pure is wholly soluble in acidulated water. Rain
+  falling on such a rock removes some of it in solution, and will
+  continue to do so until the whole is dissolved away. But where a
+  limestone is full of impurities, a weathered crust of more or less
+  insoluble particles remains after the solution of the calcareous part
+  of the stone. Hence the relative purity of limestones may be roughly
+  determined by examining their weathered surfaces, where, if they
+  contain much sand, the grains will be seen projecting from the
+  calcareous matrix, and where, should the rock be very ferruginous, the
+  yellow hydrous peroxide, or ochre, will be found as a powdery crust.
+  In limestones containing abundant encrinites, shells, or other organic
+  remains, the weathered surface commonly presents the fossils standing
+  out in relief. The crystalline arrangement of the lime in the organic
+  structures enables them to resist disintegration better than the
+  general mechanically aggregated matrix of the rock. An experienced
+  fossil collector will always search well such weathered surfaces, for
+  he often finds there, delicately picked out by the weather, minute
+  and frail fossils which are wholly invisible on a freshly broken
+  surface of the stone. Many rocks weather with a thick crust, or even
+  decay inwards for many feet or yards. Basalt, for example, often shows
+  a yellowish-brown ferruginous layer on its surface, formed by the
+  conversion of its felspar into kaolin, and the removal of its calcium
+  silicate as carbonate, by the hydration of its olivine and augite and
+  their conversion into serpentine, or some other hydrous magnesian
+  silicate, and by the conversion of its magnetite into limonite.
+  Granite sometimes shows in a most remarkable way the distance to which
+  weathering can reach. It may occasionally be dug into for a depth of
+  20 or 30 ft., the quartz crystals and veins retaining their original
+  positions, while the felspar is completely kaolinized. It is to the
+  endlessly varied effects of weathering that the abundant fantastic
+  shapes assumed by crags and other rocky masses are due. Most varieties
+  of rock have their own characteristic modes of weathering, whereby
+  they may be recognized even from a distance. To some of these features
+  reference will be made in Part VIII.
+
+  The mechanical action of rain, which is intimately bound up with its
+  chemical action, consists in washing off the fine superficial
+  particles of rocks which have been corroded and loosened by the
+  process of weathering, and in thus laying open fresh portions to the
+  same influences of decay. The detritus so removed is partly carried
+  down into the soil which is thereby enriched, partly held in
+  suspension in the little runnels into which the rain-drops gather as
+  they begin to flow over the land, partly pushed downwards along the
+  surface of sloping ground. A good deal of it finds its way into the
+  nearest brooks and rivers, which are consequently made muddy by heavy
+  rain.
+
+  It is natural that a casual consideration of the subject should lead
+  to an impression that, though the general result of the fall of rain
+  upon a land-surface must lead to some amount of disintegration and
+  lowering of that surface, the process must be so slow and slight as
+  hardly to be considered of much importance among geological
+  operations. But further attention will show such an impression to be
+  singularly erroneous. It loses sight of the fact that a change which
+  may be hardly appreciable within a human lifetime, or even within the
+  comparatively brief span of geological time embraced in the compass of
+  human history, may nevertheless become gigantic in its results in the
+  course of immensely protracted periods. An instructive lesson in the
+  erosive action of rain may be found in the pitted and channelled
+  surface of ground lying under the drip of the eaves of a cottage. The
+  fragments of stone and pebbles of gravel that form part of the soil
+  can there be seen sticking out of the ground, because being hard they
+  resist the impetus of the falling drops, protecting for a time the
+  earth beneath them, while that which surrounded and covered them is
+  washed away. From this familiar illustration the observer may advance
+  through every stage in the disappearance of material which once
+  covered the surface, until he comes to examples where once continuous
+  and thick sheets of solid rock have been reduced to a few fragments or
+  have been entirely removed. Since the whole land surface over which
+  rain falls is exposed to this waste, the superficial covering of
+  decayed rock or soil, as Hutton insisted, is constantly, though
+  imperceptibly, travelling outward and downward to the sea. In this
+  process of transport rain is an important carrying agent, while at the
+  same time it serves to connect the work of the other disintegrating
+  forces, and to make it conducive to the general degradation of the
+  land. Though this decay is general and constant, it is obviously not
+  uniform. In some places where, from the nature of the rock, from the
+  flatness of the ground, or from other causes, rain works under great
+  difficulties, the rate of waste may be extremely slow. In other places
+  it may be rapid enough to be appreciable from year to year. A survey
+  of this department of geological activity shows how unequal wasting by
+  rain, combined with the operations of brooks and rivers, has produced
+  the details of the present relief of the land, those tracts where the
+  destruction has been greatest forming hollows and valleys, others,
+  where it has been less, rising into ridges and hills (Part VIII.).
+
+  Rain-action is not merely destructive, but is accompanied with
+  reproductive effects, chief of which is the formation of soil. In
+  favourable situations it has gathered together accumulations of loam
+  and earth from neighbouring higher ground, such as the "brick-earth,"
+  "head," and "rain-wash" of the south of England--earthy deposits,
+  sometimes full of angular stones, derived from the subaerial waste of
+  the rocks of the neighbourhood.
+
+  2. _Underground Water._--Of the rain which falls upon the land one
+  portion flows off into brooks and rivers by which the water is
+  conducted back to the ocean; the larger part, however, sinks into the
+  ground and disappears. It is this latter part which has now to be
+  considered. Over and above the proportion of the rainfall which is
+  absorbed by living vegetation and by the soil, there is a continual
+  filtering down of the water from the surface into the rocks that lie
+  below, where it partly lodges in pores and interstices, and partly
+  finds its way into subterranean joints and fissures, in which it
+  performs an underground circulation, and ultimately issues once more
+  at the surface in the form of springs (q.v.). In the course of this
+  circulation the water performs an important geological task. Not only
+  carrying down with it the substances which the rain has abstracted
+  from the air, but obtaining more acids and organic matter from the
+  soil, it is enabled to effect chemical changes in the rocks
+  underneath, and especially to dissolve limestone and other calcareous
+  formations. So considerable is the extent of this solution in some
+  places that the springs which come to the surface, and begin there to
+  evaporate and lose some of their carbonic acid, contain more dissolved
+  lime than they can hold. They consequently deposit it in the form of
+  calcareous tuff or sinter (q.v.). Other subterranean waters issue with
+  a large proportion of iron-salts in solution which form deposits of
+  ochre. The various mineral springs so largely made use of for the
+  mitigation or cure of diseases owe their properties to the various
+  salts which they have dissolved out of rocks underground. As the
+  result of prolonged subterranean solution in limestone districts,
+  passages and caves (q.v.), sometimes of great width and length, are
+  formed. When these lie near the surface their roofs sometimes fall in
+  and engulf brooks and rivers, which then flow for some way underground
+  until the tunnels conduct them back again to daylight on some lower
+  ground.
+
+  Besides its chemical activity water exerts among subterranean rocks a
+  mechanical influence which leads to important changes in the
+  topography of the surface. In removing the mineral matter, either in
+  solution or as fine sediment, it sometimes loosens the support of
+  overlying masses of rock which may ultimately give way on sloping
+  ground, and rush down the declivities in the form of landslips. These
+  destructive effects are specially frequent on the sides of valleys in
+  mountainous countries and on lines of sea-cliff.
+
+  3. _Brooks and Rivers._--As geological agents the running waters on
+  the face of the land play an important part in epigene changes. Like
+  rain and springs they have both a chemical and a mechanical action.
+  The latter receives most attention, as it undoubtedly is the more
+  important; but the former ought not to be omitted in any survey of the
+  general waste of the earth's surface. The water of rivers must possess
+  the powers of a chemical solvent like rain and springs, though its
+  actual work in this respect can be less easily measured, seeing that
+  river water is directly derived from rain and springs, and necessarily
+  contains in solution mineral substances supplied to it by them and not
+  by its own operation. Nevertheless, it is sometimes easy to prove that
+  streams dissolve chemically the rocks of their channels. Thus, in
+  limestone districts the base of the cliffs of river ravines may be
+  found eaten away into tunnels, arches, and overhanging projections,
+  presenting in their smooth surfaces a great contrast to the angular
+  jointed faces of the same rock, where now exposed to the influence
+  only of the weather on the higher parts of the cliff.
+
+  The mechanical action of rivers consists (a) in transporting mud,
+  sand, gravel and blocks of stone from higher to lower levels; (b) in
+  using these loose materials to widen and deepen their channels by
+  erosion; (c) in depositing their load of detritus wherever possible
+  and thus to make new geological formations.
+
+  (a) _Transporting Power._--River-water is distinguished from that of
+  springs by being less transparent, because it contains more or less
+  mineral matter in suspension, derived mainly from what is washed down
+  by rain, or carried in by brooks, but partly also from the abrasion of
+  the water-channels by the erosive action of the rivers themselves. The
+  progress of this burden of detritus may be instructively followed from
+  the mountain-tributaries of a river down to the mouth of the main
+  stream. In the high grounds the water-courses may be observed to be
+  choked with large fragments of rock disengaged from the cliffs and
+  crags on either side. Traced downwards the blocks are seen to become
+  gradually smaller and more rounded. They are ground against each
+  other, and upon the rocky sides and bottom of the channel, getting
+  more and more reduced as they descend, and at the same time abrading
+  the rocks over or against which they are driven. Hence a great deal of
+  debris is produced, and is swept along by the onward and downward
+  movement of the water. The finer portions, such as mud and fine sand,
+  are carried in suspension, and impart the characteristic turbidity to
+  river-water; the coarser sand and gravel are driven along the
+  river-bottom. The proportion of suspended mineral matter has been
+  ascertained with more or less precision for a number of rivers. As an
+  illustrative example of a river draining a vast area with different
+  climates, forms of surface and geological structure the Mississippi
+  may be cited. The average proportion of sediment in its water was
+  ascertained by Humphreys and Abbot to be 1/1500 by weight or 1/2900 by
+  volume. These engineers found that, in addition to this suspended
+  material, coarse detritus is constantly being pushed forward along the
+  bed of the river into the Gulf of Mexico, to an amount which they
+  estimated at about 750,000,000 cubic ft. of sand, earth and gravel;
+  they concluded that the Mississippi carries into the gulf every year
+  an amount of mechanically transported sediment sufficient to make a
+  prism one square mile in area and 268 ft. in height.
+
+  (b) _Excavating Power._--It is by means of the sand, gravel and stones
+  which they drive against the sides and bottoms of their channels that
+  streams have hollowed out the beds in which they flow. Not only is the
+  coarse detritus reduced in size by the friction of the stones against
+  each other, but, at the same time, these materials abrade the rocks
+  against which they are driven by the current. Where, owing to the
+  shape of the bottom of the channel, the stones are caught in eddies,
+  and are kept whirling round there, they become more and more worn down
+  themselves, and at the same time scour out basin-shaped cavities, or
+  "pot-holes," in the solid rock below. The uneven bed of a swiftly
+  flowing stream may in this way be honeycombed with such eroded basins
+  which coalesce and thus appreciably lower the surface of the bed. The
+  steeper the channel, other conditions being equal, the more rapid will
+  be the erosion. Geological structure also affects the character and
+  rate of the excavation. Where the rocks are so arranged as to favour
+  the formation and persistence of a waterfall, a long chasm may be
+  hollowed out like that of the Niagara below the falls, where a hard
+  thick bed of nearly flat limestone lies on softer and more easily
+  eroded shales. The latter are scooped out from underneath the
+  limestone, which from time to time breaks off in large masses and the
+  waterfall gradually retreats up stream, while the ravine is
+  proportionately lengthened. To the excavating power of rivers the
+  origin of the valley systems of the dry land must be mainly assigned
+  (see Part VIII.).
+
+  (c) _Reproductive Power._--So long as a stream flows over a steep
+  declivity its velocity suffices to keep the sediment in suspension,
+  but when from any cause, such as a diminution of slope, the velocity
+  is checked, the transporting power is lessened and the sediment begins
+  to fall to the bottom and to remain there. Hence various river-formed
+  or "alluvial" deposits are laid down. These sometimes cover
+  considerable spaces at the foot of mountains. The floors of valleys
+  are strewn with detritus, and their level may thereby be sensibly
+  raised. In floods the ground inundated on either side of a stream
+  intercepts some part of the detritus, which is then spread over the
+  flood-plain and gradually heightens it. At the same time the stream
+  continues to erode the channel, and ultimately is unable to reach the
+  old flood-plain. It consequently forms a new plain at a lower level,
+  and thus, by degrees, it comes to be flanked on either side by a
+  series of successive terraces or platforms, each of which marks one of
+  its former levels. Where a river enters a large body of water its
+  current is checked. Some of its sediment is consequently dropped, and
+  by slow accumulation forms a delta (q.v.). On land, every lake in
+  mountain districts furnishes instances of this kind of alluvium. But
+  the most important deltas are those formed in the sea at the mouths of
+  the larger rivers of the globe. Off many coast-lines the detritus
+  washed from the land gathers into bars, which enclose long strips of
+  water more or less completely separated from the sea outside and known
+  as lagoons. A chain of such lagoon-barriers stretches for hundreds of
+  miles round the Gulf of Mexico and the eastern shores of the United
+  States.
+
+  4. _Lakes._--These sheets of water, considered as a whole, do not
+  belong to the normal system of drainage on the land whereby valleys
+  are excavated. On the contrary they are exceptional to it; for the
+  constant tendency of running water is to fill them up, or to drain
+  them by wearing down the barriers that contain them at their outflow.
+  Some of them are referable to movements of the terrestrial crust
+  whereby depressions arise on the surface of the land, as has been
+  noted after earthquakes. Others have arisen from solution such as that
+  of rock-salt or of limestone, the removal of which by underground
+  water causes a subsidence of the ground above. A third type of
+  lake-basin occurs in regions that are now or have once been subject to
+  the erosive action of glaciers (see under next subdivision,
+  _Terrestrial Ice_). Many small lakes or tarns have been caused by the
+  deposit of debris across a valley as by landslips or moraines.
+  Considered from a geological point of view, lakes perform an important
+  function in regulating the drainage of the ground below their outfall
+  and diminishing the destructive effects of floods, in filtering the
+  water received from their affluent streams, and in providing
+  undisturbed areas of deposit in which thick and extensive lacustrine
+  formations may be accumulated. In the inland basins of some dry
+  climates the lakes are salt, owing to excess of evaporation, and their
+  bottoms become the sites of chemical deposits, particularly of
+  chlorides of sodium and magnesium, and calcium sulphate and carbonate.
+
+  5. _Terrestrial Ice._--Each of the forms assumed by frozen water has
+  its own characteristic action in geological processes. Frost has a
+  powerful influence in breaking up damp soils and surfaces of stone in
+  the pores or cracks of which moisture has lodged. The water in
+  freezing expands, and in so doing pushes asunder the component
+  particles of soil or stone, or widens the space between the walls of
+  joints or crevices. When the ice melts the loosened grains remain
+  apart ready to be washed away by rain or blown off by wind, while by
+  the widening of joints large blocks of rock are detached from the
+  faces of cliffs. Where rivers or lakes are frozen over the ice exerts
+  a marked pressure on their banks; and when it breaks up large sheets
+  of it are driven ashore, pushing up quantities of gravel and stones
+  above the level of the water. The piling up of the disrupted ice
+  against obstructions in rivers ponds back the water, and often leads
+  to destructive floods when the ice barriers break. Where the ice has
+  formed round boulders in shallow water, or at the bottom
+  ("anchor-ice"), it may lift these up when the frost gives way, and may
+  transport them for some distance. Ice formed in the atmosphere, and
+  descending to the ground in the form of hail, often causes great
+  destruction to vegetation and not infrequently to animal life. Where
+  the frozen moisture reaches the earth as snow, it serves to protect
+  rock, soil and vegetation from the effects of frost; but on sloping
+  ground it is apt to give rise to destructive avalanches or landslips,
+  while indirectly, by its rapid melting, it may cause serious floods in
+  rivers.
+
+  But the most striking geological work performed by terrestrial ice is
+  that achieved by glaciers (q.v.) and ice-sheets. These vast masses of
+  moving ice, when they descend from mountains where the steeper rocks
+  are clear of snow, receive on their surface the debris detached by
+  frost from the declivities above, and bear these materials to lower
+  levels or to the sea. Enormous quantities of rock-rubbish are thus
+  transported in the Alps and other high mountain ranges. When the ice
+  retreats the boulders carried by it are dropped where it melts, and
+  left there as memorials of the former extension of the glaciers.
+  Evidence of this nature proves the much wider extent of the Alpine ice
+  at a comparatively recent geological date. It can also be shown that
+  detritus from Scandinavia has been ice-borne to the south-east of
+  England and far into the heart of Europe.
+
+  The ice, by means of grains of sand and pieces of stone which it drags
+  along, scores, scratches and polishes the surfaces of rock underneath
+  it, and, in this way, produces the abundant fine sediment that gives
+  the characteristic milky appearance to the rivers that issue from the
+  lower ends of glaciers. By such long-continued attrition the rocks are
+  worn down, portions of them of softer nature, or where the ice acts
+  with especial vigour, are hollowed out into cavities which, on the
+  disappearance of the ice, may be filled with water and become tarns or
+  lakes. Rocks over which land-ice has passed are marked by a peculiar
+  smooth, flowing outline, which forms a contrast to the more rugged
+  surface produced by ordinary weathering. They are covered with
+  groovings, which range from the finest striae left by sharp grains of
+  sand to deep ruts ground out by blocks of stone. The trend of these
+  markings shows the direction in which the ice flowed. By their
+  evidence the position and movement of former glaciers in countries
+  from which the ice has entirely vanished may be clearly determined
+  (see GLACIAL PERIOD).
+
+  6. _The Sea._--The physical features of the sea are discussed in
+  separate articles (see OCEAN AND OCEANOGRAPHY). The sea must be
+  regarded as the great regulator of temperature and climate over the
+  globe, and as thus exerting a profound influence on the distribution
+  of plant and animal life. Its distinctly geological work is partly
+  erosive and partly reproductive. As an eroding agent it must to some
+  extent effect chemical decompositions in the rocks and sediments over
+  which it spreads; but these changes have not yet been satisfactorily
+  studied. Undoubtedly, its chief destructive power is of a mechanical
+  kind, and arises from the action of its waves in beating upon
+  shore-cliffs. By the alternate compression and expansion of the air in
+  crevices of the rocks on which heavy breakers fall, and by the
+  hydraulic pressure which these masses of sea-water exert on the walls
+  of the fissures into which they rush, large masses of rock are
+  loosened and detached, and caves and tunnels are drilled along the
+  base of sea-cliffs. Probably still more efficacious are the blows of
+  the loose shingle, which, caught up and hurled forward by the waves,
+  falls with great force upon the shore rocks, battering them as with a
+  kind of artillery until they are worn away. The smooth surfaces of the
+  rocks within reach of the waves contrasted with their angular forms
+  above that limit bear witness to the amount of waste, while the
+  rounded forms of the boulders and shingle show that they too are being
+  continually reduced in size. Thus the sea, by its action on the
+  coasts, produces much sediment, which is swept away by its waves and
+  currents and strewn over its floor. Besides this material, it is
+  constantly receiving the fine silt and sand carried down by rivers. As
+  the floor of the ocean is thus the final receptacle for the waste of
+  the land, it becomes the chief era on the surface of the globe for the
+  accumulation of new stratified formations. And such has been one of
+  its great functions since the beginning of geological time, as is
+  proved by the rocks that form the visible part of the earth's crust,
+  and consist in great part of marine deposits. Chemical precipitates
+  take place more especially in enclosed parts of the sea, where
+  concentration of the water by evaporation can take place, and where
+  layers of sodium chloride, calcium sulphate and carbonate, and other
+  salts are laid down. But the chief marine accumulations are of
+  detrital origin. Near the land and for a variable distance extending
+  sometimes to 200 or 300 m. from shore the deposits consist chiefly of
+  sediments derived from the waste of the land, the finer silts being
+  transported farthest from their source. At greater depths and
+  distances the ocean floor receives a slow deposit of exceedingly fine
+  clay, which is believed to be derived from the decomposition of pumice
+  and volcanic dust from insular or submarine volcanoes. Wide tracts of
+  the bottom are covered with various forms of ooze derived from the
+  accumulation of the remains of minute organisms.
+
+
+  (C) _Life._
+
+  Among the agents by which geological changes are carried on upon the
+  surface of the globe living organisms must be enumerated. Both plants
+  and animals co-operate with the inorganic agents in promoting the
+  degradation of the land. In some cases, on the other hand, they
+  protect rocks from decay, while, by the accumulation of their remains,
+  they give rise to extensive formations both upon the land and in the
+  sea. Their operations may hence be described as alike destructive,
+  conservative and reproductive. Under this heading also the influence
+  of Man as a geological agent deserves notice.
+
+  (a) _Plants._--Vegetation promotes the disintegration of rocks and
+  soil in the following ways: (1) By keeping the surfaces of stone
+  moist, and thus promoting both mechanical and chemical dissolution, as
+  is especially shown by liverworts, mosses and other moisture-loving
+  plants. (2) By producing through their decay carbonic and other
+  acids, which, together with decaying organic matter taken up by
+  passing moisture, become potent in effecting the chemical
+  decomposition of rocks and in promoting the disintegration of soils.
+  (3) By inserting their roots or branches between joints of rock, which
+  are thereby loosened, so that large slices may be eventually wedged
+  off. (4) By attracting rain, as thick woods, forests and peat-mosses
+  do, and thus accelerating the general waste of a country by running
+  water. (5) By promoting the decay of diseased and dead plants and
+  animals, as when fungi overspread a damp rotting tree or the carcase
+  of a dead animal.
+
+  That plants also exert a conservative influence on the surface of the
+  land is shown in various ways. (1) The formation of a stratum of turf
+  protects the soil and rocks underneath from being rapidly
+  disintegrated and washed away by atmospheric action. (2) Many plants,
+  even without forming a layer of turf, serve by their roots or branches
+  to protect the loose sand or soil on which they grow from being
+  removed by wind. The common sand-carex and other arenaceous plants
+  bind the loose sand-dunes of our coasts, and give them a permanence,
+  which would at once be destroyed were the sand laid bare again to
+  storms. The growth of shrubs and brushwood along the course of a
+  stream not only keeps the alluvial banks from being so easily
+  undermined and removed as would otherwise be the case, but serves to
+  arrest the sediment in floods, filtering the water and thereby adding
+  to the height of the flood plain. (3) Some marine plants, like the
+  calcareous nullipores, afford protection to shore rocks by covering
+  them with a hard incrustation. The tangles and smaller Fuci which grow
+  abundantly on the littoral zone break the force of the waves or
+  diminish the effects of ground swell. (4) Forests and brushwood
+  protect the soil, especially on slopes, from being washed away by rain
+  or ploughed up by avalanches.
+
+  Plants contribute by the aggregation of their remains to the formation
+  of stratified deposits. Some marine algae which secrete carbonate of
+  lime not only encrust rocks but give rise to sheets of submarine
+  limestone. An analogous part is played in fresh-water lakes by various
+  lime-secreting plants, such as _Chara_. Long-continued growth of
+  vegetation has, in some regions, produced thick accumulations of a
+  dark loam, as in the black cotton soil (_regur_) of India, and the
+  black earth (_tchernozom_) of Russia. Peat-mosses are formed in
+  temperate and arctic climates by the growth of marsh-loving plants,
+  sometimes to a thickness of 40 or 50 ft. In tropical regions the
+  mangrove swamps on low moist shores form a dense jungle, sometimes 20
+  m. broad, which protects these shores from the sea until, by the
+  arrest of sediment and the constant contribution of decayed
+  vegetation, the spongy ground is at last turned into firm soil. Some
+  plants (diatoms) can abstract silica and build it into their
+  framework, so that their remains form a siliceous deposit or ooze
+  which covers spaces of the deep sea-floor estimated at more than ten
+  millions of square miles in extent.
+
+  (b) _Animals._--These exert a destructive influence in the following
+  ways: (1) By seriously affecting the composition and arrangement of
+  the vegetable soil. Worms bring up the lower portions of the soil to
+  the surface, and while thus promoting its fertility increase its
+  liability to be washed away by rain. Burrowing animals, by throwing up
+  the soil and subsoil, expose these to be dried and blown away by the
+  wind. At the same time their subterranean passages serve to drain off
+  the superficial water and to injure the stability of the surface of
+  the ground above them. In Britain the mole and rabbit are familiar
+  examples. (2) By interfering with or even diverting the flow of
+  streams. Thus beaver-dams check the current of water-courses,
+  intercept floating materials, and sometimes turn streams into new
+  channels. The embankments of the Mississippi are sometimes weakened to
+  such an extent by the burrowings of the cray-fish as to give way and
+  allow the river to inundate the surrounding country. Similar results
+  have happened in Europe from subterranean operations of rats. (3) Some
+  mollusca bore into stone or wood and by the number of contiguous
+  perforations greatly weaken the material. (4) Many animals exercise a
+  ruinously destructive influence upon vegetation. Of the numerous
+  plagues of this kind the locust, phylloxera and Colorado beetle may be
+  cited.
+
+  The most important geological function performed by animals is the
+  formation of new deposits out of their remains. It is chiefly by the
+  lower grades of the animal kingdom that this work is accomplished,
+  especially by molluscs, corals and foraminifera. Shell-banks are
+  formed abundantly in such comparatively shallow and enclosed basins as
+  that of the North Sea, and on a much more extensive scale on the floor
+  of the West Indian seas. By the coral polyps thick masses of
+  limestones have been built up in the warmer seas of the globe (see
+  CORAL REEFS). The floor of the Atlantic and other oceans is covered
+  with a fine calcareous ooze derived mainly from the remains of
+  foraminifera, while in other regions the bottom shows a siliceous ooze
+  formed almost entirely of radiolaria. Vertebrate animals give rise to
+  phosphatic deposits formed sometimes of their excrement, as in guano
+  and coprolites, sometimes of an accumulation of their bones.
+
+  (c) _Man._--No survey of the geological workings of plant and animal
+  life upon the surface of the globe can be complete which does not take
+  account of the influence of man--an influence of enormous and
+  increasing consequence in physical geography, for man has introduced,
+  as it were, an element of antagonism to nature. His interference shows
+  itself in his relations to climate, where he has affected the
+  meteorological conditions of different countries: (1) By removing
+  forests, and laying bare to the sun and winds areas which were
+  previously kept cool and damp under trees, or which, lying on the lee
+  side, were protected from tempests. It is supposed that the wholesale
+  destruction of the woodlands formerly existing in countries bordering
+  the Mediterranean has been in part the cause of the present
+  desiccation of these districts. (2) By drainage, whereby the
+  discharged rainfall is rapidly removed, and the evaporation is
+  lessened, with a consequent diminution of rainfall and some increase
+  in the general temperature of a country. (3) By the other processes of
+  agriculture, such as the transformation of moor and bog into
+  cultivated land, and the clothing of bare hillsides with green crops
+  or plantations of coniferous and hardwood trees.
+
+  Still more obvious are the results of human interference with the flow
+  of water: (1) By increasing or diminishing the rainfall man directly
+  affects the volume of rivers. (2) By his drainage operations he makes
+  the rain to run off more rapidly than before, and thereby increases
+  the magnitude of floods and of the destruction caused by them. (3) By
+  wells, bores, mines, or other subterranean works he interferes with
+  the underground waters, and consequently with the discharge of
+  springs. (4) By embanking rivers he confines them to narrow channels,
+  sometimes increasing their scour, and enabling them to carry their
+  sediment further seaward, sometimes causing them to deposit it over
+  the plains and raise their level. (5) By his engineering operations
+  for water-supply he abstracts water from its natural basins and
+  depletes the streams.
+
+  In many ways man alters the aspect of a country: (1) By changing
+  forest into bare mountain, or clothing bare mountains with forest. (2)
+  By promoting the growth or causing the removal of peat-mosses. (3) By
+  heedlessly uncovering sand-dunes, and thereby setting in motion a
+  process of destruction which may convert hundreds of acres of fertile
+  land into waste sand, or by prudently planting the dunes with
+  sand-loving vegetation and thus arresting their landward progress. (4)
+  By so guiding the course of rivers as to make them aid him in
+  reclaiming waste land, and bringing it under cultivation. (5) By piers
+  and bulwarks, whereby the ravages of the sea are stayed, or by the
+  thoughtless removal from the beach of stones which the waves had
+  themselves thrown up, and which would have served for a time to
+  protect the land. (6) By forming new deposits either designedly or
+  incidentally. The roads, bridges, canals, railways, tunnels, villages
+  and towns with which man has covered the surface of the land will in
+  many cases form a permanent record of his presence. Under his hand the
+  whole surface of civilized countries is very slowly covered with a
+  stratum, either formed wholly by him or due in great measure to his
+  operations and containing many relics of his presence. The soil of
+  ancient towns has been increased to a depth of many feet by their
+  successive destructions and renovations.
+
+  Perhaps the most subtle of human influences are to be seen in the
+  distribution of plant and animal life upon the globe. Some of man's
+  doings in this domain are indeed plain enough, such as the extirpation
+  of wild animals, the diminution or destruction of some forms of
+  vegetation, the introduction of plants and animals useful to himself,
+  and especially the enormous predominance given by him to the cereals
+  and to the spread of sheep and cattle. But no such extensive
+  disturbance of the normal conditions of the distribution of life can
+  take place without carrying with it many secondary effects, and
+  setting in motion a wide cycle of change and of reaction in the animal
+  and vegetable kingdoms. For example, the incessant warfare waged by
+  man against birds and beasts of prey in districts given up to the
+  chase leads sometimes to unforeseen results. The weak game is allowed
+  to live, which would otherwise be killed off and give more room for
+  the healthy remainder. Other animals which feed perhaps on the same
+  materials as the game are by the same cause permitted to live
+  unchecked, and thereby to act as a further hindrance to the spread of
+  the protected species. But the indirect results of man's interference
+  with the regime of plants and animals still require much prolonged
+  observation.
+
+
+PART V.--GEOTECTONIC OR STRUCTURAL GEOLOGY
+
+From a study of the nature and composition of minerals and rocks, and an
+investigation of the different agencies by which they are formed and
+modified, the geologist proceeds to inquire how these materials have
+been put together so as to build up the visible part of the earth's
+crust. He soon ascertains that they have not been thrown together wholly
+at random, but that they show a recognizable order of arrangement. Some
+of them, especially those of most recent growth, remain in their
+original condition and position, but, in proportion to their antiquity,
+they generally present increasing alteration, until it may no longer be
+possible to tell what was their pristine state. As by far the largest
+accessible portion of the terrestrial crust consists of stratified
+rocks, and as these furnish clear evidence of most of the modifications
+to which they have been subjected in the long course of geological
+history, it is convenient to take them into consideration first. They
+possess a number of structures which belong to the original conditions
+in which they were accumulated. They present in addition other
+structures which have been superinduced upon them, and which they share
+with the unstratified or igneous rocks.
+
+
+1. ORIGINAL STRUCTURES
+
+(a) _Stratified Rocks._--This extensive and important series is above
+all distinguished by possessing a prevailing stratified arrangement.
+Their materials have been laid down in laminae, layers and strata, or
+beds, pointing generally to the intermittent deposition of the sediments
+of which they consist. As this stratification was, as a rule, originally
+nearly or quite horizontal, it serves as a base from which to measure
+any subsequent disturbance which the rocks have undergone. The
+occurrence of false-bedding, i.e. bands of inclined layers between the
+normal planes of stratification, does not form any real exception; but
+indicates the action of shifting currents whereby the sediment was
+transported and thrown down. Other important records of the original
+conditions of deposit are supplied by ripple-marks, sun-cracks,
+rain-prints and concretions.
+
+  From the nature of the material further light is cast on the
+  geographical conditions in which the strata were accumulated. Thus,
+  conglomerates indicate the proximity of old shore-lines, sandstones
+  mark deposits in comparatively shallow water, clays and shales point
+  to the tranquil accumulation of fine silt at a greater depth and
+  further from land, while fossiliferous limestones bear witness to
+  clearer water in which organisms flourished at some distance from
+  deposits of sand and mud. Again, the alternation of different kinds of
+  sediment suggests a variability in the conditions of deposition, such
+  as a shifting of the sediment-bearing currents and of the areas of
+  muddy and clear water. A thick group of conformable strata, that is, a
+  series of deposits which show no discordance in their stratification,
+  may usually be regarded as having been laid down on a sea-floor that
+  was gently sinking. Here and there evidence is obtainable of the
+  limits or of the progress of the subsidence by what is called
+  "overlap." Of the absolute length of time represented by any strata or
+  groups of strata no satisfactory estimates can yet be formed. Certain
+  general conclusions may indeed be drawn, and comparisons may be made
+  between different series of rocks. Sandstones full of false-bedding
+  were probably accumulated more rapidly than finely-laminated shales or
+  clays. It is not uncommon in certain Carboniferous formations to find
+  coniferous and other trunks embedded in sandstone. Some of these trees
+  seem to have been carried along and to have sunk, their heavier or
+  root end touching the bottom and their upper end slanting upward in
+  the direction of the current, exactly as in the case of the snags of
+  the Mississippi. In other cases the trees have been submerged while
+  still in their positions of growth. The continuous deposit of sand at
+  last rose above the level of the trunks and buried them. It is clear
+  then that the rate of deposit must have been sometimes sufficiently
+  rapid to allow sand to accumulate to a depth of 30 ft. or more before
+  the decay of the wood. Modern instances are known where, under certain
+  circumstances, submerged trees may last for some centuries, but even
+  the most durable must decay in what, after all, is a brief space of
+  geological time. Since continuous layers of the same kind of deposit
+  suggest a persistence of geological conditions, while numerous
+  alternations of different kinds of sedimentary matter point to
+  vicissitudes or alternations of conditions, it may be supposed that
+  the time represented by a given thickness of similar strata was less
+  than that shown by the same thickness of dissimilar strata, because
+  the changes needed to bring new varieties of sediment into the area of
+  deposit would usually require the lapse of some time for their
+  completion. But this conclusion may often be erroneous. It will be
+  best supported when, from the very nature of the rocks, wide
+  variations in the character of the water-bottom can be established.
+  Thus a group of shales followed by a fossiliferous limestone would
+  almost always mark the lapse of a much longer period than an equal
+  depth of sandy strata. A thick mass of limestone, made up of organic
+  remains which lived and died upon the spot, and whose remains are
+  crowded together generation above generation, must have demanded many
+  years or centuries for its formation.
+
+  But in all speculations of this kind we must bear in mind that the
+  length of time represented by a given depth of strata is not to be
+  estimated merely from their thickness or lithological character. The
+  interval between the deposit of two successive laminae of shale may
+  have been as long as, or even longer than, that required for the
+  formation of one of the laminae. In like manner the interval needed
+  for the transition from one stratum or kind of strata to another may
+  often have been more than equal to the time required for the formation
+  of the strata on either side. But the relative chronological
+  importance of the bars or lines in the geological record can seldom be
+  satisfactorily discussed merely on lithological grounds. This must
+  mainly be decided on the evidence of organic remains, as shown in
+  Part VI., where the grouping of the stratified rocks into formations
+  and systems is described.
+
+(b) _Igneous Rocks._--As part of the earth's crust these rocks present
+characters by which they are strongly differentiated from the stratified
+series. While the broad petrographical distinctions of their several
+varieties remain persistent, they present sufficient local variations of
+type to point to the existence of what have been called petrographic
+provinces, in each of which the eruptive masses are connected by a
+general family relationship, differing more or less from that of a
+neighbouring province. In each region presenting a long chronological
+series of eruptive rocks a petrographical sequence can be traced, which
+is observed to be not absolutely the same everywhere, though its general
+features may be persistent. The earliest manifestations of eruptive
+material in any district appear to have been most frequently of an
+intermediate type between acid and basic, passing thence into a
+thoroughly acid series and concluding with an effusion of basic
+material.
+
+Considered as part of the architecture of the crust of the earth,
+igneous rocks are conveniently divisible into two great series: (1)
+those bodies of material which have been injected into the crust and
+have solidified there, and (2) those which have reached the surface and
+have been ejected there, either in a molten state as lava or in a
+fragmental form as dust, ashes and scoriae. The first of these divisions
+represents the plutonic, intrusive or subsequent phase of eruptivity;
+the second marks the volcanic, interstratified or contemporaneous phase.
+
+  1. The plutonic or intrusive rocks, which have been forced into the
+  crust and have consolidated there, present a wide range of texture
+  from the most coarse-grained granites to the most perfect natural
+  glass. Seeing that they have usually cooled with extreme slowness
+  underground, they are as a general rule more largely crystalline than
+  the volcanic series. The form assumed by each individual body of
+  intrusive material has depended upon the shape of the space into which
+  it has been injected, and where it has cooled and become solid. This
+  shape has been determined by the local structure of the earth's crust
+  on the one hand and by the energy of the eruptive force on the other.
+  It offers a convenient basis for the classification of the intrusive
+  rocks, which, as part of the framework of the crust, may thus be
+  grouped according to the shape of the cavity which received them, as
+  bosses, sills, dikes and necks.
+
+  Bosses, or stocks, are the largest and most shapeless extravasations
+  of erupted material. They include the great bodies of granite which,
+  in most countries of the world, have risen for many miles through the
+  stratified formations and have altered the rocks around them by
+  contact-metamorphism. Sills, or intrusive sheets, are bed-like masses
+  which have been thrust between the planes of sedimentary or even of
+  igneous rocks. The term laccolite has been applied to sills which are
+  connected with bosses. Intrusive sheets are distinguishable from true
+  contemporaneously intercalated lavas by not keeping always to the same
+  platform, but breaking across and altering the contiguous strata, and
+  by the closeness of their texture where they come in contact with the
+  contiguous rocks, which, being cold, chilled the molten material and
+  caused it to consolidate on its outer margins more rapidly than in its
+  interior. Dikes or veins are vertical walls or ramifying branches of
+  intrusive material which has consolidated in fissures or irregular
+  clefts of the crust. Necks are volcanic chimneys which have been
+  filled up with erupted material, and have now been exposed at the
+  surface after prolonged denudation has removed not only the
+  superficial volcanic masses originally associated with them, but also
+  more or less of the upper part of the vents. Plutonic rocks do not
+  present evidence of their precise geological age. All that can be
+  certainly affirmed from them is that they must be younger than the
+  rocks into which they have been intruded. From their internal
+  structure, however, and from the evidence of the rocks associated with
+  them, some more or less definite conjectures may be made as to the
+  limits of time within which they were probably injected.
+
+  2. The interstratified or volcanic series is of special importance in
+  geology, inasmuch as it contains the records of volcanic action during
+  the past history of the globe. It was pointed out in Part I. that
+  while towards the end of the 18th and in the beginning of the 19th
+  century much attention was paid by Hutton and his followers to the
+  proofs of intrusion afforded by what they called the "unerupted lavas"
+  within the earth's crust, these observers lost sight of the
+  possibility that some of these rocks might have been erupted at the
+  surface, and might thus be chronicles of volcanic action in former
+  geological periods. It is not always possible to satisfactorily
+  discriminate between the two types of contemporaneously intercalated
+  and subsequently injected material. But rocks of the former type have
+  not broken into or involved the overlying strata, and they are usually
+  marked by the characteristic structures of superficial lavas and by
+  their association with volcanic tuffs. By means of the evidence which
+  they supply, it has been ascertained that volcanic action has been
+  manifested in the globe since the earliest geological periods. In the
+  British Isles, for example, the volcanic record is remarkably full for
+  the long series of ages from Cambrian to Permian time, and again for
+  the older Tertiary period.
+
+
+2. SUBSEQUENTLY INDUCED STRUCTURES
+
+After their accumulation, whether as stratified or eruptive masses, all
+kinds of rocks have been subject to various changes, and have acquired
+in consequence a variety of superinduced structures. It has been pointed
+out in the part of this article dealing with dynamical geology that one
+of the most important forms of energy in the evolution of geological
+processes is to be found in the movements that take place within the
+crust of the earth. Some of these movements are so slight as to be only
+recognizable by means of delicate instruments; but from this inferior
+limit they range up to gigantic convulsions by which mountain-chains are
+upheaved. The crust must be regarded as in a perpetual state of strain,
+and its component materials are therefore subject to all the effects
+which flow from that condition. It is the one great object of the
+geotectonic division of geology to study the structures which have been
+developed in consequence of earth-movements, and to discover from this
+investigation the nature of the processes whereby the rocks of the crust
+have been brought into the condition and the positions in which we now
+find them. The details of this subject will be found in separate
+articles descriptive of each of the technical terms applied to the
+several kinds of superinduced structures. All that need be offered here
+is a general outline connecting the several portions of the subject
+together.
+
+  One of the most universal of these later structures is to be seen in
+  the divisional planes, usually vertical or highly inclined, by which
+  rocks are split into quadrangular or irregularly shaped blocks. To
+  these planes the name of joints has been given. They are of prime
+  importance from an industrial point of view, seeing that the art of
+  quarrying consists mainly in detecting and making proper use of them.
+  Their abundance in all kinds of rocks, from those of recent date up to
+  those of the highest antiquity, affords a remarkable testimony to the
+  strains which the terrestrial crust has suffered. They have arisen
+  sometimes from tension, such as that caused by contraction from the
+  drying and consolidation of an aqueous sediment or from the cooling of
+  a molten mass; sometimes from torsion during movements of the crust.
+
+  Although the stratified rocks were originally deposited in a more or
+  less nearly horizontal position on the floor of the sea, where now
+  visible on the dry land they are seldom found to have retained their
+  flatness. On the contrary, they are seen to have been generally tilted
+  up at various angles, sometimes even placed on end (crop, dip,
+  strike). When a sufficiently large area of ground is examined, the
+  inclination into which the strata have been thrown may be observed not
+  to continue far in the same direction, but to turn over to the
+  opposite or another quarter. It can then be seen that in reality the
+  rocks have been thrown into undulations. From the lowest and flattest
+  arches where the departure from horizontality may be only trifling,
+  every step may be followed up to intense curvature, where the strata
+  have been compressed and plicated as if they had been piles of soft
+  carpets (anticline, syncline, monocline, geo-anticline, geo-syncline,
+  isoclinal, plication, curvature, quaquaversal). It has further
+  happened abundantly all over the surface of the globe that relief from
+  internal strain in the crust has been obtained by fracture, and the
+  consequent subsidence or elevation of one or both sides of the
+  fissure. The differential movement between the two sides may be
+  scarcely perceptible in the feeblest dislocation, but in the extreme
+  cases it may amount to many thousand feet (fault, fissure,
+  dislocation, hade, slickensides). The great faults in a country are
+  among its most important structural features, and as they not
+  infrequently continue to be lines of weakness in the crust along which
+  sudden slipping may from time to time take place, they become the
+  lines of origin of earthquakes. The San Francisco earthquake of 1906,
+  already cited, affords a memorable illustration of this connexion.
+
+  It is in a great mountain-chain that the extraordinary complication of
+  plicated and faulted structures in the crust of the earth can be most
+  impressively beheld. The combination of overturned folds with rupture
+  has been already referred to as a characteristic feature in the Alps
+  (Part IV.). The gigantic folds have in many places been pushed over
+  each other so as to lie almost flat, while the upper limb has not
+  infrequently been driven for many miles beyond the lower by a rupture
+  along the axis. In this way successive slices of a thick series of
+  formations have been carried northwards on the northern slope of the
+  Alps, and have been piled so abnormally above each other that some of
+  their oldest members recur several times on different thrust-planes,
+  the whole being underlain by Tertiary strata (see ALPS). Further
+  proof of the colossal compression to which the rocks have been
+  subjected is afforded by their intense crumpling and corrugation, and
+  by the abundantly faulted and crushed condition to which they have
+  been reduced. Similar evidence as to stresses in the terrestrial crust
+  and the important changes which they produce among the rocks may also
+  be obtained on a smaller scale in many non-mountainous countries.
+
+  Another marked result of the compression of the terrestrial crust has
+  been induced in some rocks by the production of the fissile structure
+  which is typically shown in roofing-slate (cleavage). Closely
+  connected with this internal rearrangement has been the development of
+  microscopic microlites or crystals (rutile, mica, &c.) in argillaceous
+  slates which were undoubtedly originally fine marine mud and silt.
+  From this incipient form of metamorphism successive stages may be
+  traced through the various kinds of argillite and phyllite into
+  mica-schist, and thence into more crystalline gneissoid varieties
+  (foliation, slate, mica-schist, gneiss). The Alps afford excellent
+  illustrations of these transformations.
+
+  The fissures produced in the crust are sometimes clean, sharply
+  defined divisional planes, like cracks across a pane of glass. Much
+  more usually, however, the rocks on either side have been broken up by
+  the friction of movement, and the fault is marked by a variable
+  breadth of this broken material. Sometimes the walls have separated
+  and molten rock has risen from below and solidified between them as a
+  dike. Occasionally the fissures have opened to the surface, and have
+  been filled in from above with detritus, as in the sandstone-dikes of
+  Colorado and California. In mineral districts the fissures have been
+  filled with various spars and ores, forming what are known as mineral
+  veins.
+
+  Where one series of rocks is covered by another without any break or
+  discordance in the stratification they are said to be conformable. But
+  where the older series has been tilted up or visibly denuded before
+  being overlain by the younger, the latter is termed unconformable.
+  This relation is one of the greatest value in structural geology, for
+  it marks a gap in the geological record, which may represent a vast
+  lapse of time not there recorded by strata.
+
+
+PART VI.--PALEONTOLOGICAL GEOLOGY
+
+This division of the science deals with fossils, or the traces of plants
+and animals preserved in the rocks of the earth's crust, and endeavours
+to gather from them information as to the history of the globe and its
+inhabitants. The term "fossil" (Lat. _fossilis_, from _fodere_, to dig
+up), meaning literally anything "dug up," was formerly applied
+indiscriminately to any mineral substance taken out of the earth's
+crust, whether organized or not. Since the time of Lamarck, however, the
+meaning of the word has been restricted, so as to include only the
+remains or traces of plants and animals preserved in any natural
+formation whether hard rock or superficial deposit. It includes not
+merely the petrified structures of organisms, but whatever was directly
+connected with or produced by these organisms. Thus the resin which was
+exuded from trees of long-perished forests is as much a fossil as any
+portion of the stem, leaves, flowers or fruit, and in some respects is
+even more valuable to the geologist than more determinable remains of
+its parent trees, because it has often preserved in admirable perfection
+the insects which flitted about in the woodlands. The burrows and trails
+of a worm preserved in sandstone and shale claim recognition as fossils,
+and indeed are commonly the only indications to be met with of the
+existence of annelid life among old geological formations. The droppings
+of fishes and reptiles, called coprolites, are excellent fossils, and
+tell their tale as to the presence and food of vertebrate life in
+ancient waters. The little agglutinated cases of the caddis-worm remain
+as fossils in formations from which, perchance, most other traces of
+life may have passed away. Nay, the very handiwork of man, when
+preserved in any natural manner, is entitled to rank among fossils; as
+where his flint-implements have been dropped into the pre-historic
+gravels of river-valleys or where his canoes have been buried in the
+silt of lake-bottoms.
+
+  A study of the land-surfaces and sea-floors of the present time shows
+  that there are so many chances against the conservation of the remains
+  of either terrestrial or marine animals and plants that if, as is
+  probable, the same conditions existed in former geological periods, we
+  should regard the occurrence of organic remains among the stratified
+  formations of the earth's crust as generally the result of various
+  fortunate accidents.
+
+  Let us consider, in the first place, the chances for the preservation
+  of remains of the present fauna and flora of a country. The surface of
+  the land may be densely clothed with forest and abundantly peopled
+  with animal life. But the trees die and moulder into soil. The
+  animals, too, disappear, generation after generation, and leave few or
+  no perceptible traces of their existence. If we were not aware from
+  authentic records that central and northern Europe were covered with
+  vast forests at the beginning of our era, how could we know this fact?
+  What has become of the herds of wild oxen, the bears, wolves and other
+  denizens of primeval Europe? How could we prove from the examination
+  of the surface soil of any country that those creatures had once
+  abounded there? The conditions for the preservation of any relics of
+  the plant and animal life of a terrestrial surface must obviously be
+  always exceptional. They are supplied only where the organic remains
+  can be protected from the air and superficial decay. Hence they may be
+  observed in (1) the deposits on the floors of lakes; (2) in
+  peat-mosses; (3) in deltas at river-mouths; and (4) under the
+  stalagmite of caverns in limestone districts. But in these and other
+  favourable places a mere infinitesimal fraction of the fauna or flora
+  of a land-surface is likely to be entombed or preserved.
+
+  In the second place, although in the sea the conditions for the
+  preservation of organic remains are in many respects more favourable
+  than on land, they are apt to be frustrated by many adverse
+  circumstances. While the level of the land remains stationary, there
+  can be but little effective entombment of marine organisms in littoral
+  deposits; for only a limited accumulation of sediment will be formed
+  until subsidence of the sea-floor takes place. In the trifling beds of
+  sand or gravel thrown up on a stationary shore, only the harder and
+  more durable forms of life, such as gastropods and lamellibranchs,
+  which can withstand the triturating effects of the beach waves, are
+  likely to remain uneffaced.
+
+  Below tide-marks, along the margin of the land where sediment is
+  gradually deposited, the conditions are more favourable for the
+  preservation of marine organisms. In the sheets of sand and mud there
+  laid down the harder parts of many forms of life may be entombed and
+  protected from decay. But only a small proportion of the total marine
+  fauna may be expected to appear in such deposits. At the best, merely
+  littoral and shallow-water forms will occur, and, even under the most
+  favourable conditions, they will represent but a fraction of the whole
+  assemblage of life in these juxta-terrestrial parts of the ocean. As
+  we recede from the land the rate of deposition of sediment on the
+  sea-floor must become feebler, until, in the remote central abysses,
+  it reaches a hardly appreciable minimum. Except, therefore, where some
+  kind of ooze or other deposit is accumulating in these more pelagic
+  regions, the conditions must be on the whole unfavourable for the
+  preservation of any adequate representation of the deep-sea fauna.
+  Hard durable objects, such as teeth and bones, may slowly accumulate,
+  and be protected by a coating of peroxide of manganese, or of some of
+  the silicates now forming here and there over the deep-sea bottom; or
+  the rate of growth of the abysmal deposit may be so tardy that most of
+  the remains of at least the larger animals will disappear, owing to
+  decay, before they can be covered up and preserved. Any such deep-sea
+  formation, if raised into land, would supply but a meagre picture of
+  the whole life of the sea.
+
+  It would thus appear that the portion of the sea-floor best suited for
+  receiving and preserving the most varied assemblage of marine organic
+  remains is the area in front of the land, to which rivers and currents
+  bring continual supplies of sediment. The most favourable conditions
+  for the accumulation of a thick mass of marine fossiliferous strata
+  will arise when the area of deposit is undergoing a gradual
+  subsidence. If the rate of depression and that of deposit were equal,
+  or nearly so, the movement might proceed for a vast period without
+  producing any great apparent change in marine geography, and even
+  without seriously affecting the distribution of life over the
+  sea-floor within the area of subsidence. Hundreds or thousands of feet
+  of sedimentary strata might in this way be heaped up round the
+  continents, containing a fragmentary series of organic remains
+  belonging to those forms of comparatively shallow-water life which had
+  hard parts capable of preservation. There can be little doubt that
+  such has, in fact, been the history of the main mass of stratified
+  formations in the earth's crust. By far the largest proportion of
+  these piles of marine strata has unquestionably been laid down in
+  water of no great depth within the area of deposit of terrestrial
+  sediment. The enormous thickness to which they attain seems only
+  explicable by prolonged and repeated movements of subsidence,
+  interrupted, however, as we know, by other movements of a contrary
+  kind.
+
+  Since the conditions for the preservation of organic remains exist
+  more favourably under the sea than on land, marine organisms must be
+  far more abundantly conserved than those of the land. This is true
+  to-day, and has, as far as known, been true in all past geological
+  time. Hence for the purposes of the geologist the fossil remains of
+  marine forms of life far surpass all others in value. Among them there
+  will necessarily be a gradation of importance, regulated chiefly by
+  their relative abundance. Now, of all the marine tribes which live
+  within the juxta-terrestrial belt of sedimentation, unquestionably the
+  Mollusca stand in the place of pre-eminence as regards their aptitude
+  for becoming fossils. They almost all possess a hard, durable shell,
+  capable of resisting considerable abrasion and readily passing into a
+  mineralized condition. They are extremely abundant both as to
+  individuals and genera. They occur on the shore within tide mark, and
+  range thence down into the abysses. Moreover, they appear to have
+  possessed these qualifications from early geological times. In the
+  marine Mollusca, therefore, we have a common ground of comparison
+  between the stratified formations of different periods. They have been
+  styled the alphabet of palaeontological inquiry.
+
+There are two main purposes to which fossils may be put in geological
+research: (1) to throw light upon former conditions of physical
+geography, such as the presence of land, rivers, lakes and seas, in
+places where they do not now exist, changes of climate, and the former
+distribution of plants and animals; and (2) to furnish a guide in
+geological chronology whereby rocks may be classified according to
+relative date, and the facts of geological history may be arranged and
+interpreted as a connected record of the earth's progress.
+
+  1. As examples of the first of these two directions of inquiry
+  reference may be made to (a) former land-surfaces revealed by the
+  occurrence of layers of soil with tree-stumps and roots still in the
+  position of growth (see PURBECKIAN); (b) ancient lakes proved by beds
+  of marl or limestone full of lacustrine shells; (c) old sea-bottoms
+  marked by the occurrence of marine organisms; (d) variations in the
+  quality of the water, such as freshness or saltness, indicated by
+  changes in the size and shape of the fossils; (e) proximity to former
+  land, suggested by the occurrence of abundant drift-wood in the
+  strata; (f) former conditions of climate, different from the present,
+  as evidenced by such organisms as tropical types of plants and animals
+  intercalated among the strata of temperate or northern countries.
+
+  2. In applying fossils to the determination of geological chronology
+  it is first necessary to ascertain the order of superposition of the
+  rocks. Obviously, in a continuous series of undisturbed sedimentary
+  deposits the lowest must necessarily be the oldest, and the plants or
+  animals which they contain must have lived and died before any of the
+  organisms that occur in the overlying strata. This order of
+  superposition having been settled in a series of formations, it is
+  found that the fossils at the bottom are not quite the same as those
+  at the top of the series. Tracing the beds upward, we discover that
+  species after species of the lowest platforms disappears, until
+  perhaps not one of them is found. With the cessation of these older
+  species others make their entrance. These, in turn, are found to die
+  out, and to be replaced by newer forms. After patient examination of
+  the rocks, it has been ascertained that every well-marked "formation,"
+  or group of strata, is characterized by its own species or genera, or
+  by a general assemblage, or _facies_, of organic forms. Such a
+  generalization can only, of course, be determined by actual practical
+  experience over an area of some size. When the typical fossils of a
+  formation are known, they serve to identify that formation in its
+  progress across a country. Thus, in tracts where the true order of
+  superposition cannot be determined, owing to the want of sections or
+  to the disturbed condition of the rocks, fossils serve as a means of
+  identification and furnish a guide to the succession of the rocks.
+  They even demonstrate that in some mountainous ground the beds have
+  been turned completely upside down, where it can be shown that the
+  fossils in what are now the uppermost strata ought properly to lie
+  underneath those in the beds below them.
+
+  It is by their characteristic fossils that the stratified rocks of the
+  earth's crust can be most satisfactorily subdivided into convenient
+  groups of strata and classed in chronological order. Each "formation"
+  is distinguished by its own peculiar assemblage of organic remains, by
+  means of which it can be followed and recognized, even amid the
+  crumplings and dislocations of a disturbed region. The same general
+  succession of organic types can be observed over a large part of the
+  world, though, of course, with important modifications in different
+  countries. This similarity of succession has been termed _homotaxis_,
+  a term which expresses the fact that the order in which the leading
+  types of organized existence have appeared upon the earth has been
+  similar even in widely separated regions. It is evident that, in this
+  way, a reliable method of comparison is furnished, whereby the
+  stratified formations of different parts of the earth's crust can be
+  brought into relation with each other. Had the geologist continued to
+  remain, as in the days of Werner, hampered by the limitations imposed
+  by a reliance on mere lithological characters, he would have made
+  little or no progress in deciphering the record of the successive
+  phases of the history of the globe chronicled in the crust. Just as,
+  at the present time, sheets of gravel in one place are contemporaneous
+  with sheets of mud at another, so in the past all kinds of
+  sedimentation have been in progress simultaneously, and those of one
+  period may not be distinguishable in themselves from those of another.
+  Little or no reliance can be placed upon lithological resemblances or
+  differences in comparing the sedimentary formations of different
+  countries.
+
+  In making use of fossil evidence for the purpose of subdividing the
+  stratified rocks of the earth's crust, it is found to be applicable to
+  the smaller details of stratigraphy as well as to the definition of
+  large groups of strata. Thus a particular stratum may be marked by the
+  occurrence in it of various fossils, one or more of which may be
+  distinctive, either from occurring in no other bed above and below or
+  from special abundance in that stratum. One or more of these species
+  is therefore used as a guide to the occurrence of the bed in
+  question, which is called by the name of the most abundant species. In
+  this way what is called a "geological horizon," or "zone," is marked
+  off, and its exact position in the series of formations is fixed.
+
+  Perhaps the most distinctive feature in the progress of
+  palaeontological geology during the last half century has been the
+  recognition and wide application of this method of zonal stratigraphy,
+  which, in itself, was only a further development of William Smith's
+  famous idea, "Strata identified by Organized Fossils." It was first
+  carried out in detail by various palaeontologists in reference to the
+  Jurassic formations, notably by F.A. von Quenstedt and C.A. Oppel in
+  Germany and A.D. d'Orbigny in France. The publication of Oppel's
+  classic work _Die Juraformation Englands, Frankreichs und des
+  sudwestlichen Deutschlands_ (1856-1858) marked an epoch in the
+  development of stratigraphical geology. Combining what had been done
+  by various observers with his own laborious researches in France,
+  England, Wurttemberg and Bavaria, he drew up a classification of the
+  Jurassic system, grouping its several formations into zones, each
+  characterized by some distinctly predominant fossil after which it was
+  named (see LIAS). The same method of classification was afterwards
+  extended to the Cretaceous series by A.D. d'Orbigny, E. Hebert and
+  others, until the whole Mesozoic rocks from the Trias to the top of
+  the Chalk has now been partitioned into zones, each named after some
+  characteristic species or genus of fossils. More recently the
+  principle has been extended to the Palaeozoic formations, though as
+  yet less fully than to the younger parts of the geological record. It
+  has been successfully applied by Professor C. Lapworth to the
+  investigation of the Silurian series (see SILURIAN; ORDOVICIAN
+  SYSTEM). He found that the species of graptolites have each a
+  comparatively narrow vertical range, and they may consequently be used
+  for stratigraphical purposes. Applying the method, in the first
+  instance, to the highly plicated Silurian rocks of the south of
+  Scotland, he found that by means of graptolites he was able to work
+  out the structure of the ground. Each great group of strata was seen
+  to possess its own graptolitic zones, and by their means could be
+  identified not only in the original complex Scottish area, but in
+  England and Wales and in Ireland. It was eventually ascertained that
+  the succession of zones in Great Britain could be recognized on the
+  Continent, in North America and even in Australia. The brachiopods and
+  trilobites have likewise been made use of for zonal purposes among the
+  oldest sedimentary formations. The most ancient of the Palaeozoic
+  systems has as its fitting base the _Olenellus_ zone.
+
+  Within undefined and no doubt variable geographical limits
+  palaeontological zones have been found to be remarkably persistent.
+  They follow each other in the same general order, but not always with
+  equal definiteness. The type fossil may appear in some districts on a
+  higher or a lower platform than it does in others. Only to a limited
+  degree is there any coincidence between lithological variations in the
+  strata and the sequence of the zones. In the Jurassic formations,
+  indeed, where frequent alternations of different sedimentary materials
+  are to be met with, it is in some cases possible to trace a definite
+  upward or downward limit for a zone by some abrupt change in the
+  sedimentation, such as from limestone to shale. But such a precise
+  demarcation is impossible where no distinct bands of different
+  sediments are to be seen. The zones can then only be vaguely
+  determined by finding their characteristic fossils, and noting where
+  these begin to appear in the strata and where they cease. It would
+  seem, therefore, that the sequence of palaeontological zones, or
+  life-horizons, has not depended merely upon changes in the nature of
+  the conditions under which the organisms lived. We should naturally
+  expect that these changes would have had a marked influence; that, for
+  instance, a difference should be perceptible between the character of
+  the fossils in a limestone and that of those in a shale or a
+  sandstone. The environment, when a limestone was in course of
+  deposition, would generally be one of clear water, favourable for a
+  more vigorous and more varied fauna than where a shale series was
+  accumulating, when the water would be discoloured, and only such
+  animals would continue to live in it, or on the bottom, as could
+  maintain themselves in the midst of mud. But no such lithological
+  reason, betokening geographical changes that would affect living
+  creatures, can be adduced as a universally applicable explanation of
+  the occurrence and limitation of palaeontological zones. One of these
+  zones may be only a few inches, or feet or yards in vertical extent,
+  and no obvious lithological or other cause can be seen why its
+  specially characteristic fossils should not be found just as
+  frequently in the similar strata above and below. There is often
+  little or no evidence of any serious change in the conditions of
+  sedimentation, still less of any widespread physical disturbance, such
+  as the catastrophes by which the older geologists explained the
+  extinction of successive types of life.
+
+  It has been suggested that, where the life-zones are well defined,
+  sedimentation has been extremely slow, and that though these zones
+  follow each other with no break in the sedimentation, they were really
+  separated by prolonged intervals of time during which organic
+  evolution could come effectively into play. But it is not easy to
+  explain how, for example in the Lower Lias, there could have been a
+  succession of prodigious intervals, when practically no sediment was
+  laid down, and yet that the strata should show no sign of
+  contemporaneous disturbance or denudation, but succeed each other as
+  if they had been accumulated by one continuous process of deposit. It
+  must be admitted that the problem of life-zones in stratigraphical
+  geology has not yet been solved.
+
+  As Darwin first cogently showed, the history of life has been very
+  imperfectly registered in the stratified parts of the earth's crust.
+  Apart from the fact that, even under the most favourable conditions,
+  only a small proportion of the total flora and fauna of any period
+  would be preserved in the fossil state, enormous gaps occur where no
+  record has survived at all. It is as if whole chapters and books were
+  missing from a historical work. Some of these lacunae are sufficiently
+  obvious. Thus, in some cases, powerful dislocations have thrown
+  considerable portions of the rocks out of sight. Sometimes extensive
+  metamorphism has so affected them that their original characters,
+  including their organic contents, have been destroyed. Oftenest of
+  all, denudation has come into play, and vast masses of fossiliferous
+  rock have been entirely worn away, as is demonstrated by the abundant
+  unconformabilities in the structure of the earth's crust.
+
+  While the mere fact that one series of rocks lies unconformably on
+  another proves the lapse of a considerable interval between their
+  respective dates, the relative length of this interval may sometimes
+  be proved by means of fossil evidence, and by this alone. Let us
+  suppose, for example, that a certain group of formations has been
+  disturbed, upraised, denuded and covered unconformably by a second
+  group. In lithological characters the two may closely resemble each
+  other, and there may be nothing to show that the gap represented by
+  their unconformability is of an important character. In many cases,
+  indeed, it would be quite impossible to pronounce any well-grounded
+  judgment as to the amount of interval, even measured by the vague
+  relative standards of geological chronology. But if each group
+  contains a well-preserved suite of organic remains, it may not only be
+  possible, but easy, to say exactly how much of the geological record
+  has been left out between the two sets of formations. By comparing the
+  fossils with those obtained from regions where the geological record
+  is more complete, it may be ascertained, perhaps, that the lower rocks
+  belong to a certain platform or stage in geological history which for
+  our present purpose we may call D, and that the upper rocks can in
+  like manner be paralleled with stage H. It would be then apparent that
+  at this locality the chronicles of three great geological periods E,
+  F, and G were wanting, which are elsewhere found to be intercalated
+  between D and H. The lapse of time represented by this
+  unconformability would thus be equivalent to that required for the
+  accumulation of the three missing formations in those regions where
+  sedimentation was more continuous.
+
+  Fossil evidence may be made to prove the existence of gaps which are
+  not otherwise apparent. As has been already remarked, changes in
+  organic forms must, on the whole, have been extremely slow in the
+  geological past. The whole species of a sea-floor could not pass
+  entirely away, and be replaced by other forms, without the lapse of
+  long periods of time. If then among the conformable stratified
+  formations of former ages we encounter sudden and abrupt changes in
+  the _facies_ of the fossils, we may be certain that these must mark
+  omissions in the record, which we may hope to fill in from a more
+  perfect series elsewhere. The complete biological contrasts between
+  the fossil contents of unconformable strata are sufficiently
+  explicable. It is not so easy to give a satisfactory account of those
+  which occur where the beds are strictly conformable, and where no
+  evidence can be observed of any considerable change of physical
+  conditions at the time of deposit. A group of strata having the same
+  general lithological characters throughout may be marked by a great
+  discrepance between the fossils above and below a certain line. A few
+  species may pass from the one into the other, or perhaps every species
+  may be different. In cases of this kind, when proved to be not merely
+  local but persistent over wide areas, we must admit, notwithstanding
+  the apparently undisturbed and continuous character of the original
+  deposition of the strata, that the abrupt transition from the one
+  _facies_ of fossils to the other represents a long interval of time
+  which has not been recorded by the deposit of strata. A.C. Ramsay, who
+  called attention to these gaps, termed them "breaks in the succession
+  of organic remains." He showed that they occur abundantly among the
+  Palaeozoic and Secondary rocks of England. It is obvious, of course,
+  that such breaks, even though traceable over wide regions, were not
+  general over the whole globe. There have never been any universal
+  interruptions in the continuity of the chain of being, so far as
+  geological evidence can show. But the physical changes which caused
+  the breaks may have been general over a zoological district or minor
+  region. They no doubt often caused the complete extinction of genera
+  and species which had a small geographical range.
+
+  From all these facts it is clear that the geological record, as it now
+  exists, is at the best but an imperfect chronicle of geological
+  history. In no country is it complete. The lacunae of one region must
+  be supplied from another. Yet in proportion to the geographical
+  distance between the localities where the gaps occur and those whence
+  the missing intervals are supplied, the element of uncertainty in our
+  reading of the record is increased. The most desirable method of
+  research is to exhaust the evidence for each area or province, and to
+  compare the general order of its succession as a whole with that which
+  can be established for other provinces.
+
+
+PART VII.--STRATIGRAPHICAL GEOLOGY
+
+This branch of the science arranges the rocks of the earth's crust in
+the order of their appearance, and interprets the sequence of events of
+which they form the records. Its province is to cull from the other
+departments of geology the facts which may be needed to show what has
+been the progress of our planet, and of each continent and country, from
+the earliest times of which the rocks have preserved any memorial. Thus
+from mineralogy and petrography it contains information regarding the
+origin and subsequent mutations of minerals and rocks. From dynamical
+geology it learns by what agencies the materials of the earth's crust
+have been formed, altered, broken, upheaved and melted. From geotectonic
+geology it understands the various processes whereby these materials
+were put together so as to build up the complicated crust of the earth.
+From palaeontological geology it receives in well-determined fossil
+remains a clue by which to discriminate the different stratified
+formations, and to trace the grand onward march of organized existence
+upon this planet. Stratigraphical geology thus gathers up the sum of all
+that is made known by the other departments of the science, and makes it
+subservient to the interpretation of the geological history of the
+earth.
+
+The leading principles of stratigraphy may be summed up as follows:
+
+1. In every stratigraphical research the fundamental requisite is to
+establish the order of superposition of the strata. Until this is
+accomplished it is impossible to arrange the dates, and make out the
+sequence of geological history.
+
+2. The stratified portion of the earth's crust, or what has been called
+the "geological record," can be subdivided into natural groups, or
+series of strata, characterized by distinctive organic remains and
+recognizable by these remains, in spite of great changes in lithological
+character from place to place. A bed, or a number of beds, linked
+together by containing one or more distinctive species or genera of
+fossils is termed a _zone_ or _horizon_, and usually bears the name of
+one of its more characteristic fossils, as the _Planorbis_-zone of the
+Lower Lias, which is so called from the prevalence in it of the ammonite
+_Psiloceras planorbis_. Two or more such zones related to each other by
+the possession of a number of the same characteristic species or genera
+have been designated _beds_ or an _assise_. Two or more sets of beds or
+assises similarly related form a _group_ or _stage_; a number of groups
+or stages make a _series_, _formation_ or _section_, and a succession of
+formations may be united into a _system_.
+
+3. Some living species of plants and animals can be traced downwards
+through the more recent geological formations; but the number which can
+be so followed grows smaller as the examination is pursued into more
+ancient deposits. With their disappearance other species or genera
+present themselves which are no longer living. These in turn may be
+traced backward into earlier formations, till they too cease and their
+places are taken by yet older forms. It is thus shown that the
+stratified rocks contain the records of a gradual progression of organic
+forms. A species which has once died out does not seem ever to have
+reappeared.
+
+4. When the order of succession of organic remains among the stratified
+rocks has been determined, they become an invaluable guide in the
+investigation of the relative age of rocks and the structure of the
+land. Each zone and formation, being characterized by its own species or
+genera, may be recognized by their means, and the true succession of
+strata may thus be confidently established even in a country wherein the
+rocks have been shattered by dislocation, folded, inverted or
+metamorphosed.
+
+5. Though local differences exist in regard to the precise zone in which
+a given species of organism may make its first appearance, the general
+order of succession of the organic forms found in the rocks is never
+inverted. The record is nowhere complete in any region, but the portions
+represented, even though extremely imperfect, always follow each other
+in their proper chronological order, unless where disturbance of the
+crust has intervened to destroy the original sequence.
+
+6. The relative chronological value of the divisions of the geological
+record is not to be measured by mere depth of strata. While it may be
+reasonably assumed that, in general, a great thickness of stratified
+rock must mark the passage of a long period of time, it cannot safely be
+affirmed that a much less thickness elsewhere must represent a
+correspondingly diminished period. The need for this caution may
+sometimes be made evident by an unconformability between two sets of
+rocks, as has already been explained. The total depth of both groups
+together may be, say 1000 ft. Elsewhere we may find a single unbroken
+formation reaching a depth of 10,000 ft.; but it would be unwarrantable
+to assume that the latter represents ten times the length of time
+indicated by the former two. So far from this being the case, it might
+not be difficult to show that the minor thickness of rock really denotes
+by far the longer geological interval. If, for instance, it could be
+proved that the upper part of both the sections lies on one and the same
+geological platform, but that the lower unconformable series in the one
+locality belongs to a far lower and older system of rocks than the base
+of the thick conformable series in the other, then it would be clear
+that the gap marked by the unconformability really indicates a longer
+period than the massive succession of deposits.
+
+7. Fossil evidence furnishes the chief means of comparing the relative
+value of formations and groups of rock. A "break in the succession of
+organic remains," as already explained, marks an interval of time often
+unrepresented by strata at the place where the break is found. The
+relative importance of these breaks, and therefore, probably, the
+comparative intervals of time which they mark, may be estimated by the
+difference of the _facies_ or general character of the fossils on each
+side. If, for example, in one case we find every species to be
+dissimilar above and below a certain horizon, while in another locality
+only half of the species on each side are peculiar, we naturally infer,
+if the total number of species seems large enough to warrant the
+inference, that the interval marked by the former break was much longer
+than that marked by the second. But we may go further and compare by
+means of fossil evidence the relation between breaks in the succession
+of organic remains and the depth of strata between them.
+
+  Three formations of fossiliferous strata, A, C, and H, may occur
+  conformably above each other. By a comparison of the fossil contents
+  of all parts of A, it may be ascertained that, while some species are
+  peculiar to its lower, others to its higher portions, yet the majority
+  extend throughout the formation. If now it is found that of the total
+  number of species in the upper portion of A only one-third passes up
+  into C, it may be inferred with some plausibility that the time
+  represented by the break between A and C was really longer than that
+  required for the accumulation of the whole of the formation A. It
+  might even be possible to discover elsewhere a thick intermediate
+  formation B filling up the gap between A and C. In like manner were it
+  to be discovered that, while the whole of the formation C is
+  characterized by a common suite of fossils, not one of the species and
+  only one half of the genera pass up into H, the inference could hardly
+  be resisted that the gap between the two formations marks the passage
+  of a far longer interval than was needed for the deposition of the
+  whole of C. And thus we reach the remarkable conclusion that, thick
+  though the stratified formations of a country may be, in some cases
+  they may not represent so long a total period of time as do the gaps
+  in their succession,--in other words, that non-deposition was more
+  frequent and prolonged than deposition, or that the intervals of time
+  which have been recorded by strata have not been so long as those
+  which have not been so recorded.
+
+In all speculations of this nature, however, it is necessary to reason
+from as wide a basis of observation as possible, seeing that so much of
+the evidence is negative. Especially needful is it to bear in mind that
+the cessation of one or more species at a certain line among the rocks
+of a particular district may mean nothing more than that, onward from
+the time marked by that line, these species, owing to some change in the
+conditions of life, were compelled to migrate or became locally extinct
+or, from some alteration in the conditions of fossilization, were no
+longer imbedded and preserved as fossils. They may have continued to
+flourish abundantly in neighbouring districts for a long period
+afterward. Many examples of this obvious truth might be cited. Thus in a
+great succession of mingled marine, brackish-water and terrestrial
+strata, like that of the Carboniferous Limestone series of Scotland,
+corals, crinoids and brachiopods abound in the limestones and
+accompanying shales, but disappear as the sandstones, ironstones, clays,
+coals and bituminous shales supervene. An observer meeting for the first
+time with an instance of this disappearance, and remembering what he had
+read about breaks in succession, might be tempted to speculate about the
+extinction of these organisms, and their replacement by other and later
+forms of life, such as the ferns, lycopods, estuarine or fresh-water
+shells, ganoid fishes and other fossils so abundant in the overlying
+strata. But further research would show him that high above the
+plant-bearing sandstones and coals other limestones and shales might be
+observed, once more charged with the same marine fossils as before, and
+still farther overlying groups of sandstones, coals and carbonaceous
+beds followed by yet higher marine limestones. He would thus learn that
+the same organisms, after being locally exterminated, returned again and
+again to the same area. After such a lesson he would probably pause
+before too confidently asserting that the highest bed in which we can
+detect certain fossils marks their final appearance in the history of
+life. Some breaks in the succession may thus be extremely local, one set
+of organisms having been driven to a different part of the same region,
+while another set occupied their place until the first was enabled to
+return.
+
+8. The geological record is at the best but an imperfect chronicle of
+the geological history of the earth. It abounds in gaps, some of which
+have been caused by the destruction of strata owing to metamorphism,
+denudation or otherwise, others by original non-deposition, as above
+explained. Nevertheless from this record alone can the progress of the
+earth be traced. It contains the registers of the appearance and
+disappearance of tribes of plants and animals which have from time to
+time flourished on the earth. Only a small proportion of the total
+number of species which have lived in past time have been thus
+chronicled, yet by collecting the broken fragments of the record an
+outline at least of the history of life upon the earth can be
+deciphered.
+
+It cannot be too frequently stated, nor too prominently kept in view,
+that, although gaps occur in the succession of organic remains as
+recorded in the rocks, they do not warrant the conclusion that any such
+blank intervals ever interrupted the progress of plant and animal life
+upon the globe. There is every reason to believe that the march of life
+has been unbroken, onward and upward. Geological history, therefore, if
+its records in the stratified formations were perfect, ought to show a
+blending and gradation of epoch with epoch. But the progress has been
+constantly interrupted, now by upheaval, now by volcanic outbursts, now
+by depression. These interruptions serve as natural divisions in the
+chronicle, and enable the geologist to arrange his history into periods.
+As the order of succession among stratified rocks was first made out in
+Europe, and as many of the gaps in that succession were found to be
+widespread over the European area, the divisions which experience
+established for that portion of the globe came to be regarded as
+typical, and the names adopted for them were applied to the rocks of
+other and far distant regions. This application has brought out the fact
+that some of the most marked breaks in the European series do not exist
+elsewhere, and, on the other hand, that some portions of that series are
+much more complete than the corresponding sections in other regions.
+Hence, while the general similarity of succession may remain, different
+subdivisions and nomenclature are required as we pass from continent to
+continent.
+
+The nomenclature adopted for the subdivisions of the geological record
+bears witness to the rapid growth of geology. It is a patch-work in
+which no system nor language has been adhered to, but where the
+influences by which the progress of the science has been moulded may be
+distinctly traced. Some of the earliest names are lithological, and
+remind us of the fact that mineralogy and petrography preceded geology
+in the order of birth--Chalk, Oolite, Greensand, Millstone Grit. Others
+are topographical, and often recall the labours of the early geologists
+of England--London Clay, Oxford Clay, Purbeck, Portland, Kimmeridge
+beds. Others are taken from local English provincial names, and remind
+us of the debt we owe to William Smith, by whom so many of them were
+first used--Lias, Gault, Crag, Cornbrash. Others of later date recognize
+an order of superposition as already established among formations--Old
+Red Sandstone, New Red Sandstone. By common consent it is admitted that
+names taken from the region where a formation or group of rocks is
+typically developed are best adapted for general use. Cambrian,
+Silurian, Devonian, Permian, Jurassic are of this class, and have been
+adopted all over the globe.
+
+But whatever be the name chosen to designate a particular group of
+strata, it soon comes to be used as a chronological or homotaxial term,
+apart altogether from the stratigraphical character of the strata to
+which it is applied. Thus we speak of the Chalk or Cretaceous system,
+and embrace under that term formations which may contain no chalk; and
+we may describe as Silurian a series of strata utterly unlike in
+lithological characters to the formations in the typical Silurian
+country. In using these terms we unconsciously allow the idea of
+relative date to arise prominently before us. Hence such a word as
+"chalk" or "cretaceous" does not suggest so much to us the group of
+strata so called as the interval of geological history which these
+strata represent. We speak of the Cretaceous, Jurassic, and Cambrian
+periods, and of the Cretaceous fauna, the Jurassic flora, the Cambrian
+trilobites, as if these adjectives denoted simply epochs of geological
+time.
+
+The stratified formations of the earth's crust, or geological record,
+are classified into five main divisions, which in their order of
+antiquity are as follows: (1) Archean or Pre-Cambrian, called also
+sometimes Azoic (lifeless) or Eozoic (dawn of life); (2) Palaeozoic
+(ancient life) or Primary; (3) Mesozoic (middle life) or Secondary; (4)
+Cainozoic (recent life) or Tertiary; (5) Quaternary or Post-Tertiary.
+These divisions are further ranged into systems, formations, groups or
+stages, assises and zones. Accounts of the various subdivisions named
+are given in separate articles under their own headings. In order,
+however, that the sequence of the formations and their parallelism in
+Europe and North America may be presented together a stratigraphical
+table is given on next page.
+
+
+PART VIII.--PHYSIOGRAPHICAL GEOLOGY
+
+This department of geological inquiry investigates the origin and
+history of the present topographical features of the land. As these
+features must obviously be related to those of earlier time which are
+recorded in the rocks of the earth's crust, they cannot be
+satisfactorily studied until at least the main outlines of the history
+of these rocks have been traced. Hence physiographical research comes
+appropriately after the other branches of the science have been
+considered.
+
+From the stratigraphy of the terrestrial crust we learn that by far the
+largest part of the area of dry land is built up of marine formations;
+and therefore that the present land is not an aboriginal portion of the
+earth's surface, but has been overspread by the sea in which its rocks
+were mainly accumulated. We further discover that this submergence of
+the land did not happen once only, but again and again in past ages and
+in all parts of the world. Yet although the terrestrial areas varied
+much from age to age in their extent and in their distribution, being at
+one time more continental, at another more insular, there is reason to
+believe that these successive diminutions and expansions have on the
+whole been effected within, or not far outside, the limits of the
+existing continents. There is no evidence that any portion of the
+present land ever lay under the deeper parts of the ocean. The abysmal
+deposits of the ocean-floor have no true representatives among the
+sedimentary formations anywhere visible on the land. Nor, on the other
+hand, can it be shown that any part of the existing ocean abysses ever
+rose above sea-level into dry land. Hence geologists have drawn the
+inference that the ocean basins have probably been always where they now
+are; and that although the continental areas have often been narrowed by
+submergence and by denudation, there has probably seldom or never been a
+complete disappearance of land. The fact that the sedimentary formations
+of each successive geological period consist to so large an extent of
+mechanically formed terrigenous detritus, affords good evidence of the
+coexistence of tracts of land as well as of extensive denudation.
+
+
+  _The Geological Record or Order of Succession of the Stratified
+  Formations of the Earth's Crust._
+
+  +---+---+-------------------------------------------+----------------------------------+
+  |   |   |                  Europe.                  |          North America.          |
+  +---+---+-------------------------------------------+----------------------------------+
+  | Q |    \      Historic, up to the present time.   | Similar to the European          |
+  | u |     \     Prehistoric, comprising deposits of |   development, but with scantier |
+  | a |      \     the Iron, Bronze, and later        |   traces of the presence of man. |
+  | t |       \     Stone Ages.                       |                                  |
+  | e |        \    Neolithic--alluvium, peat, lake-  |                                  |
+  | r | Recent, \   dwellings, loess, &c.             |                                  |
+  | n | Post-   | Palaeolithic--river-gravels, cave-  |                                  |
+  | a | glacial |   deposits, &c.                     |                                  |
+  | r | or      |                                     |                                  |
+  | y | Human.  |                                     |                                  |
+  |   |         |                                     |                                  |
+  | o |         |                                     |                                  |
+  | r +---------+-------------------------------------+----------------------------------+
+  |   | Pleist- | Older Loess and valley-gravels;     | As in Europe, it is hardly       |
+  | P | ocene   |   cave-deposits.                    |   possible to assign a definite  |
+  | o | or      | Strand-lines or raised beaches;     |   chronological place to each of |
+  | s | Glacial.|   youngest moraines.                |   the various deposits of this   |
+  | t |         | Upper Boulder-clays; eskers; marine |   period, terrestrial and marine.|
+  | | |         |   sands and clays.                  |   They generally resemble the    |
+  | T |         | Interglacial deposits.              |   European series. The           |
+  | e |         | Lower boulder-clay or Till, with    |   characteristic marine,         |
+  | r |         |   striated rock-surfaces below.     |   fluviatile and lacustrine      |
+  | t |         |                                     |   terraces, which overlie the    |
+  | i |        /                                      |   older drifts, have been        |
+  | a |       /                                       |   classed as the Champlain Group.|
+  | r |      /                                        |                                  |
+  | y |     /                                         |                                  |
+  | . |    /                                          |                                  |
+  +---+---+-------------------------------------------+----------------------------------+
+  |   | P | Newer:--English Forest-Bed Group; Red and | On the Atlantic border           |
+  |   | l |   Norwich Crag; Amstelian and Scaldesian  |   represented by the marine      |
+  |   | i |   groups of Belgium and Holland; Sicilian |   Floridian series; in the       |
+  |   | o |   and Astian of France and Italy.         |   interior by a subaerial and    |
+  |   | c | Older:--English Coralline Crag; Diestian  |   lacustrine series; and on the  |
+  |   | e |   of Belgium; Plaisancian of southern     |   Pacific border by the thick    |
+  |   | n |   France and Italy.                       |   marine series of San Francisco.|
+  |   | e |                                           |                                  |
+  |   | . |                                           |                                  |
+  |   +---+-------------------------------------------+----------------------------------+
+  |   | M | Wanting in Britain; well developed in     | Represented in the Eastern States|
+  |   | i |   France, S. E. Europe and Italy;         |   by a marine series (Yorktown or|
+  | C | o |   divisible into the following groups in  |   Chesapeake, Chipola and        |
+  | a | c |   descending order: (1) Pontian; (2)      |   Chattahoochee groups), and in  |
+  | i | e |   Sarmatian; (3) Tortonian; (4) Helvetian;|   the interior by the lacustrine |
+  | n | n |   (5) Langhian (Burdigalian).             |   Loup Fork (Nebraska), Deep     |
+  | o | e |                                           |   River, and John Day groups.    |
+  | z | . |                                           |                                  |
+  | o +---+-------------------------------------------+----------------------------------+
+  | i |   | In Britain the "fluvio-marine series" of  | On the Atlantic border no        |
+  | c | O |   the Isle of Wight; also the volcanic    |   equivalents have been          |
+  |   | l |   plateaux of Antrim and Inner Hebrides   |   satisfactorily recognised, but |
+  | o | i |   and those of the Faeroe Isles and       |   on the Pacific side there are  |
+  | r | g |   Iceland. In continental Europe the      |   marine deposits in N. W.       |
+  |   | o |   following subdivisions have been        |   Oregon, which may represent    |
+  | T | c |   established in descending order: (1)    |   this division. In the interior |
+  | e | e |   Aquitanian, (2) Stampian (Rupelian),    |   the equivalent is believed to  |
+  | r | n |   (3) Tongrain (Sannoisian).              |   be the fresh-water White River |
+  | t | e |                                           |   series, including (1)          |
+  | i | . |                                           |   _Protoceras_ beds, (2)         |
+  | a |   |                                           |   _Oreodon_ beds, and (3)        |
+  | r |   |                                           |   _Titanothervum_ beds.          |
+  | y +---+-------------------------------------------+----------------------------------+
+  | . |   | Barton sands and clays; Ludian series of  | Woodstock and Aquia Creek groups |
+  |   |   |   France.                                 |   of Potomac River; Vicksburg,   |
+  |   |   | Bracklesham Beds; Lutetian (Calcaire      |   Jackson, Claiborne, Buhrstone, |
+  |   | E |   grossier and Caillasses) of Paris       |   and Lignitic groups of         |
+  |   | o |   basin.                                  |   Mississippi.                   |
+  |   | c | London clay, Woolwich and Reading Beds;   | In the interior a thick series of|
+  |   | e |   Thanet sands; Ypresian or Londinian of  |   fresh-water formations,        |
+  |   | n |   N. France and Belgium; Sparnacian and   |   comprising, in descending      |
+  |   | e |   Thanetian groups.                       |   order, the Uinta, Bridger,     |
+  |   | . |                                           |   Wind River, Wasatch, Torrejon, |
+  |   |   |                                           |   and Puerco groups.             |
+  |   |   |                                           | On the Pacific side the marine   |
+  |   |   |                                           |   Tejon series of Oregon and     |
+  |   |   |                                           |   California.                    |
+  |---+---+-------------------------------------------+----------------------------------|
+  |   |   |     Upper                                 | On the Atlantic border both      |
+  |   |   |     =====                                 |   marine strata and others       |
+  |   |   | Danian--wanting in Britain; uppermost     |   containing a terrestrial flora |
+  |   |   |   limestone of Denmark.                   |   represent the Cretaceous series|
+  |   |   | Senonian--Upper Chalk with Flints of      |   of formations.                 |
+  |   |   |   England; Aturian and Emscherian stages  | In the interior there is also a  |
+  |   |   |   on the European continent.              |   commingling of marine with     |
+  |   |   | Turonian--Middle Chalk with few flints,   |   lacustrine deposits. At the top|
+  |   |   |   and comprising the Angoumian and stages.|   lies the Laramie or Lignitic   |
+  |   | C | Cenomanian--Lower Chalk and Chalk Marl.   |   series with an abundant        |
+  |   | r |                                           |   terrestrial flora, passing down|
+  |   | e |     Lower                                 |   into the lacustrine and        |
+  |   | t |     =====                                 |   brackish-water Montana series. |
+  |   | a | Albian--Upper Greensand and Gault.        |   Of older date, the Colorado    |
+  |   | c | Aptian--Lower Greensand; Marls and        |   series contains an abundant    |
+  |   | e |   limestones of Provence, &c.             |   marine fauna, yet includes also|
+  |   | o | Urgonian (Barremian)--Atherfield clay;    |   some Niobrara marls and        |
+  |   | u |   massive Hippurite limestones of         |   limestones are likewise of     |
+  |   | s |   southern France.                        |   marine origin, but the lower   |
+  |   | . | Neocomian--Weald clay and Hastings sand;  |   members of the series (Benton  |
+  |   |   |   Hauterivian and Valanginian sub-stages  |   and Dakota) show another great |
+  |   |   |   of Switzerland and France.              |   representation of fresh-water  |
+  | M |   |                                           |   sedimentation with lignites and|
+  | e |   |                                           |   coals.                         |
+  | s |   |                                           | In California a vast succession  |
+  | o |   |                                           |   of marine deposits (Shasta-    |
+  | z |   |                                           |   Chico) represents the          |
+  | o |   |                                           |   Cretaceous system; and in      |
+  | i |   |                                           |   western British N. America     |
+  | c |   |                                           |   coal-seams also occur.         |
+  |   +---+-------------------------------------------+----------------------------------+
+  | o |   | Purbeckian--Purbeck beds; Munder Mergel;  | Representatives of the Middle and|
+  | r |   |   largely present in Westphalia.          |   lower Jurassic formations have |
+  |   |   | Portlandian--Portland group of England,   |   been found in California and   |
+  | S |   |   represented in S. France by the thick   |   Oregon, and farther north among|
+  | e |   |   Tithonian limestones.                   |   the Arctic islands.            |
+  | c |   | Kimmeridgian--Kimmeridge Clay of England; | Strata containing Lower Jurassic |
+  | o |   |   Virgulian and Pterocerian groups of N.  |   marine fossils appear in       |
+  | n | J |   France; represented by thick limestones |   Wyoming and Dakota; and above  |
+  | d | u |   in the Mediterranean basin.             |   them come the _Atlantosaurus_  |
+  | a | r | Corallian--Coral Rag, Coralline Oolite;   |   and _Baptanodon_ beds, which   |
+  | r | a |   Sequanian stages of the Continent,      |   have yielded so large a        |
+  | y | s |   comprising the sub-stages of Astartian  |   variety of deinosaurs and other|
+  | . | s |   and Rauracian.                          |   vertebrates, and especially the|
+  |   | i | Oxfordian--Oxford Clay; Axgovian and      |   remains of a number of genera  |
+  |   | c |   Neuvizyan stages.                       |   of small mammals.              |
+  |   | . | Callovian--Kellaways Rock, Divesian       |                                  |
+  |   |   |   sub-stage of N. France.                 |                                  |
+  |   |   | Bathonian--series of English strata from  |                                  |
+  |   |   |   Cornbrash down to Fuller's Earth.       |                                  |
+  |   |   | Bajocian--Inferior Oolite of England.     |                                  |
+  |   |   | Lassic--divisible into (1) Upper Lias     |                                  |
+  |   |   |   or Toarcian, (2) Middle Lias, Marlstone |                                  |
+  |   |   |   or Charmouthian, (3) Lower Lias of      |                                  |
+  |   |   |   Sinemurian and Hettangian.              |                                  |
+  |   +---+-------------------------------------------+----------------------------------+
+  |   |   | In Germany and western Europe this        | In New York, Connecticut, New    |
+  |   | T |   division represents the deposits of     |   Brunswick, and Nova Scotia     |
+  |   | r |   inland seas or lagoons, and is divisible|   a series of red sandstone      |
+  |   | i |   into the following stages in descending |   (Newark series) contains land- |
+  |   | a |   order: (1) Rhaetic, (2) Keuper, (3)     |   plants and labyrinthodonts     |
+  |   | s |   Muschelkalk, (4) Bunter.  In the        |   like the lagoon type of central|
+  |   | s |   eastern Alps and the Mediterranean      |   and western Europe. On the     |
+  |   | i |   basin the contemporaneous sedimentary   |   Pacific slope, however, marine |
+  |   | c |   formations are those of open clear      |   equivalents occur, representing|
+  |   | . |   sea, in which a thickness of many       |   the pelagic type of south-     |
+  |   |   |   thousand feet of strata was accumulated.|   eastern Europe.                |
+  +---+---+-------------------------------------------+----------------------------------+
+  |   | P | Thuringian--Zechstein, Magnesian          | To this division of the geologi- |
+  |   | e |   Limestone; named from its development   |   cal record the Upper Barren    |
+  |   | r |   in Thuringia; well represented          |   Measures of the coal-fields of |
+  |   | m |   also in Saxony, Bavaria and Bohemia.    |   Pennsylvania, Prince Edward    |
+  |   | i | Saxonian--Rothliegendes Group; Red        |   Island, Nova Scotia and        |
+  |   | a |   Sandstones, &c.                         |   New Brunswick have been        |
+  |   | n | Autunian--where the strata present the    |   assigned.                      |
+  |   | . |   lagoon facies, well displayed at Autun  | Farther south in Kansas, Texas,  |
+  |   |   |   in France; where the marine type is     |   and Nebraska the representa-   |
+  |   |   |   predominant, as in Russia, the group    |   tives of the division have an  |
+  |   |   |   has been termed Artinskian.             |   abundant marine fauna.         |
+  |   +---+-------------------------------------------+----------------------------------+
+  |   | C | Stephanian or Uralian--represented in     | Upper productive Coal-measures.  |
+  |   | a |   Russia by marine formations, and in     | Lower Barren measures.           |
+  |   | r |   central and western Europe by numerous  | Lower productive Coal-measures.  |
+  |   | b |   small basins containing a peculiar      | Pottsville conglomerate.         |
+  |   | o |   flora and in some places a great variety| Mauch Chunk shales; limestones   |
+  |   | n |   of insects.                             |   of Chester, St Louis, &c.      |
+  |   | i | Westphalian or Moscovian--Coal-measures,  | Pocono series; Kinderhook        |
+  |   | f |   Millstone Grit.                         |   limestone.                     |
+  |   | e | Culm or Dinantian--Carboniferous Limestone|                                  |
+  |   | r |   and Calciferous Sandstone series.       |                                  |
+  |   | o |                                           |                                  |
+  |   | u |                                           |                                  |
+  |   | s |                                           |                                  |
+  |   | . |                                           |                                  |
+  |   +---+-------------------------------------------+----------------------------------+
+  |   |                            Devonian and Old Red Sandstone.                       |
+  | P +----------------------+------------------------+----------------------------------+
+  | a |     Devonian type.   |  Old Red Sandstone     |                                  |
+  | l |                      |       type.            |                                  |
+  | a +----------------------+------------------------+ / Catskill red sandstone; Old    |
+  | e |         / Famennian. | Yellow and red         | |   Red Sandstone type: the      |
+  | o | Upper  <             |   sandstone with       |<    strata below show the        |
+  | z |         \ Frasnian.  |   _Holoptychius_,      | |   Devonian type.               |
+  | o |                      |   _Bothriolepis_,&c.   | | Chemung Group.                 |
+  | i |                      |                        | \ Genesee   "                    |
+  | c |                      |                        |                                  |
+  |   |         / Givetian.  | Caithness Flagstones   |                                  |
+  | o | Middle <             |   with _Osteolepus_,   | / Hamilton Group.                |
+  | r |         \ Eifelian.  |   _Dipterus_,          | \ Marcellus  "                   |
+  |   |                      |   _Homosteus_, &c.     |                                  |
+  | P |                      |                        |                                  |
+  | r |                      | Red and purple         | / Corniferous Limestone. / Upper |
+  | i |        /Coblentizian.|   sandstones and       | |                        | Held- |
+  | m | Lower <              |   conglomerates with   |<  Onondaga Limestone.   <  erberg|
+  | a |        \Gedinnian.   |   _Cephalaspis_,       | |                        \ Group.|
+  | r |                      |   _Pteraspis_, &c.     | \ Oriskany Sandstone.            |
+  | y +---+------------------+------------------------+----------------------------------+
+  | . |   |                                           |  / Lower Helderberg Group.       |
+  |   | S |               / Ludlow  Group.            |  | Water-Lime.                   |
+  |   | i |   Upper      <  Wenlock   "               | <  Niagara Shale and Limestone.  |
+  |   | l |               \ Llandovery"               |  | Clinton Group.                |
+  |   | u |                                           |  \ Medina    "                   |
+  |   | r |                                           |                                  |
+  |   | i |                                           |  / Cincinnati Group.             |
+  |   | a |   Lower       / Caradoc or Bala Group.    |  | Utica        "                |
+  |   | n | (Ordovician) <  Llandeilo         "       | <  Trenton      "                |
+  |   | . |               \ Arenig            "       |  | Chazy        "                |
+  |   |   |                                           |  \ Calciferous  "                |
+  |   +---+-------------------------------------------+----------------------------------+
+  |   | C | Upper or _Olenus_ series--Tremadoc        | Upper or Potsdam series with     |
+  |   | a |   slates and _Lingula_ Flags.             |   _Olenus_ and _Dicelocephalus_  |
+  |   | m | Middle or _Pardoxides_ series--Menevian   |   fauna.                         |
+  |   | b |   Group.                                  | Middle or Acadian series with    |
+  |   | r | Lower or _Olenellus_ series--Llanberis    |   _Paradoxides_ fauna.           |
+  |   | i |   and Harlech Group, and _Olenellus_-     | Lower or Georgian series with    |
+  |   | a |   zone.                                   |   _Olenellus_ fauna.             |
+  |   | n |                                           |                                  |
+  |   | . |                                           |                                  |
+  +---+---+-------------------------------------------+----------------------------------+
+  |   |   |                       Archean, Pre-Cambrian, Eozoic.                         |
+  +---+---+-------------------------------------------+----------------------------------+
+  |   |   | In Scotland, underneath the Cambrian      | In Canada and the Lake Superior  |
+  |   |   |   Olenellus group, lies unconformably     |   region of the United States    |
+  |   |   |   a mass of red sandstone and con-        |   a vast succession of rocks of  |
+  |   |   |   glomerate (Torridonian) 8000 or 10,000  |   Pre-Cambrian age has been      |
+  |   |   |   ft. thick, which rests with a strong    |   grouped into the following     |
+  |   |   |   gneisses and schists (Lewisian).  A     |   subdivisions in descending     |
+  |   |   |   thick series of slates and phyllites    |   order: (1) Keweenwan, lying    |
+  |   |   |   lies below the oldest Palaeozoic rocks  |   unconformably on (2) Animikie, |
+  |   |   |   in central Europe, with coarse          |   separated by a strong          |
+  |   |   |   gneisses below.                         |   unconformability from (3) Upper|
+  |   |   |                                           |   Huronian, (4) Lower Huronian   |
+  |   |   |                                           |   with an unconformable base, (5)|
+  |   |   |                                           |   Goutchiching, (6) Laurentian.  |
+  |   |   |                                           |   In the eastern part of Canada, |
+  |   |   |                                           |   Newfoundland, &c., and also in |
+  |   |   |                                           |   Montana, sedimentary formations|
+  |   |   |                                           |   of great thickness below the   |
+  |   |   |                                           |   lowest Cambrian zone have been |
+  |   |   |                                           |   found to contain some obscure  |
+  |   |   |                                           |   organisms.                     |
+  +---+---+-------------------------------------------+----------------------------------+
+
+From these general considerations we proceed to inquire how the existing
+topographical features of the land arose. Obviously the co-operation of
+the two great geological agencies of hypogene and epigene energy, which
+have been at work from the beginning of our globe's decipherable
+history, must have been the cause to which these features are to be
+assigned; and the task of the geologist is to ascertain, if possible,
+the part that has been taken by each. There is a natural tendency to see
+in a stupendous piece of scenery, such as a deep ravine, a range of
+hills, a line of precipice or a chain of mountains, evidence only of
+subterranean convulsion; and before the subject was taken up as a matter
+of strict scientific induction, an appeal to former cataclysms was
+considered a sufficient solution of the problems presented by such
+features of landscape. The rise of the modern Huttonian school, however,
+led to a more careful examination of these problems. The important share
+taken by erosion in the determination of the present features of
+landscape was then recognized, while a fuller appreciation of the
+relative parts played by the hypogene and epigene causes has gradually
+been reached.
+
+1. The study of the progress of denudation at the present time has led
+to the conclusion that even if the rate of waste were not more rapid
+than it is to-day, it would yet suffice in a comparatively brief
+geological period to reduce the dry land to below the sea-level. But not
+only would the area of the land be diminished by denudation, it could
+hardly fail to be more or less involved in those widespread movements of
+subsidence, during which the thick sedimentary formations of the crust
+appear to have been accumulated. It is thus manifest that there must
+have been from time to time during the history of our globe upward
+movements of the crust, whereby the balance between land and sea was
+redressed. Proofs of such movements have been abundantly preserved among
+the stratified formations. We there learn that the uplifts have usually
+followed each other at long intervals between which subsidence
+prevailed, and thus that there has been a prolonged oscillation of the
+crust over the great continental areas of the earth's surface.
+
+An examination of that surface leads to the recognition of two great
+types of upheaval. In the one, the sea-floor, with all its thick
+accumulations of sediment, has been carried upwards, sometimes for
+several thousand feet, so equably that the strata retain their original
+flatness with hardly any sensible disturbance for hundreds of square
+miles. In the other type the solid crust has been plicated, corrugated
+and dislocated, especially along particular lines, and has attained its
+most stupendous disruption in lofty chains of mountains. Between these
+two phases of uplift many intermediate stages have been developed,
+according to the direction and intensity of the subterranean force and
+the varying nature and disposition of the rocks Of the crust.
+
+(a) Where the uplift has extended over wide spaces, without appreciable
+deformation of the crust, the flat strata have given rise to low plains,
+or if the amount of uprise has been great enough, to high plains,
+plateaux or tablelands. The plains of Russia, for example, lie for the
+most part on such tracts of equably uplifted strata. The great plains of
+the western interior of the United States form a great plateau or
+tableland, 5000 or 6000 ft. above the sea, and many thousands of square
+miles in extent, on which the Rocky Mountains have been ridged up.
+
+(b) It is in a great mountain-chain that the complicated structures
+developed during disturbances of the earth's crust can best be studied
+(see Parts IV. and V. of this article), and where the influence of these
+structures on the topography of the surface is most effectively
+displayed. Such a chain may be the result of one colossal disturbance;
+but those of high geological antiquity usually furnish proofs of
+successive uplifts with more or less intervening denudation. Formed
+along lines of continental displacement in the crust, they have again
+and again given relief from the strain of compression by fresh
+crumpling, fracture and uprise. The chief guide in tracing these
+successive stages of growth is supplied by unconformability. If, for
+example, a mountain-range consists of upraised Silurian rocks, upon the
+upturned and denuded edges of which the Carboniferous Limestone lies
+transgressively, it is clear that its original upheaval must have taken
+place in the period of geological time represented by the interval
+between the Silurian and the Carboniferous Limestone formations. If, as
+the range is followed along its course, the Carboniferous Limestone is
+found to be also highly inclined and covered unconformably by the Upper
+Coal-measures, a second uplift of that portion of the ground can be
+proved to have taken place between the time of the Limestone and that of
+the Upper Coal-measures. By this simple and obvious kind of evidence the
+relative ages of different mountain-chains may be compared. In most
+great chains, however, the rocks have been so intensely crumpled, and
+even inverted, that much labour may be required before their true
+relations can be determined.
+
+The Alps furnish an instructive example of the long series of
+revolutions through which a great mountain-system may have passed before
+reaching its present development. The first beginnings of the chain may
+have been upraised before the oldest Palaeozoic formations were laid
+down. There are at least traces of land and shore-lines in the
+Carboniferous period. Subsequent submergences and uplifts appear to have
+occurred during the Mesozoic periods. There is evidence that thereafter
+the whole region sank deep under the sea, in which the older Tertiary
+sediments were accumulated, and which seems to have spread right across
+the heart of the Old World. But after the deposition of the Eocene
+formations came the gigantic disruptions whereby all the rocks of the
+Alpine region were folded over each other, crushed, corrugated,
+fractured and displaced, some of their older portions, including the
+fundamental gneisses and schists, being squeezed up, torn off, and
+pushed horizontally for many miles over the younger rocks. But this
+upheaval, though the most momentous, was not the last which the chain
+has undergone, for at a later epoch in Tertiary time renewed disturbance
+gave rise to a further series of ruptures and plications. The chain thus
+successively upheaved has been continuously exposed to denudation and
+has consequently lost much of its original height. That it has been left
+in a state of instability is indicated by the frequent earthquakes of
+the Alpine region, which doubtless arise from the sudden snapping of
+rocks under intense strain.
+
+A distinct type of mountain due to direct hypogene action is to be seen
+in a volcano. It has been already pointed out (Part IV. sect. 1) that at
+the vents which maintain a communication between the molten magma of the
+earth's interior and the surface, eruptions take place whereby
+quantities of lava and fragmentary materials are heaped round each
+orifice of discharge. A typical volcanic mountain takes the form of a
+perfect cone, but as it grows in size and its main vent is choked, while
+the sides of the cone are unable to withstand the force of the
+explosions or the pressure of the ascending column of lava, eruptions
+take place laterally, and numerous parasitic cones arise on the flanks
+of the parent mountain. Where lava flows out from long fissures, it may
+pile up vast sheets of rock, and bury the surrounding country under
+several thousand feet of solid stone, covering many hundreds of square
+miles. In this way volcanic tablelands have been formed which, attacked
+by the denuding forces, are gradually trenched by valleys and ravines,
+until the original level surface of the lava-field may be almost or
+wholly lost. As striking examples of this physiographical type reference
+may be made to the plateau of Abyssinia, the Ghats of India, the
+plateaux of Antrim, the Inner Hebrides and Iceland, and the great
+lava-plains of the western territories of the United States.
+
+2. But while the subterranean movements have upraised portions of the
+surface of the lithosphere above the level of the ocean, and have thus
+been instrumental in producing the existing tracts of land, the detailed
+topographical features of a landscape are not solely, nor in general
+even chiefly, attributable to these movements. From the time that any
+portion of the sea-floor appears above sea-level, it undergoes erosion
+by the various epigene agents. Each climate and geological region has
+its own development of these agents, which include air, aridity, rapid
+and frequent alternations of wetness and dryness or of heat and cold,
+rain, springs, frosts, rivers, glaciers, the sea, plant and animal life.
+In a dry climate subject to great extremes of temperature the character
+and rate of decay will differ from those of a moist or an arctic
+climate. But it must be remembered that, however much they may vary in
+activity and in the results which they effect, the epigene forces work
+without intermission, while the hypogene forces bring about the upheaval
+of land only after long intervals. Hence, trifling as the results during
+a human life may appear, if we realize the multiplying influence of time
+we are led to perceive that the apparently feeble superficial agents
+can, in the course of ages, achieve stupendous transformations in the
+aspect of the land. If this efficacy may be deduced from what can be
+seen to be in progress now, it may not less convincingly be shown, from
+the nature of the sedimentary rocks of the earth's crust, to have been
+in progress from the early beginnings of geological history. Side by
+side with the various upheavals and subsidences, there has been a
+continuous removal of materials from the land, and an equally persistent
+deposit of these materials under water, with the consequent growth of
+new rocks. Denudation has been aptly compared to a process of
+sculpturing wherein, while each of the implements employed by nature,
+like a special kind of graving tool, produces its own characteristic
+impress on the land, they all combine harmoniously towards the
+achievement of their one common task. Hence the present contours of the
+land depend partly on the original configuration of the ground, and the
+influence it may have had in guiding the operations of the erosive
+agents, partly on the vigour with which these agents perform their work,
+and partly on the varying structure and powers of resistance possessed
+by the rocks on which the erosion is carried on.
+
+Where a new tract of land has been raised out of the sea by such an
+energetic movement as broke up the crust and produced the complicated
+structure and tumultuous external forms of a great mountain chain, the
+influence of the hypogene forces on the topography attains its highest
+development. But even the youngest existing chain has suffered so
+greatly from denudation that the aspect which it presented at the time
+of its uplift can only be dimly perceived. No more striking illustration
+of this feature can be found than that supplied by the Alps, nor one
+where the geotectonic structures have been so fully studied in detail.
+On the outer flanks of these mountains the longitudinal ridges and
+valleys of the Jura correspond with lines of anticline and syncline. Yet
+though the dominant topographical elements of the region have obviously
+been produced by the plication of the stratified formations, each ridge
+has suffered so large an amount of erosion that the younger rocks have
+been removed from its crest where the older members of the series are
+now exposed to view, while on every slope proofs may be seen of
+extensive denudation. If from these long wave-like undulations of the
+ground, where the relations between the disposition of the rocks below
+and the forms of the surface are so clearly traceable, the observer
+proceeds inwards to the main chain, he finds that the plications and
+displacements of the various formations assume an increasingly
+complicated character; and that although proofs of great denudation
+continue to abound, it becomes increasingly difficult to form any
+satisfactory conjecture as to the shape of the ground when the upheaval
+ended or any reliable estimate of the amount of material which has since
+then been removed. Along the central heights the mountains lift
+themselves towards the sky like the storm-swept crests of vast
+earth-billows. The whole aspect of the ground suggests intense
+commotion, and the impression thus given is often much intensified by
+the twisted and crumpled strata, visible from a long distance, on the
+crags and crests. On this broken-up surface the various agents of
+denudation have been ceaselessly engaged since it emerged from the sea.
+They have excavated valleys, sometimes along depressions provided for
+them by the subterranean disturbances, sometimes down the slopes of the
+disrupted blocks of ground. So powerful has been this erosion that
+valleys cut out along lines of anticline, which were natural ridges,
+have sometimes become more important than those in lines of syncline,
+which were structurally depressions. The same subaerial forces have
+eroded lake-basins, dug out corries or cirques, notched the ridges,
+splintered the crests and furrowed the slopes, leaving no part of the
+original surface of the uplifted chain unmodified.
+
+It has often been noted with surprise that features of underground
+structure which, it might have been confidently anticipated, should have
+exercised a marked influence on the topography of the surface have not
+been able to resist the levelling action of the denuding agents, and do
+not now affect the surface at all. This result is conspicuously seen in
+coal-fields where the strata are abundantly traversed by faults. These
+dislocations, having sometimes a displacement of several hundred feet,
+might have been expected to break up the surface into a network of
+cliffs and plains; yet in general they do not modify the level character
+of the ground above. One of the most remarkable faults in Europe is the
+great thrust which bounds the southern edge of the Belgian coal-field
+and brings the Devonian rocks above the Coal-measures. It can be traced
+across Belgium into the Boulonnais, and may not improbably run beneath
+the Secondary and Tertiary rocks of the south of England. It is crossed
+by the valleys of the Meuse and other northerly-flowing streams. Yet so
+indistinctly is it marked in the Meuse valley that no one would suspect
+its existence from any peculiarity in the general form of the ground,
+and even an experienced geologist, until he had learned the structure of
+the district, would scarcely detect any fault at all.
+
+Where faults have influenced the superficial topography, it is usually
+by giving rise to a hollow along which the subaerial agents and
+especially running water can act effectively. Such a hollow may be
+eventually widened and deepened into a valley. On bare crags and crests,
+lines of fault are apt to be marked by notches or clefts, and they thus
+help to produce the pinnacles and serrated outlines of these exposed
+uplands.
+
+It was cogently enforced by Hutton and Playfair, and independently by
+Lamarck, that no co-operation of underground agency is needed to produce
+such topography as may be seen in a great part of the world, but that if
+a tract of sea-floor were upraised into a wide plain, the fall of rain
+and the circulation of water over its surface would in the end carve out
+such a system of hills and valleys as may be seen on the dry land now.
+No such plain would be a dead-level. It would have inequalities on its
+surface which would serve as channels to guide the drainage from the
+first showers of rain. And these channels would be slowly widened and
+deepened until they would become ravines and valleys, while the ground
+between them would be left projecting as ridges and hills. Nor would the
+erosion of such a system of water-courses require a long series of
+geological periods for its accomplishment. From measurements and
+estimates of the amount of erosion now taking place in the basin of the
+Mississippi river it has been computed that valleys 800 ft. deep might
+be carved out in less than a million years. In the vast tablelands of
+Colorado and other western regions of the United States an impressive
+picture is presented of the results of mere subaerial erosion on
+undisturbed and nearly level strata. Systems of stream-courses and
+valleys, river gorges unexampled elsewhere in the world for depth and
+length, vast winding lines of escarpment, like ranges of sea-cliffs,
+terraced slopes rising from plateau to plateau, huge buttresses and
+solitary stacks standing like islands out of the plains, great
+mountain-masses towering into picturesque peaks and pinnacles cleft by
+innumerable gullies, yet everywhere marked by the parallel bars of the
+horizontal strata out of which they have been carved--these are the
+orderly symmetrical characteristics of a country where the scenery is
+due entirely to the action of subaerial agents on the one hand and the
+varying resistance of perfectly regular stratified rocks on the other.
+
+The details of the sculpture of the land have mainly depended on the
+nature of the materials on which nature's erosive tools have been
+employed. The joints by which all rocks are traversed have been
+especially serviceable as dominant lines down which the rain has
+filtered, up which the springs have risen and into which the frost
+wedges have been driven. On the high bare scarps of a lofty mountain the
+inner structure of the mass is laid open, and there the system of joints
+even more than faults is seen to have determined the lines of crest, the
+vertical walls of cliff and precipice, the forms of buttress and recess,
+the position of cleft and chasm, the outline of spire and pinnacle. On
+the lower slopes, even under the tapestry of verdure which nature
+delights to hang where she can over her naked rocks, we may detect the
+same pervading influence of the joints upon the forms assumed by ravines
+and crags. Each kind of stone, too, gives rise to its own characteristic
+form of scenery. Massive crystalline rocks, such as granite, break up
+along their joints and often decay into sand or earth along their
+exposed surfaces, giving rise to rugged crags with long talus slopes at
+their base. The stratified rocks besides splitting at their joints are
+especially distinguished by parallel ledges, cornices and recesses,
+produced by the irregular decay of their component strata, so that they
+often assume curiously architectural types of scenery. But besides this
+family feature they display many minor varieties of aspect according to
+their lithological composition. A range of sandstone hills, for example,
+presents a marked contrast to one of limestone, and a line of chalk
+downs to the escarpments formed by alternating bands of harder and
+softer clays and shales.
+
+It may suffice here merely to allude to a few of the more important
+parts of the topography of the land in their relation to physiographical
+geology. A true mountain-chain, viewed from the geological side, is a
+mass of high ground which owes its prominence to a ridging-up of the
+earth's crust, and the intense plication and rupture of the rocks of
+which it is composed. But ranges of hills almost mountainous in their
+bulk may be formed by the gradual erosion of valleys out of a mass of
+original high ground, such as a high plateau or tableland. Eminences
+which have been isolated by denudation from the main mass of the
+formations of which they originally formed part are known as "outliers"
+or "hills of circumdenudation."
+
+Tablelands, as already pointed out, may be produced either by the
+upheaval of tracts of horizontal strata from the sea-floor into land; or
+by the uprise of plains of denudation, where rocks of various
+composition, structure and age have been levelled down to near or below
+the level of the sea by the co-operation of the various erosive agents.
+Most of the great tablelands of the globe are platforms of
+little-disturbed strata which have been upraised bodily to a
+considerable elevation. No sooner, however, are they placed in that
+position than they are attacked by running water, and begin to be
+hollowed out into systems of valleys. As the valleys sink, the platforms
+between them grow into narrower and more definite ridges, until
+eventually the level tableland is converted into a complicated network
+of hills and valleys, wherein, nevertheless, the key to the whole
+arrangement is furnished by a knowledge of the disposition and effects
+of the flow of water. The examples of this process brought to light in
+Colorado, Wyoming, Nevada and the other western regions by Newberry,
+King, Hayden, Powell and other explorers, are among the most striking
+monuments of geological operations in the world.
+
+Examples of ancient and much decayed tablelands formed by the denudation
+of much disturbed rocks are furnished by the Highlands of Scotland and
+of Norway. Each of these tracts of high ground consists of some of the
+oldest and most dislocated formations of Europe, which at a remote
+period were worn down into a plain, and in that condition may have lain
+long submerged under the sea and may possibly have been overspread there
+with younger formations. Having at a much later time been raised several
+thousand feet above sea-level the ancient platforms of Britain and
+Scandinavia have been since exposed to denudation, whereby each of them
+has been so deeply channeled into glens and fjords that it presents
+to-day a surface of rugged hills, either isolated or connected along the
+flanks, while only fragments of the general surface of the tableland can
+here and there be recognized amidst the general destruction.
+
+Valleys have in general been hollowed out by the greater erosive action
+of running water along the channels of drainage. Their direction has
+been probably determined in the great majority of cases by
+irregularities of the surface along which the drainage flowed on the
+first emergence of the land. Sometimes these irregularities have been
+produced by folds of the terrestrial crust, sometimes by faults,
+sometimes by the irregularities on the surface of an uplifted platform
+of deposition or of denudation. Two dominant trends may be observed
+among them. Some are longitudinal and run along the line of flexures in
+the upraised tract of land, others are transverse where the drainage has
+flowed down the slopes of the ridges into the longitudinal valleys or
+into the sea. The forms of valleys have been governed partly by the
+structure and composition of the rocks, and partly by the relative
+potency of the different denuding agents. Where the influence of rain
+and frost has been slight, and the streams, supplied from distant
+sources, have had sufficient declivity, deep, narrow, precipitous
+ravines or gorges have been excavated. The canyons of the arid region of
+the Colorado are a magnificent example of this result. Where, on the
+other hand, ordinary atmospheric action has been more rapid, the sides
+of the river channels have been attacked, and open sloping glens and
+valleys have been hollowed out. A gorge or defile is usually due to the
+action of a waterfall, which, beginning with some abrupt declivity or
+precipice in the course of the river when it first commenced to flow, or
+caused by some hard rock crossing the channel, has eaten its way
+backward.
+
+Lakes have been already referred to, and their modes of origin have been
+mentioned. As they are continually being filled up with the detritus
+washed into them from the surrounding regions they cannot be of any
+great geological antiquity, unless where by some unknown process their
+basins are from time to time widened and deepened.
+
+In the general subaerial denudation of a country, innumerable minor
+features are worked out as the structure of the rocks controls the
+operations of the eroding agents. Thus, among comparatively undisturbed
+strata, a hard bed resting upon others of a softer kind is apt to form
+along its outcrop a line of cliff or escarpment. Though a long range of
+such cliffs resembles a coast that has been worn by the sea, it may be
+entirely due to mere atmospheric waste. Again, the more resisting
+portions of a rock may be seen projecting as crags or knolls. An igneous
+mass will stand out as a bold hill from amidst the more decomposable
+strata through which it has risen. These features, often so marked on
+the lower grounds, attain their most conspicuous development among the
+higher and barer parts of the mountains, where subaerial disintegration
+is most rapid. The torrents tear out deep gullies from the sides of the
+declivities. Corries or cirques are scooped out on the one hand and
+naked precipices are left on the other. The harder bands of rock project
+as massive ribs down the slopes, shoot up into prominent _aiguilles_, or
+help to give to the summits the notched saw-like outlines they so often
+present.
+
+The materials worn from the surface of the higher are spread out over
+the lower grounds. The streams as they descend begin to drop their
+freight of sediment when, by the lessening of their declivity, their
+carrying power is diminished. The great plains of the earth's surface
+are due to this deposit of gravel, sand and loam. They are thus
+monuments at once of the destructive and reproductive processes which
+have been in progress unceasingly since the first land rose above the
+sea and the first shower of rain fell. Every pebble and particle of
+their soil, once part of the distant mountains, has travelled slowly and
+fitfully to lower levels. Again and again have these materials been
+shifted, ever moving downward and sea-ward. For centuries, perhaps, they
+have taken their share in the fertility of the plains and have
+ministered to the nurture of flower and tree, of the bird of the air,
+the beast of the field and of man himself. But their destiny is still
+the great ocean. In that bourne alone can they find undisturbed repose,
+and there, slowly accumulating in massive beds, they will remain until,
+in the course of ages, renewed upheaval shall raise them into future
+land, there once more to pass through the same cycle of change. (A. Ge.)
+
+  LITERATURE.--_Historical_: The standard work is Karl A. von Zittel's
+  _Geschichte der Geologie und Palaontologie_ (1899), of which there is
+  an abbreviated, but still valuable, English translation; D'Archiac,
+  _Histoire des progres de la geologie_, deals especially with the
+  period 1834-1850; Keferstein, _Geschichte und Literatur der
+  Geognosie_, gives a summary up to 1840; while Sir A. Geikie's
+  _Founders of Geology_ (1897; 2nd ed., 1906) deals more particularly
+  with the period 1750-1820. General treatises: Sir Charles Lyell's
+  _Principles of Geology_ is a classic. Of modern English works, Sir A.
+  Geikie's _Text Book of Geology_ (4th ed., 1903) occupies the first
+  place; the work of T.C. Chamberlin and R.D. Salisbury, _Geology; Earth
+  History_ (3 vols., 1905-1906), is especially valuable for American
+  geology. A. de Lapparent's _Traite de geologie_ (5th ed., 1906), is
+  the standard French work. H. Credner's _Elemente der Geologie_ has
+  gone through several editions in Germany. Dynamical and
+  physiographical geology are elaborately treated by E. Suess, _Das
+  Antlitz der Erde_, translated into English, with the title _The Face
+  of the Earth_. The practical study of the science is treated of by F.
+  von Richthofen, _Fuhrer fur Forschungsreisende_ (1886); G.A. Cole,
+  _Aids in Practical Geology_ (5th ed., 1906); A. Geikie, _Outlines of
+  Field Geology_ (5th ed., 1900). The practical applications of Geology
+  are discussed by J.V. Elsden, _Applied Geology_ (1898-1899). The
+  relations of Geology to scenery are dealt with by Sir A. Geikie,
+  _Scenery of Scotland_ (3rd ed., 1901); J.E. Marr, _The Scientific
+  Study of Scenery_ (1900); Lord Avebury, _The Scenery of Switzerland_
+  (1896); _The Scenery of England_ (1902); and J. Geikie, _Earth
+  Sculpture_ (1898). A detailed bibliography is given in Sir A. Geikie's
+  _Text Book of Geology_. See also the separate articles on geological
+  subjects for special references to authorities.
+
+
+FOOTNOTES:
+
+  [1] In De Luc's _Lettres physiques et morales sur les montagnes_
+    (1778), the word "cosmology" is used for our science, the author
+    stating that "geology" is more appropriate, but it "was not a word in
+    use." In a completed edition, published in 1779, the same statement
+    is made, but "geology" occurs in the text; in the same year De
+    Saussure used the word without any explanation, as if it were well
+    known.
+
+  [2] The subject of the age of the earth has also been discussed by
+    Professor J. Joly and Professor W.J. Sollas. The former geologist,
+    approaching the question from a novel point of view, has estimated
+    the total quantity of sodium in the water of the ocean and the
+    quantity of that element received annually by the ocean from the
+    denudation of the land. Dividing the one sum by the other, he arrives
+    at the result that the probable age of the earth is between 90 and
+    100 millions of years (_Trans. Roy. Dublin Soc._ ser. ii. vol. vii.,
+    1899, p. 23: _Geol. Mag._, 1900, p. 220). Professor Sollas believes
+    that this limit exceeds what is required for the evolution of
+    geological history, that the lower limit assigned by Lord Kelvin
+    falls short of what the facts demand, and that geological time will
+    probably be found to have been comprised within some indeterminate
+    period between these limits. (Address to Section C, _Brit. Assoc.
+    Report_, 1900; _Age of the Earth_, London, 1905.)
+
+
+
+
+GEOMETRICAL CONTINUITY. In a report of the Institute prefixed to Jean
+Victor Poncelet's _Traite des proprietes projectives des figures_
+(Paris, 1822), it is said that he employed "ce qu'il appelle le principe
+de continuite." The law or principle thus named by him had, he tells us,
+been tacitly assumed as axiomatic by "les plus savans geometres." It had
+in fact been enunciated as "lex continuationis," and "la loi de la
+continuite," by Gottfried Wilhelm Leibnitz (Oxf. N.E.D.), and previously
+under another name by Johann Kepler in cap. iv. 4 of his _Ad Vitellionem
+paralipomena quibus astronomiae pars optica traditur_ (Francofurti,
+1604). Of sections of the cone, he says, there are five species from the
+"recta linea" or line-pair to the circle. From the line-pair we pass
+through an infinity of hyperbolas to the parabola, and thence through an
+infinity of ellipses to the circle. Related to the sections are certain
+remarkable points which have no name. Kepler calls them foci. The circle
+has one focus at the centre, an ellipse or hyperbola two foci
+equidistant from the centre. The parabola has one focus within it, and
+another, the "caecus focus," which may be imagined to be _at infinity_
+on the axis _within or without the curve_. The line from it to any point
+of the section is parallel to the axis. To carry out the analogy we must
+speak paradoxically, and say that the line-pair likewise has foci, which
+in this case coalesce as in the circle and fall upon the lines
+themselves; for our geometrical terms should be subject to analogy.
+Kepler dearly loves analogies, his most trusty teachers, acquainted with
+all the secrets of nature, "_omnium naturae arcanorum conscios_." And
+they are to be especially regarded in geometry as, by the use of
+"however absurd expressions," classing extreme limiting forms with an
+infinity of intermediate cases, and placing the whole essence of a thing
+clearly before the eyes.
+
+Here, then, we find formulated by Kepler the doctrine of the concurrence
+of parallels at a single point at infinity and the principle of
+continuity (under the name analogy) in relation to the infinitely great.
+Such conceptions so strikingly propounded in a famous work could not
+escape the notice of contemporary mathematicians. Henry Briggs, in a
+letter to Kepler from Merton College, Oxford, dated "10 Cal. Martiis
+1625," suggests improvements in the _Ad Vitellionem paralipomena_, and
+gives the following construction: Draw a line CBADC, and let an ellipse,
+a parabola, and a hyperbola have B and A for focus and vertex. Let CC
+be the other foci of the ellipse and the hyperbola. Make AD equal to AB,
+and with centres CC and radius in each case equal to CD describe
+circles. Then any point of the ellipse is equidistant from the focus B
+and one circle, and any point of the hyperbola from the focus B and the
+other circle. Any point P of the parabola, in which the second focus is
+missing or infinitely distant, is equidistant from the focus B and the
+line through D which we call the directrix, this taking the place of
+either circle when its centre C is at infinity, and every line CP being
+then parallel to the axis. Thus Briggs, and we know not how many "savans
+geometres" who have left no record, had already taken up the new
+doctrine in geometry in its author's lifetime. Six years after Kepler's
+death in 1630 Girard Desargues, "the Monge of his age," brought out the
+first of his remarkable works founded on the same principles, a short
+tract entitled _Methode universelle de mettre en perspective les objets
+donnes reellement ou en devis_ (Paris, 1636); but "Le privilege etoit de
+1630." (Poudra, _[OE]uvres de Des._, i. 55). Kepler as a modern geometer
+is best known by his _New Stereometry of Wine Casks_ (Lincii, 1615), in
+which he replaces the circuitous Archimedean method of exhaustion by a
+direct "royal road" of infinitesimals, treating a vanishing arc as a
+straight line and regarding a curve as made up of a succession of short
+chords. Some 2000 years previously one Antipho, probably the well-known
+opponent of Socrates, has regarded a circle in like manner as the
+limiting form of a many-sided inscribed rectilinear figure. Antipho's
+notion was rejected by the men of his day as unsound, and when
+reproduced by Kepler it was again stoutly opposed as incapable of any
+sort of geometrical demonstration--not altogether without reason, for it
+rested on an assumed law of continuity rather than on palpable proof.
+
+To complete the theory of continuity, the one thing needful was the idea
+of imaginary points implied in the algebraical geometry of Rene
+Descartes, in which equations between variables representing
+co-ordinates were found often to have imaginary roots. Newton, in his
+two sections on "Inventio orbium" (_Principia_ i. 4, 5), shows in his
+brief way that he is familiar with the principles of modern geometry. In
+two propositions he uses an auxiliary line which is supposed to cut the
+conic in X and Y, but, as he remarks at the end of the second (prop.
+24), it may not cut it at all. For the sake of brevity he passes on at
+once with the observation that the required constructions are evident
+from the case in which the line cuts the trajectory. In the scholium
+appended to prop. 27, after saying that an asymptote is a tangent at
+infinity, he gives an unexplained general construction for the axes of a
+conic, which seems to imply that it has asymptotes. In all such cases,
+having equations to his loci in the background, he may have thought of
+elements of the figure as passing into the imaginary state in such
+manner as not to vitiate conclusions arrived at on the hypothesis of
+their reality.
+
+Roger Joseph Boscovich, a careful student of Newton's works, has a full
+and thorough discussion of geometrical continuity in the third and last
+volume of his _Elementa universae matheseos_ (ed. prim. Venet, 1757),
+which contains _Sectionum conicarum elementa nova quadam methodo
+concinnata et dissertationem de transformatione locorum geometricorum,
+ubi de continuitatis lege, et de quibusdam infiniti mysteriis_. His
+first principle is that all varieties of a defined locus have the same
+properties, so that what is demonstrable of one should be demonstrable
+in like manner of all, although some artifice may be required to bring
+out the underlying analogy between them. The opposite extremities of an
+infinite straight line, he says, are to be regarded as joined, as if the
+line were a circle having its centre at the infinity on either side of
+it. This leads up to the idea of a _veluti plus quam infinita extensio_,
+a line-circle containing, as we say, the line infinity. Change from the
+real to the imaginary state is contingent upon the passage of some
+element of a figure through zero or infinity and never takes place _per
+saltum_. Lines being some positive and some negative, there must be
+negative rectangles and negative squares, such as those of the exterior
+diameters of a hyperbola. Boscovich's first principle was that of
+Kepler, by whose _quantumvis absurdis locutionibus_ the boldest
+applications of it are covered, as when we say with Poncelet that all
+concentric circles in a plane touch one another in two imaginary fixed
+points at infinity. In G.K. Ch. von Staudt's _Geometrie der Lage and
+Beitrage zur G. der L._ (Nurnberg, 1847, 1856-1860) the geometry of
+position, including the extension of the field of pure geometry to the
+infinite and the imaginary, is presented as an independent science,
+"welche des Messens nicht bedarf." (See GEOMETRY: _Projective_.)
+
+Ocular illusions due to distance, such as Roger Bacon notices in the
+_Opus majus_ (i. 126, ii. 108, 497; Oxford, 1897), lead up to or
+illustrate the mathematical uses of the infinite and its reciprocal the
+infinitesimal. Specious objections can, of course, be made to the
+anomalies of the law of continuity, but they are inherent in the higher
+geometry, which has taught us so much of the "secrets of nature."
+Kepler's excursus on the "analogy" between the conic sections
+hereinbefore referred to is given at length in an article on "The
+Geometry of Kepler and Newton" in vol. xviii. of the _Transactions of
+the Cambridge Philosophical Society_ (1900). It had been generally
+overlooked, until attention was called to it by the present writer in a
+note read in 1880 (_Proc. C.P.S._ iv. 14-17), and shortly afterwards in
+_The Ancient and Modern Geometry of Conics, with Historical Notes and
+Prolegomena_ (Cambridge 1881).     (C. T.*)
+
+
+
+
+GEOMETRY, the general term for the branch of mathematics which has for
+its province the study of the properties of space. From experience, or
+possibly intuitively, we characterize existent space by certain
+fundamental qualities, termed axioms, which are insusceptible of proof;
+and these axioms, in conjunction with the mathematical entities of the
+point, straight line, curve, surface and solid, appropriately defined,
+are the premises from which the geometer draws conclusions. The
+geometrical axioms are merely conventions; on the one hand, the system
+may be based upon inductions from experience, in which case the deduced
+geometry may be regarded as a branch of physical science; or, on the
+other hand, the system may be formed by purely logical methods, in which
+case the geometry is a phase of pure mathematics. Obviously the geometry
+with which we are most familiar is that of existent space--the
+three-dimensional space of experience; this geometry may be termed
+Euclidean, after its most famous expositor. But other geometries exist,
+for it is possible to frame systems of axioms which definitely
+characterize some other kind of space, and from these axioms to deduce a
+series of non-contradictory propositions; such geometries are called
+non-Euclidean.
+
+It is convenient to discuss the subject-matter of geometry under the
+following headings:
+
+I. _Euclidean Geometry_: a discussion of the axioms of existent space
+and of the geometrical entities, followed by a synoptical account of
+Euclid's Elements.
+
+II. _Projective Geometry_: primarily Euclidean, but differing from I. in
+employing the notion of geometrical continuity (q.v.)--points and lines
+at infinity.
+
+III. _Descriptive Geometry_: the methods for representing upon planes
+figures placed in space of three dimensions.
+
+IV. _Analytical Geometry_: the representation of geometrical figures and
+their relations by algebraic equations.
+
+V. _Line Geometry_: an analytical treatment of the line regarded as the
+space element.
+
+VI. _Non-Euclidean Geometry_: a discussion of geometries other than that
+of the space of experience.
+
+VII. _Axioms of Geometry_: a critical analysis of the foundations of
+geometry.
+
+  Special subjects are treated under their own headings: e.g.
+  PROJECTION, PERSPECTIVE; CURVE, SURFACE; CIRCLE, CONIC SECTION;
+  TRIANGLE, POLYGON, POLYHEDRON; there are also articles on special
+  curves and figures, e.g. ELLIPSE, PARABOLA, HYPERBOLA; TETRAHEDRON,
+  CUBE, OCTAHEDRON, DODECAHEDRON, ICOSAHEDRON; CARDIOID, CATENARY,
+  CISSOID, CONCHOID, CYCLOID, EPICYCLOID, LIMACON, OVAL, QUADRATRIX,
+  SPIRAL, &c.
+
+_History._--The origin of geometry (Gr. [Greek: ge], earth, [Greek:
+metron], a measure) is, according to Herodotus, to be found in the
+etymology of the word. Its birthplace was Egypt, and it arose from the
+need of surveying the lands inundated by the Nile floods. In its
+infancy it therefore consisted of a few rules, very rough and
+approximate, for computing the areas of triangles and quadrilaterals;
+and, with the Egyptians, it proceeded no further, the geometrical
+entities--the point, line, surface and solid--being only discussed in so
+far as they were involved in practical affairs. The point was realized
+as a mark or position, a straight line as a stretched string or the
+tracing of a pole, a surface as an area; but these units were not
+abstracted; and for the Egyptians geometry was only an art--an auxiliary
+to surveying.[1] The first step towards its elevation to the rank of a
+science was made by Thales (q.v.) of Miletus, who transplanted the
+elementary Egyptian mensuration to Greece. Thales clearly abstracted the
+notions of points and lines, founding the geometry of the latter unit,
+and discovering _per saltum_ many propositions concerning areas, the
+circle, &c. The empirical rules of the Egyptians were corrected and
+developed by the Ionic School which he founded, especially by
+Anaximander and Anaxagoras, and in the 6th century B.C. passed into the
+care of the Pythagoreans. From this time geometry exercised a powerful
+influence on Greek thought. Pythagoras (q.v.), seeking the key of the
+universe in arithmetic and geometry, investigated logically the
+principles underlying the known propositions; and this resulted in the
+formulation of definitions, axioms and postulates which, in addition to
+founding a _science_ of geometry, permitted a crystallization,
+fractional, it is true, of the amorphous collection of material at hand.
+Pythagorean geometry was essentially a geometry of areas and solids; its
+goal was the regular solids--the tetrahedron, cube, octahedron,
+dodecahedron and icosahedron--which symbolized the five elements of
+Greek cosmology. The geometry of the circle, previously studied in Egypt
+and much more seriously by Thales, was somewhat neglected, although this
+curve was regarded as the most perfect of all plane figures and the
+sphere the most perfect of all solids. The circle, however, was taken up
+by the Sophists, who made most of their discoveries in attempts to solve
+the classical problems of squaring the circle, doubling the cube and
+trisecting an angle. These problems, besides stimulating pure geometry,
+i.e. the geometry of constructions made by the ruler and compasses,
+exercised considerable influence in other directions. The first problem
+led to the discovery of the method of _exhaustion_ for determining
+areas. Antiphon inscribed a square in a circle, and on each side an
+isosceles triangle having its vertex on the circle; on the sides of the
+octagon so obtained, isosceles triangles were again constructed, the
+process leading to inscribed polygons of 8, 16 and 32 sides; and the
+areas of these polygons, which are easily determined, are successive
+approximations to the area of the circle. Bryson of Heraclea took an
+important step when he circumscribed, in addition to inscribing,
+polygons to a circle, but he committed an error in treating the circle
+as the mean of the two polygons. The method of Antiphon, in assuming
+that by continued division a polygon can be constructed coincident with
+the circle, demanded that magnitudes are not infinitely divisible. Much
+controversy ranged about this point; Aristotle supported the doctrine of
+infinite divisibility; Zeno attempted to show its absurdity. The
+mechanical tracing of loci, a principle initiated by Archytas of
+Tarentum to solve the last two problems, was a frequent subject for
+study, and several mechanical curves were thus discovered at subsequent
+dates (cissoid, conchoid, quadratrix). Mention may be made of
+Hippocrates, who, besides developing the known methods, made a study of
+similar figures, and, as a consequence, of proportion. This step is
+important as bringing into line discontinuous number and continuous
+magnitude.
+
+A fresh stimulus was given by the succeeding Platonists, who, accepting
+in part the Pythagorean cosmology, made the study of geometry
+preliminary to that of philosophy. The many discoveries made by this
+school were facilitated in no small measure by the clarification of the
+axioms and definitions, the logical sequence of propositions which was
+adopted, and, more especially, by the formulation of the analytic
+method, i.e. of assuming the truth of a proposition and then reasoning
+to a known truth. The main strength of the Platonist geometers lies in
+stereometry or the geometry of solids. The Pythagoreans had dealt with
+the sphere and regular solids, but the pyramid, prism, cone and cylinder
+were but little known until the Platonists took them in hand. Eudoxus
+established their mensuration, proving the pyramid and cone to have
+one-third the content of a prism and cylinder on the same base and of
+the same height, and was probably the discoverer of a proof that the
+volumes of spheres are as the cubes of their radii. The discussion of
+sections of the cone and cylinder led to the discovery of the three
+curves named the parabola, ellipse and hyperbola (see CONIC SECTION); it
+is difficult to over-estimate the importance of this discovery; its
+investigation marks the crowning achievement of Greek geometry, and led
+in later years to the fundamental theorems and methods of modern
+geometry.
+
+The presentation of the subject-matter of geometry as a connected and
+logical series of propositions, prefaced by [Greek: Horoi] or
+foundations, had been attempted by many; but it is to Euclid that we owe
+a complete exposition. Little indeed in the _Elements_ is probably
+original except the arrangement; but in this Euclid surpassed such
+predecessors as Hippocrates, Leon, pupil of Neocleides, and Theudius of
+Magnesia, devising an apt logical model, although when scrutinized in
+the light of modern mathematical conceptions the proofs are riddled with
+fallacies. According to the commentator Proclus, the _Elements_ were
+written with a twofold object, first, to introduce the novice to
+geometry, and secondly, to lead him to the regular solids; conic
+sections found no place therein. What Euclid did for the line and
+circle, Apollonius did for the conic sections, but there we have a
+discoverer as well as editor. These two works, which contain the
+greatest contributions to ancient geometry, are treated in detail in
+Section I. _Euclidean Geometry_ and the articles EUCLID; CONIC SECTION;
+APPOLONIUS. Between Euclid and Apollonius there flourished the
+illustrious Archimedes, whose geometrical discoveries are mainly
+concerned with the mensuration of the circle and conic sections, and of
+the sphere, cone and cylinder, and whose greatest contribution to
+geometrical method is the elevation of the method of exhaustion to the
+dignity of an instrument of research. Apollonius was followed by
+Nicomedes, the inventor of the conchoid; Diocles, the inventor of the
+cissoid; Zenodorus, the founder of the study of isoperimetrical figures;
+Hipparchus, the founder of trigonometry; and Heron the elder, who wrote
+after the manner of the Egyptians, and primarily directed attention to
+problems of practical surveying.
+
+Of the many isolated discoveries made by the later Alexandrian
+mathematicians, those of Menelaus are of importance. He showed how to
+treat spherical triangles, establishing their properties and determining
+their congruence; his theorem on the products of the segments in which
+the sides of a triangle are cut by a line was the foundation on which
+Carnot erected his theory of transversals. These propositions, and also
+those of Hipparchus, were utilized and developed by Ptolemy (q.v.), the
+expositor of trigonometry and discoverer of many isolated propositions.
+Mention may be made of the commentator Pappus, whose _Mathematical
+Collections_ is valuable for its wealth of historical matter; of Theon,
+an editor of Euclid's _Elements_ and commentator of Ptolemy's
+_Almagest_; of Proclus, a commentator of Euclid; and of Eutocius, a
+commentator of Apollonius and Archimedes.
+
+The Romans, essentially practical and having no inclination to study
+science _qua_ science, only had a geometry which sufficed for surveying;
+and even here there were abundant inaccuracies, the empirical rules
+employed being akin to those of the Egyptians and Heron. The Hindus,
+likewise, gave more attention to computation, and their geometry was
+either of Greek origin or in the form presented in trigonometry, more
+particularly connected with arithmetic. It had no logical foundations;
+each proposition stood alone; and the results were empirical. The Arabs
+more closely followed the Greeks, a plan adopted as a sequel to the
+translation of the works of Euclid, Apollonius, Archimedes and many
+others into Arabic. Their chief contribution to geometry is exhibited in
+their solution of algebraic equations by intersecting conics, a step
+already taken by the Greeks in isolated cases, but only elevated into a
+_method_ by Omar al Hayyami, who flourished in the 11th century. During
+the middle ages little was added to Greek and Arabic geometry. Leonardo
+of Pisa wrote a _Practica geometriae_ (1220), wherein Euclidean methods
+are employed; but it was not until the 14th century that geometry,
+generally Euclid's _Elements_, became an essential item in university
+curricula. There was, however, no sign of original development, other
+branches of mathematics, mainly algebra and trigonometry, exercising a
+greater fascination until the 16th century, when the subject again came
+into favour.
+
+The extraordinary mathematical talent which came into being in the 16th
+and 17th centuries reacted on geometry and gave rise to all those
+characters which distinguish modern from ancient geometry. The first
+innovation of moment was the formulation of the principle of geometrical
+continuity by Kepler. The notion of infinity which it involved permitted
+generalizations and systematizations hitherto unthought of (see
+GEOMETRICAL CONTINUITY); and the method of indefinite division applied
+to rectification, and quadrature and cubature problems avoided the
+cumbrous method of exhaustion and provided more accurate results.
+Further progress was made by Bonaventura Cavalieri, who, in his
+_Geometria indivisibilibus continuorum_ (1620), devised a method
+intermediate between that of exhaustion and the infinitesimal calculus
+of Leibnitz and Newton. The logical basis of his system was corrected by
+Roberval and Pascal; and their discoveries, taken in conjunction with
+those of Leibnitz, Newton, and many others in the fluxional calculus,
+culminated in the branch of our subject known as differential geometry
+(see INFINITESIMAL CALCULUS; CURVE; SURFACE).
+
+A second important advance followed the recognition that conics could be
+regarded as projections of a circle, a conception which led at the hands
+of Desargues and Pascal to modern _projective geometry_ and
+_perspective_. A third, and perhaps the most important, advance attended
+the application of algebra to geometry by Descartes, who thereby founded
+_analytical geometry_. The new fields thus opened up were diligently
+explored, but the calculus exercised the greatest attraction and
+relatively little progress was made in geometry until the beginning of
+the 19th century, when a new era opened.
+
+Gaspard Monge was the first important contributor, stimulating
+analytical and differential geometry and founding _descriptive geometry_
+in a series of papers and especially in his lectures at the Ecole
+polytechnique. Projective geometry, founded by Desargues, Pascal, Monge
+and L.N.M. Carnot, was crystallized by J.V. Poncelet, the creator of the
+modern methods. In his _Traite des proprietes des figures_ (1822) the
+line and circular points at infinity, imaginaries, polar reciprocation,
+homology, cross-ratio and projection are systematically employed. In
+Germany, A.F. Mobius, J. Plucker and J. Steiner were making far-reaching
+contributions. Mobius, in his _Barycentrische Calcul_ (1827), introduced
+homogeneous co-ordinates, and also the powerful notion of geometrical
+transformation, including the special cases of collineation and duality;
+Plucker, in his _Analytisch-geometrische Entwickelungen_ (1828-1831),
+and his _System der analytischen Geometrie_ (1835), introduced the
+abridged notation, line and plane co-ordinates, and the conception of
+generalized space elements; while Steiner, besides enriching geometry in
+numerous directions, was the first to systematically generate figures by
+projective pencils. We may also notice M. Chasles, whose _Apercu
+historique_ (1837) is a classic. Synthetic geometry, characterized by
+its fruitfulness and beauty, attracted most attention, and it so
+happened that its originally weak logical foundations became replaced by
+a more substantial set of axioms. These were found in the anharmonic
+ratio, a device leading to the liberation of synthetic geometry from
+metrical relations, and in involution, which yielded rigorous
+definitions of imaginaries. These innovations were made by K.J.C. von
+Staudt. Analytical geometry was stimulated by the algebra of invariants,
+a subject much developed by A. Cayley, G. Salmon, S.H. Aronhold, L.O.
+Hesse, and more particularly by R.F.A. Clebsch.
+
+The introduction of the line as a space element, initiated by H.
+Grassmann (1844) and Cayley (1859), yielded at the hands of Plucker a
+new geometry, termed _line geometry_, a subject developed more notably
+by F. Klein, Clebsch, C.T. Reye and F.O.R. Sturm (see Section V., _Line
+Geometry_).
+
+_Non-euclidean geometries_, having primarily their origin in the
+discussion of Euclidean parallels, and treated by Wallis, Saccheri and
+Lambert, have been especially developed during the 19th century. Four
+lines of investigation may be distinguished:--the naive-synthetic,
+associated with Lobatschewski, Bolyai, Gauss; the metric differential,
+studied by Riemann, Helmholtz, Beltrami; the projective, developed by
+Cayley, Klein, Clifford; and the critical-synthetic, promoted chiefly by
+the Italian mathematicians Peano, Veronese, Burali-Forte, Levi Civitta,
+and the Germans Pasch and Hilbert.     (C. E.*)
+
+
+I. EUCLIDEAN GEOMETRY
+
+This branch of the science of geometry is so named since its methods and
+arrangement are those laid down in Euclid's _Elements_.
+
+S 1. _Axioms._--The object of geometry is to investigate the properties
+of space. The first step must consist in establishing those fundamental
+properties from which all others follow by processes of deductive
+reasoning. They are laid down in the Axioms, and these ought to form
+such a system that nothing need be added to them in order fully to
+characterize space, and that nothing may be omitted without making the
+system incomplete. They must, in fact, completely "define" space.
+
+S 2. _Definitions._--The axioms of Euclidean Geometry are obtained from
+inspection of existent space and of solids in existent space,--hence
+from experience. The same source gives us the notions of the geometrical
+entities to which the axioms relate, viz. solids, surfaces, lines or
+curves, and points. A solid is directly given by experience; we have
+only to abstract all material from it in order to gain the notion of a
+geometrical solid. This has shape, size, position, and may be moved. Its
+boundary or boundaries are called surfaces. They separate one part of
+space from another, and are said to have no thickness. Their boundaries
+are curves or lines, and these have length only. Their boundaries,
+again, are points, which have no magnitude but only position. We thus
+come in three steps from solids to points which have no magnitude; in
+each step we lose one extension. Hence we say a solid has three
+dimensions, a surface two, a line one, and a point none. Space itself,
+of which a solid forms only a part, is also said to be of three
+dimensions. The same thing is intended to be expressed by saying that a
+solid has length, breadth and thickness, a surface length and breadth, a
+line length only, and a point no extension whatsoever.
+
+Euclid gives the essence of these statements as definitions:--
+
+  Def. 1, I. _A point is that which has no parts, or which has no
+  magnitude._
+
+  Def. 2, I. _A line is length without breadth._
+
+  Def. 5, I. _A superficies is that which has only length and breadth._
+
+  Def. 1, XI. _A solid is that which has length, breadth and thickness._
+
+It is to be noted that the synthetic method is adopted by Euclid; the
+analytical derivation of the successive ideas of "surface," "line," and
+"point" from the experimental realization of a "solid" does not find a
+place in his system, although possessing more advantages.
+
+If we allow motion in geometry, we may generate these entities by moving
+a point, a line, or a surface, thus:--
+
+  The path of a moving point is a line.
+
+  The path of a moving line is, in general, a surface.
+
+  The path of a moving surface is, in general, a solid.
+
+And we may then assume that the lines, surfaces and solids, as defined
+before, can all be generated in this manner. From this generation of the
+entities it follows again that the boundaries--the first and last
+position of the moving element--of a line are points, and so on; and
+thus we come back to the considerations with which we started.
+
+Euclid points this out in his definitions,--Def. 3, I., Def. 6, I., and
+Def. 2, XI. He does not, however, show the connexion which these
+definitions have with those mentioned before. When points and lines have
+been defined, a statement like Def. 3, I., "The extremities of a line
+are points," is a proposition which either has to be proved, and then it
+is a theorem, or which has to be taken for granted, in which case it is
+an axiom. And so with Def. 6, I., and Def. 2, XI.
+
+S 3. Euclid's definitions mentioned above are attempts to describe, in a
+few words, notions which we have obtained by inspection of and
+abstraction from solids. A few more notions have to be added to these,
+principally those of the simplest line--the straight line, and of the
+simplest surface--the flat surface or plane. These notions we possess,
+but to define them accurately is difficult. Euclid's Definition 4, I.,
+"A straight line is that which lies evenly between its extreme points,"
+must be meaningless to any one who has not the notion of straightness in
+his mind. Neither does it state a property of the straight line which
+can be used in any further investigation. Such a property is given in
+Axiom 10, I. It is really this axiom, together with Postulates 2 and 3,
+which characterizes the straight line.
+
+Whilst for the straight line the verbal definition and axiom are kept
+apart, Euclid mixes them up in the case of the plane. Here the
+Definition 7, I., includes an axiom. It defines a plane as a surface
+which has the property that every straight line which joins any two
+points in it lies altogether in the surface. But if we take a straight
+line and a point in such a surface, and draw all straight lines which
+join the latter to all points in the first line, the surface will be
+fully determined. This construction is therefore sufficient as a
+definition. That every other straight line which joins any two points in
+this surface lies altogether in it is a further property, and to assume
+it gives another axiom.
+
+Thus a number of Euclid's axioms are hidden among his first definitions.
+A still greater confusion exists in the present editions of Euclid
+between the postulates and axioms so called, but this is due to later
+editors and not to Euclid himself. The latter had the last three axioms
+put together with the postulates [Greek: (aitemata)], so that these were
+meant to include all assumptions relating to space. The remaining
+assumptions, which relate to magnitudes in general, viz. the first eight
+"axioms" in modern editions, were called "common notions" [Greek:
+(koivai ennoiai)]. Of the latter a few may be said to be definitions.
+Thus the eighth might be taken as a definition of "equal," and the
+seventh of "halves." If we wish to collect the axioms used in Euclid's
+_Elements_, we have therefore to take the three postulates, the last
+three axioms as generally given, a few axioms hidden in the definitions,
+and an axiom used by Euclid in the proof of Prop. 4, I, and on a few
+other occasions, viz. that figures may be moved in space without change
+of shape or size.
+
+  S 4. _Postulates._--The assumptions actually made by Euclid may be
+  stated as follows:--
+
+  (1) Straight lines exist which have the property that any one of them
+  may be produced both ways without limit, that through any two points
+  in space such a line may be drawn, and that any two of them coincide
+  throughout their indefinite extensions as soon as two points in the
+  one coincide with two points in the other. (This gives the contents of
+  Def. 4, part of Def. 35, the first two Postulates, and Axiom 10.)
+
+  (2) Plane surfaces or planes exist having the property laid down in
+  Def. 7, that every straight line joining any two points in such a
+  surface lies altogether in it.
+
+  (3) Right angles, as defined in Def. 10, are possible, and all right
+  angles are equal; that is to say, wherever in space we take a plane,
+  and wherever in that plane we construct a right angle, all angles thus
+  constructed will be equal, so that any one of them may be made to
+  coincide with any other. (Axiom 11.)
+
+  (4) The 12th Axiom of Euclid. This we shall not state now, but only
+  introduce it when we cannot proceed any further without it.
+
+  (5) Figures maybe freely moved in space without change of shape or
+  size. This is assumed by Euclid, but not stated as an axiom.
+
+  (6) In any plane a circle may be described, having any point in that
+  plane as centre, and its distance from any other point in that plane
+  as radius. (Postulate 3.)
+
+The definitions which have not been mentioned are all "nominal
+definitions," that is to say, they fix a name for a thing described.
+Many of them overdetermine a figure.
+
+S 5. Euclid's _Elements_ (see EUCLID) are contained in thirteen books.
+Of these the first four and the sixth are devoted to "plane geometry,"
+as the investigation of figures in a plane is generally called. The 5th
+book contains the theory of proportion which is used in Book VI. The
+7th, 8th and 9th books are purely arithmetical, whilst the 10th contains
+a most ingenious treatment of geometrical irrational quantities. These
+four books will be excluded from our survey. The remaining three books
+relate to figures in space, or, as it is generally called, to "solid
+geometry." The 7th, 8th, 9th, 10th, 13th and part of the 11th and 12th
+books are now generally omitted from the school editions of the
+_Elements_. In the first four and in the 6th book it is to be understood
+that all figures are drawn in a plane.
+
+
+  BOOK I. OF EUCLID'S "ELEMENTS."
+
+  S 6. According to the third postulate it is possible to draw in any
+  plane a circle which has its centre at any given point, and its radius
+  equal to the distance of this point from any other point given in the
+  plane. This makes it possible (Prop. 1) to construct on a given line
+  AB an equilateral triangle, by drawing first a circle with A as centre
+  and AB as radius, and then a circle with B as centre and BA as radius.
+  The point where these circles intersect--that they intersect Euclid
+  quietly assumes--is the vertex of the required triangle. Euclid does
+  not suppose, however, that a circle may be drawn which has its radius
+  equal to the distance between any two points unless one of the points
+  be the centre. This implies also that we are not supposed to be able
+  to make any straight line equal to any other straight line, or to
+  carry a distance about in space. Euclid therefore next solves the
+  problem: It is required along a given straight line from a point in it
+  to set off a distance equal to the length of another straight line
+  given anywhere in the plane. This is done in two steps. It is shown in
+  Prop. 2 how a straight line may be drawn from a given point equal in
+  length to another given straight line not drawn from that point. And
+  then the problem itself is solved in Prop. 3, by drawing first through
+  the given point some straight line of the required length, and then
+  about the same point as centre a circle having this length as radius.
+  This circle will cut off from the given straight line a length equal
+  to the required one. Nowadays, instead of going through this long
+  process, we take a pair of compasses and set off the given length by
+  its aid. This assumes that we may move a length about without changing
+  it. But Euclid has not assumed it, and this proceeding would be fully
+  justified by his desire not to take for granted more than was
+  necessary, if he were not obliged at his very next step actually to
+  make this assumption, though without stating it.
+
+  S 7. We now come (in Prop. 4) to the first theorem. It is the
+  fundamental theorem of Euclid's whole system, there being only a very
+  few propositions (like Props. 13, 14, 15, I.), except those in the 5th
+  book and the first half of the 11th, which do not depend upon it. It
+  is stated very accurately, though somewhat clumsily, as follows:--
+
+  _If two triangles have two sides of the one equal to two sides of the
+  other, each to each, and have also the angles contained by those sides
+  equal to one another, they shall also have their bases or third sides
+  equal; and the two triangles shall be equal; and their other angles
+  shall be equal, each to each, namely, those to which the equal sides
+  are opposite._
+
+  That is to say, the triangles are "identically" equal, and one may be
+  considered as a copy of the other. The proof is very simple. The first
+  triangle is taken up and placed on the second, so that the parts of
+  the triangles which are known to be equal fall upon each other. It is
+  then easily seen that also the remaining parts of one coincide with
+  those of the other, and that they are therefore equal. This process of
+  applying one figure to another Euclid scarcely uses again, though many
+  proofs would be simplified by doing so. The process introduces motion
+  into geometry, and includes, as already stated, the axiom that figures
+  may be moved without change of shape or size.
+
+  If the last proposition be applied to an isosceles triangle, which has
+  two sides equal, we obtain the theorem (Prop. 5), _if two sides of a
+  triangle are equal, then the angles opposite these sides are equal_.
+
+  Euclid's proof is somewhat complicated, and a stumbling-block to many
+  schoolboys. The proof becomes much simpler if we consider the
+  isosceles triangle ABC (AB = AC) twice over, once as a triangle BAC,
+  and once as a triangle CAB; and now remember that AB, AC in the first
+  are equal respectively to AC, AB in the second, and the angles
+  included by these sides are equal. Hence the triangles are equal, and
+  the angles in the one are equal to those in the other, viz. those
+  which are opposite equal sides, i.e. angle ABC in the first equals
+  angle ACB in the second, as they are opposite the equal sides AC and
+  AB in the two triangles.
+
+  There follows the converse theorem (Prop. 6). _If two angles in a
+  triangle are equal, then the sides opposite them are equal_,--i.e. the
+  triangle is isosceles. The proof given consists in what is called a
+  _reductio ad absurdum_, a kind of proof often used by Euclid, and
+  principally in proving the converse of a previous theorem. It assumes
+  that the theorem to be proved is wrong, and then shows that this
+  assumption leads to an absurdity, i.e. to a conclusion which is in
+  contradiction to a proposition proved before--that therefore the
+  assumption made cannot be true, and hence that the theorem is true. It
+  is often stated that Euclid invented this kind of proof, but the
+  method is most likely much older.
+
+  S 8. It is next proved that _two triangles which have the three sides
+  of the one equal respectively to those of the other are identically
+  equal, hence that the angles of the one are equal respectively to
+  those of the other, those being equal which are opposite equal sides_.
+  This is Prop. 8, Prop. 7 containing only a first step towards its
+  proof.
+
+  These theorems allow now of the solution of a number of problems,
+  viz.:--
+
+  _To bisect a given angle_ (Prop. 9).
+
+  _To bisect a given finite straight line_ (Prop. 10).
+
+  _To draw a straight line perpendicularly to a given straight line
+  through a given point in it_ (Prop. 11), _and also through a given
+  point not in it_ (Prop. 12).
+
+  The solutions all depend upon properties of isosceles triangles.
+
+  S 9. The next three theorems relate to angles only, and might have
+  been proved before Prop. 4, or even at the very beginning. The first
+  (Prop. 13) says, _The angles which one straight line makes with
+  another straight line on one side of it either are two right angles or
+  are together equal to two right angles_. This theorem would have been
+  unnecessary if Euclid had admitted the notion of an angle such that
+  its two limits are in the same straight line, and had besides defined
+  the sum of two angles.
+
+  Its converse (Prop. 14) is of great use, inasmuch as it enables us in
+  many cases to prove that two straight lines drawn from the same point
+  are one the continuation of the other. So also is
+
+  Prop. 15. _If two straight lines cut one another, the vertical or
+  opposite angles shall be equal._
+
+  S 10. Euclid returns now to properties of triangles. Of great
+  importance for the next steps (though afterwards superseded by a more
+  complete theorem) is
+
+  Prop. 16. _If one side of a triangle be produced, the exterior angle
+  shall be greater than either of the interior opposite angles._
+
+  Prop. 17. _Any two angles of a triangle are together less than two
+  right angles, is an immediate consequence of it._ By the aid of these
+  two, the following fundamental properties of triangles are easily
+  proved:--
+
+  Prop. 18. _The greater side of every triangle has the greater angle
+  opposite to it_;
+
+  Its converse, Prop. 19. _The greater angle of every triangle is
+  subtended by the greater side, or has the greater side opposite to
+  it_;
+
+  Prop. 20. _Any two sides of a triangle are together greater than the
+  third side_;
+
+  And also Prop. 21. _If from the ends of the side of a triangle there
+  be drawn two straight lines to a point within the triangle, these
+  shall be less than the other two sides of the triangle, but shall
+  contain a greater angle._
+
+  S 11. Having solved two problems (Props. 22, 23), he returns to two
+  triangles which have two sides of the one equal respectively to two
+  sides of the other. It is known (Prop. 4) that if the included angles
+  are equal then the third sides are equal; and conversely (Prop. 8), if
+  the third sides are equal, then the angles included by the first sides
+  are equal. From this it follows that if the included angles are not
+  equal, the third sides are not equal; and conversely, that if the
+  third sides are not equal, the included angles are not equal. Euclid
+  now completes this knowledge by proving, that "_if the included angles
+  are not equal, then the third side in that triangle is the greater
+  which contains the greater angle_"; and conversely, that "_if the
+  third sides are unequal, that triangle contains the greater angle
+  which contains the greater side_." These are Prop. 24 and Prop. 25.
+
+  S 12. The next theorem (Prop. 26) says that _if two triangles have one
+  side and two angles of the one equal respectively to one side and two
+  angles of the other, viz. in both triangles either the angles adjacent
+  to the equal side, or one angle adjacent and one angle opposite it,
+  then the two triangles are identically equal_.
+
+  This theorem belongs to a group with Prop. 4 and Prop. 8. Its first
+  case might have been given immediately after Prop. 4, but the second
+  case requires Prop. 16 for its proof.
+
+  S 13. We come now to the investigation of parallel straight lines,
+  i.e. of straight lines which lie in the same plane, and cannot be made
+  to meet however far they be produced either way. The investigation
+  which starts from Prop. 16, will become clearer if a few names be
+  explained which are not all used by Euclid. If two straight lines be
+  cut by a third, the latter is now generally called a "transversal" of
+  the figure. It forms at the two points where it cuts the given lines
+  four angles with each. Those of the angles which lie between the given
+  lines are called interior angles, and of these, again, any two which
+  lie on opposite sides of the transversal but one at each of the two
+  points are called "alternate angles."
+
+  We may now state Prop. 16 thus:--_If two straight lines which meet are
+  cut by a transversal, their alternate angles are unequal_. For the
+  lines will form a triangle, and one of the alternate angles will be an
+  exterior angle to the triangle, the other interior and opposite to it.
+
+  From this follows at once the theorem contained in Prop. 27. _If two
+  straight lines which are cut by a transversal make alternate angles
+  equal, the lines cannot meet, however far they be produced, hence they
+  are parallel._ This proves the existence of parallel lines.
+
+  Prop. 28 states the same fact in different forms. _If a straight line,
+  falling on two other straight lines, make the exterior angle equal to
+  the interior and opposite angle on the same side of the line, or make_
+  _the interior angles on the same side together equal to two right
+  angles, the two straight lines shall be parallel to one another_.
+
+  Hence we know that, "if two straight lines which are cut by a
+  transversal meet, their alternate angles are not equal"; and hence
+  that, "if alternate angles are equal, then the lines are parallel."
+
+  The question now arises, Are the propositions converse to these true
+  or not? That is to say, "If alternate angles are unequal, do the lines
+  meet?" And "if the lines are parallel, are alternate angles
+  necessarily equal?"
+
+  The answer to either of these two questions implies the answer to the
+  other. But it has been found impossible to prove that the negation or
+  the affirmation of either is true.
+
+  The difficulty which thus arises is overcome by Euclid assuming that
+  the first question has to be answered in the affirmative. This gives
+  his last axiom (12), which we quote in his own words.
+
+  Axiom 12.--_If a straight line meet two straight lines, so as to make
+  the two interior angles on the same side of it taken together less
+  than two right angles, these straight lines, being continually
+  produced, shall at length meet on that side on which are the angles
+  which are less than two right angles._
+
+  The answer to the second of the above questions follows from this, and
+  gives the theorem Prop. 29:--_If a straight line fall on two parallel
+  straight lines, it makes the alternate angles equal to one another,
+  and the exterior angle equal to the interior and opposite angle on the
+  same side, and also the two interior angles on the same side together
+  equal to two right angles_.
+
+  S 14. With this a new part of elementary geometry begins. The earlier
+  propositions are independent of this axiom, and would be true even if
+  a wrong assumption had been made in it. They all relate to figures in
+  a plane. But a plane is only one among an infinite number of
+  conceivable surfaces. We may draw figures on any one of them and study
+  their properties. We may, for instance, take a sphere instead of the
+  plane, and obtain "spherical" in the place of "plane" geometry. If on
+  one of these surfaces lines and figures could be drawn, answering to
+  all the definitions of our plane figures, and if the axioms with the
+  exception of the last all hold, then all propositions up to the 28th
+  will be true for these figures. This is the case in spherical geometry
+  if we substitute "shortest line" or "great circle" for "straight
+  line," "small circle" for "circle," and if, besides, we limit all
+  figures to a part of the sphere which is less than a hemisphere, so
+  that two points on it cannot be opposite ends of a diameter, and
+  therefore determine always one and only one great circle.
+
+  For spherical triangles, therefore, all the important propositions 4,
+  8, 26; 5 and 6; and 18, 19 and 20 will hold good.
+
+  This remark will be sufficient to show the impossibility of proving
+  Euclid's last axiom, which would mean proving that this axiom is a
+  consequence of the others, and hence that the theory of parallels
+  would hold on a spherical surface, where the other axioms do hold,
+  whilst parallels do not even exist.
+
+  It follows that the axiom in question states an inherent difference
+  between the plane and other surfaces, and that the plane is only fully
+  characterized when this axiom is added to the other assumptions.
+
+  S 15. The introduction of the new axiom and of parallel lines leads to
+  a new class of propositions.
+
+  After proving (Prop. 30) that "_two lines which are each parallel to a
+  third are parallel to each other_," we obtain the new properties of
+  triangles contained in Prop. 32. Of these the second part is the most
+  important, viz. the theorem, _The three interior angles of every
+  triangle are together equal to two right angles_.
+
+  As easy deductions not given by Euclid but added by Simson follow the
+  propositions about the angles in polygons, they are given in English
+  editions as corollaries to Prop. 32.
+
+  These theorems do not hold for spherical figures. The sum of the
+  interior angles of a spherical triangle is always greater than two
+  right angles, and increases with the area.
+
+  S 16. The theory of parallels as such may be said to be finished with
+  Props. 33 and 34, which state properties of the parallelogram, i.e. of
+  a quadrilateral formed by two pairs of parallels. They are--
+
+  Prop. 33. _The straight lines which join the extremities of two equal
+  and parallel straight lines towards the same parts are themselves
+  equal and parallel_; and
+
+  Prop. 34. _The opposite sides and angles of a parallelogram are equal
+  to one another, and the diameter (diagonal) bisects the parallelogram,
+  that is, divides it into two equal parts._
+
+  S 17. The rest of the first book relates to areas of figures.
+
+  The theory is made to depend upon the theorems--
+
+  Prop. 35. _Parallelograms on the same base and between the same
+  parallels are equal to one another_; and
+
+  Prop. 36. _Parallelograms on equal bases and between the same
+  parallels are equal to one another_.
+
+  As each parallelogram is bisected by a diagonal, the last theorems
+  hold also if the word parallelogram be replaced by "triangle," as is
+  done in Props. 37 and 38.
+
+  It is to be remarked that Euclid proves these propositions only in the
+  case when the parallelograms or triangles have their bases in the same
+  straight line.
+
+  The theorems converse to the last form the contents of the next three
+  propositions, viz.: Props, 40 and 41.--_Equal triangles, on the same
+  or on equal bases, in the same straight line, and on the same side of
+  it, are between the same parallels_.
+
+  That the two cases here stated are given by Euclid in two separate
+  propositions proved separately is characteristic of his method.
+
+  S 18. To compare areas of other figures, Euclid shows first, in Prop.
+  42, how _to draw a parallelogram which is equal in area to a given
+  triangle, and has one of its angles equal to a given angle_. If the
+  given angle is right, then the problem is solved _to draw a
+  "rectangle" equal in area to a given triangle_.
+
+  Next this parallelogram is transformed into another parallelogram,
+  _which has one of its sides equal to a given straight line_, whilst
+  its angles remain unaltered. This may be done by aid of the theorem in
+
+  Prop. 43. _The complements of the parallelograms which are about the
+  diameter of any parallelogram are equal to one another._
+
+  Thus the problem (Prop. 44) is solved to _construct a parallelogram on
+  a given line, which is equal in area to a given triangle, and which
+  has one angle equal to a given angle_ (generally a right angle).
+
+  As every polygon can be divided into a number of triangles, we can now
+  construct a parallelogram having a given angle, say a right angle, and
+  being equal in area to a given polygon. For each of the triangles into
+  which the polygon has been divided, a parallelogram may be
+  constructed, having one side equal to a given straight line and one
+  angle equal to a given angle. If these parallelograms be placed side
+  by side, they may be added together to form a single parallelogram,
+  having still one side of the given length. This is done in Prop. 45.
+
+  Herewith a means is found to compare areas of different polygons. We
+  need only construct two rectangles equal in area to the given
+  polygons, and having each one side of given length. By comparing the
+  unequal sides we are enabled to judge whether the areas are equal, or
+  which is the greater. Euclid does not state this consequence, but the
+  problem is taken up again at the end of the second book, where it is
+  shown how to construct a square equal in area to a given polygon.
+
+  Prop. 46 is: _To describe a square on a given straight line_.
+
+  S 19. The first book concludes with one of the most important theorems
+  in the whole of geometry, and one which has been celebrated since the
+  earliest times. It is stated, but on doubtful authority, that
+  Pythagoras discovered it, and it has been called by his name. If we
+  call that side in a right-angled triangle which is opposite the right
+  angle the hypotenuse, we may state it as follows:--
+
+  Theorem of Pythagoras (Prop. 47).--_In every right-angled triangle the
+  square on the hypotenuse is equal to the sum of the squares of the
+  other sides._
+
+  And conversely--
+
+  Prop. 48. _If the square described on one of the sides of a triangle
+  be equal to the squares described on the other sides, then the angle
+  contained by these two sides is a right angle._
+
+  On this theorem (Prop. 47) almost all geometrical measurement depends,
+  which cannot be directly obtained.
+
+
+  BOOK II.
+
+  S 20. The propositions in the second book are very different in
+  character from those in the first; they all relate to areas of
+  rectangles and squares. Their true significance is best seen by
+  stating them in an algebraic form. This is often done by expressing
+  the lengths of lines by aid of numbers, which tell how many times a
+  chosen unit is contained in the lines. If there is a unit to be found
+  which is contained an exact number of times in each side of a
+  rectangle, it is easily seen, and generally shown in the teaching of
+  arithmetic, that the rectangle contains a number of unit squares equal
+  to the product of the numbers which measure the sides, a unit square
+  being the square on the unit line. If, however, no such unit can be
+  found, this process requires that connexion between lines and numbers
+  which is only established by aid of ratios of lines, and which is
+  therefore at this stage altogether inadmissible. But there exists
+  another way of connecting these propositions with algebra, based on
+  modern notions which seem destined greatly to change and to simplify
+  mathematics. We shall introduce here as much of it as is required for
+  our present purpose.
+
+  At the beginning of the second book we find a definition according to
+  which "a rectangle is said to be 'contained' by the two sides which
+  contain one of its right angles"; in the text this phraseology is
+  extended by speaking of rectangles contained by any two straight
+  lines, meaning the rectangle which has two adjacent sides equal to the
+  two straight lines.
+
+  We shall denote a finite straight line by a single small letter, a, b,
+  c, ... x, and the area of the rectangle contained by two lines a and b
+  by ab, and this we shall call the product of the two lines a and b. It
+  will be understood that this definition has nothing to do with the
+  definition of a product of numbers.
+
+  We define as follows:--
+
+  The _sum_ of two straight lines a and b means a straight line c which
+  may be divided in two parts equal respectively to a and b. This sum is
+  denoted by a + b.
+
+  The _difference_ of two lines a and b (in symbols, a-b) means a line c
+  which when added to b gives a; that is,
+
+    a - b = c if b + c = a.
+
+  The _product_ of two lines a and b (in symbols, ab) means the area of
+  the rectangle contained by the lines a and b. For aa, which means the
+  square on the line a, we write a^2.
+
+  S 21. The first ten of the fourteen propositions of the second book
+  may then be written in the form of formulae as follows:--
+
+    Prop. 1. a(b + c + d + ... ) = ab + ac + ad + ...
+
+      "   2. ab + ac = a^2 if b + c = a.
+
+      "   3. a(a + b) = a^2 + ab.
+
+      "   4. (a + b)^2 = a^2 + 2ab + b^2.
+
+      "   5. (a + b)(a - b) + b^2 = a^2.
+
+      "   6. (a + b)(a - b) + b^2 = a^2.
+
+      "   7. a^2 + (a - b)^2 = 2a(a - b) + b^2.
+
+      "   8. 4(a + b)a + b^2 = (2a + b)^2.
+
+      "   9. (a + b)^2 + (a - b)^2 = 2a^2 + 2b^2.
+
+      "  10. (a + b)^2 + (a - b)^2 = 2a^2 + 2b^2.
+
+  It will be seen that 5 and 6, and also 9 and 10, are identical. In
+  Euclid's statement they do not look the same, the figures being
+  arranged differently.
+
+  If the letters a, b, c, ... denoted numbers, it follows from algebra
+  that each of these formulae is true. But this does not prove them in
+  our case, where the letters denote lines, and their products areas
+  without any reference to numbers. To prove them we have to discover
+  the laws which rule the operations introduced, viz. addition and
+  multiplication of segments. This we shall do now; and we shall find
+  that these laws are the same with those which hold in algebraical
+  addition and multiplication.
+
+  S 22. In a sum of numbers we may change the order in which the numbers
+  are added, and we may also add the numbers together in groups and then
+  add these groups. But this also holds for the sum of segments and for
+  the sum of rectangles, as a little consideration shows. That the sum
+  of rectangles has always a meaning follows from the Props. 43-45 in
+  the first book. These laws about addition are reducible to the two--
+
+    a + b = b + a            (1),
+
+    a + (b + c) = a + b + c  (2);
+
+  or, when expressed for rectangles,
+
+    ab + ed = ed + ab              (3),
+
+    ab + (cd + ef) = ab + cd + ef  (4).
+
+  The brackets mean that the terms in the bracket have been added
+  together before they are added to another term. The more general cases
+  for more terms may be deduced from the above.
+
+  For the product of two numbers we have the law that it remains
+  unaltered if the factors be interchanged. This also holds for our
+  geometrical product. For if ab denotes the area of the rectangle which
+  has a as base and b as altitude, then ba will denote the area of the
+  rectangle which has b as base and a as altitude. But in a rectangle we
+  may take either of the two lines which contain it as base, and then
+  the other will be the altitude. This gives
+
+    ab = ba       (5).
+
+  In order further to multiply a sum by a number, we have in algebra the
+  rule:--Multiply each term of the sum, and add the products thus
+  obtained. That this holds for our geometrical products is shown by
+  Euclid in his first proposition of the second book, where he proves
+  that the area of a rectangle whose base is the sum of a number of
+  segments is equal to the sum of rectangles which have these segments
+  separately as bases. In symbols this gives, in the simplest case,
+
+     a(b + c) = ab + ac   \
+                           >  (6).
+  and (b + c)a = ba + ca  /
+
+  To these laws, which have been investigated by Sir William Hamilton
+  and by Hermann Grassmann, the former has given special names. He calls
+  the laws expressed in
+
+    (1) and (3) the commutative law for addition;
+
+            (5)          "       "   "  multiplication;
+
+    (2) and (4) the associative laws for addition;
+
+            (6) the distributive law.
+
+  S 23. Having proved that these six laws hold, we can at once prove
+  every one of the above propositions in their algebraical form.
+
+  The first is proved geometrically, it being one of the fundamental
+  laws. The next two propositions are only special cases of the first.
+  Of the others we shall prove one, viz. the fourth:--
+
+         (a + b)^2 = (a + b)(a + b) = (a + b)a + (a + b)b   by (6).
+
+  But                     (a + b)a = aa + ba       by (6),
+                                   = aa + ab       by (5);
+
+  and                     (a + b)b = ab + bb       by (6).
+
+  Therefore     (a + b)^2 = aa + ab + (ab + bb) \
+                         = aa + (ab + ab) + bb   >  by (4).
+                         = aa + 2ab + bb        /
+
+  This gives the theorem in question.
+
+  In the same manner every one of the first ten propositions is proved.
+
+  It will be seen that the operations performed are exactly the same as
+  if the letters denoted numbers.
+
+  Props. 5 and 6 may also be written thus--
+
+    (a + b)(a - b) = a^2 - b^2.
+
+  Prop. 7, which is an easy consequence of Prop. 4, may be transformed.
+  If we denote by c the line a + b, so that
+
+    c = a + b, a = c - b,
+
+  we get
+
+    c^2 + (c - b)^2 = 2c(c - b) + b^2
+                    = 2c^2 - 2bc + b^2.
+
+  Subtracting c^2 from both sides, and writing a for c, we get
+
+    (a - b)^2 = a^2 - 2ab + b^2.
+
+  In Euclid's _Elements_ this form of the theorem does not appear, all
+  propositions being so stated that the notion of subtraction does not
+  enter into them.
+
+  S 24. The remaining two theorems (Props. 12 and 13) connect the square
+  on one side of a triangle with the sum of the squares on the other
+  sides, in case that the angle between the latter is acute or obtuse.
+  They are important theorems in trigonometry, where it is possible to
+  include them in a single theorem.
+
+  S 25. There are in the second book two problems, Props. 11 and 14.
+
+  If written in the above symbolic language, the former requires to find
+  a line x such that a(a - x) = x^2. Prop. 11 contains, therefore, the
+  solution of a quadratic equation, which we may write x^2 + ax = a^2.
+  The solution is required later on in the construction of a regular
+  decagon.
+
+  More important is the problem in the last proposition (Prop. 14). It
+  requires the construction of a square equal in area to a given
+  rectangle, hence a solution of the equation
+
+    x^2 = ab.
+
+  In Book I., 42-45, it has been shown how a rectangle may be
+  constructed equal in area to a given figure bounded by straight lines.
+  By aid of the new proposition we may therefore now determine a line
+  such that the square on that line is equal in area to any given
+  rectilinear figure, or we can _square_ any such figure.
+
+  As of two squares that is the greater which has the greater side, it
+  follows that now the comparison of two areas has been reduced to the
+  comparison of two lines.
+
+  The problem of reducing other areas to squares is frequently met with
+  among Greek mathematicians. We need only mention the problem of
+  squaring the circle (see CIRCLE).
+
+  In the present day the comparison of areas is performed in a simpler
+  way by reducing all areas to rectangles having a common base. Their
+  altitudes give then a measure of their areas.
+
+  The construction of a rectangle having the base u, and being equal in
+  area to a given rectangle, depends upon Prop. 43, I. This therefore
+  gives a solution of the equation
+
+    ab = ux,
+
+  where x denotes the unknown altitude.
+
+
+  BOOK III.
+
+  S 26. The third book of the _Elements_ relates exclusively to
+  properties of the circle. A circle and its circumference have been
+  defined in Book I., Def. 15. We restate it here in slightly different
+  words:--
+
+  _Definition_.--The circumference of a circle is a plane curve such
+  that all points in it have the same distance from a fixed point in the
+  plane. This point is called the "centre" of the circle.
+
+  Of the new definitions, of which eleven are given at the beginning of
+  the third book, a few only require special mention. The first, which
+  says that circles with equal radii are equal, is in part a theorem,
+  but easily proved by applying the one circle to the other. Or it may
+  be considered proved by aid of Prop. 24, equal circles not being used
+  till after this theorem.
+
+  In the second definition is explained what is meant by a line which
+  "touches" a circle. Such a line is now generally called a tangent to
+  the circle. The introduction of this name allows us to state many of
+  Euclid's propositions in a much shorter form.
+
+  For the same reason we shall call a straight line joining two points
+  on the circumference of a circle a "chord."
+
+  Definitions 4 and 5 may be replaced with a slight generalization by
+  the following:--
+
+  _Definition_.--By the distance of a point from a line is meant the
+  length of the perpendicular drawn from the point to the line.
+
+  S 27. From the definition of a circle it follows that every circle has
+  a centre. Prop. 1 requires to find it when the circle is given, i.e.
+  when its circumference is drawn.
+
+  To solve this problem a chord is drawn (that is, any two points in the
+  circumference are joined), and through the point where this is
+  bisected a perpendicular to it is erected. Euclid then proves, first,
+  that no point off this perpendicular can be the centre, hence that the
+  centre must lie in this line; and, secondly, that of the points on the
+  perpendicular one only can be the centre, viz. the one which bisects
+  the parts of the perpendicular bounded by the circle. In the second
+  part Euclid silently assumes that the perpendicular there used does
+  cut the circumference in two, and only in two points. The proof
+  therefore is incomplete. The proof of the first part, however, is
+  exact. By drawing two non-parallel chords, and the perpendiculars
+  which bisect them, the centre will be found as the point where these
+  perpendiculars intersect.
+
+  S 28. In Prop. 2 it is proved that a chord of a circle lies altogether
+  within the circle.
+
+  What we have called the first part of Euclid's solution of Prop. 1 may
+  be stated as a theorem:--
+
+  _Every straight line which bisects a chord, and is at right angles to
+  it, passes through the centre of the circle._
+
+  The converse to this gives Prop. 3, which may be stated thus:--
+
+  _If a straight line through the centre of a circle bisect a chord,
+  then it is perpendicular to the chord, and if it be perpendicular to
+  the chord it bisects it._
+
+  An easy consequence of this is the following theorem, which is
+  essentially the same as Prop. 4:--
+
+  _Two chords of a circle, of which neither passes through the centre,
+  cannot bisect each other._
+
+  These last three theorems are fundamental for the theory of the
+  circle. It is to be remarked that Euclid never proves that a straight
+  line cannot have more than two points in common with a circumference.
+
+  S 29. The next two propositions (5 and 6) might be replaced by a
+  single and a simpler theorem, viz:--
+
+  _Two circles which have a common centre, and whose circumferences have
+  one point in common, coincide._
+
+  Or, more in agreement with Euclid's form:--
+
+  _Two different circles, whose circumferences have a point in common,
+  cannot have the same centre._
+
+  That Euclid treats of two cases is characteristic of Greek
+  mathematics.
+
+  The next two propositions (7 and 8) again belong together. They may be
+  combined thus:--
+
+  _If from a point in a plane of a circle, which is not the centre,
+  straight lines be drawn to the different points of the circumference,
+  then of all these lines one is the shortest, and one the longest, and
+  these lie both in that straight line which joins the given point to
+  the centre. Of all the remaining lines each is equal to one and only
+  one other, and these equal lines lie on opposite sides of the shortest
+  or longest, and make equal angles with them._
+
+  Euclid distinguishes the two cases where the given point lies within
+  or without the circle, omitting the case where it lies in the
+  circumference.
+
+  From the last proposition it follows that if from a point more than
+  two equal straight lines can be drawn to the circumference, this point
+  must be the centre. This is Prop. 9.
+
+  As a consequence of this we get
+
+  _If the circumferences of the two circles have three points in common
+  they coincide._
+
+  For in this case the two circles have a common centre, because from
+  the centre of the one three equal lines can be drawn to points on the
+  circumference of the other. But two circles which have a common
+  centre, and whose circumferences have a point in common, coincide.
+  (Compare above statement of Props. 5 and 6.)
+
+  This theorem may also be stated thus:--
+
+  _Through three points only one circumference may be drawn; or, Three
+  points determine a circle._
+
+  Euclid does not give the theorem in this form. He proves, however,
+  _that the two circles cannot cut another in more than two points_
+  (Prop. 10), and _that two circles cannot touch one another in more
+  points than one_ (Prop. 13).
+
+  S 30. Propositions 11 and 12 assert that _if two circles touch, then
+  the point of contact lies on the line joining their centres_. This
+  gives two propositions, because the circles may touch either
+  internally or externally.
+
+  S 31. Propositions 14 and 15 relate to the length of chords. The first
+  says _that equal chords are equidistant from the centre, and that
+  chords which are equidistant from the centre are equal_;
+
+  Whilst Prop. 15 compares unequal chords, viz. _Of all chords the
+  diameter is the greatest, and of other chords that is the greater
+  which is nearer to the centre_; and conversely, _the greater chord is
+  nearer to the centre_.
+
+  S 32. In Prop. 16 the tangent to a circle is for the first time
+  introduced. The proposition is meant to show that the straight line at
+  the end point of the diameter and at right angles to it is a tangent.
+  The proposition itself does not state this. It runs thus:--
+
+  Prop. 16. _The straight line drawn at right angles to the diameter of
+  a circle, from the extremity of it, falls without the circle; and no
+  straight line can be drawn from the extremity, between that straight
+  line and the circumference, so as not to cut the circle._
+
+  _Corollary_.--The straight line at right angles to a diameter drawn
+  through the end point of it touches the circle.
+
+  The statement of the proposition and its whole treatment show the
+  difficulties which the tangents presented to Euclid.
+
+  Prop. 17 solves the problem _through a given point, either in the
+  circumference or without it, to draw a tangent to a given circle_.
+
+  Closely connected with Prop. 16 are Props. 18 and 19, which state
+  (Prop. 18), _that the line joining the centre of a circle to the point
+  of contact of a tangent is perpendicular to the tangent_; and
+  conversely (Prop. 19), _that the straight line through the point of
+  contact of, and perpendicular to, a tangent to a circle passes through
+  the centre of the circle_.
+
+  S 33. The rest of the book relates to angles connected with a circle,
+  viz. angles which have the vertex either at the centre or on the
+  circumference, and which are called respectively angles at the centre
+  and angles at the circumference. Between these two kinds of angles
+  exists the important relation expressed as follows:--
+
+  Prop. 20. _The angle at the centre of a circle is double of the angle
+  at the circumference on the same base, that is, on the same arc._
+
+  This is of great importance for its consequences, of which the two
+  following are the principal:--
+
+  Prop. 21. _The angles in the same segment of a circle are equal to one
+  another_;
+
+  Prop. 22. _The opposite angles of any quadrilateral figure inscribed
+  in a circle are together equal to two right angles._
+
+  Further consequences are:--
+
+  Prop. 23. _On the same straight line, and on the same side of it,
+  there cannot be two similar segments of circles, not coinciding with
+  one another_;
+
+  Prop. 24. _Similar segments of circles on equal straight lines are
+  equal to one another._
+
+  The problem Prop. 25. _A segment of a circle being given to describe
+  the circle of which it is a segment_, may be solved much more easily
+  by aid of the construction described in relation to Prop. 1, III., in
+  S 27.
+
+  S 34. There follow four theorems connecting the angles at the centre,
+  the arcs into which they divide the circumference, and the chords
+  subtending these arcs. They are expressed for angles, arcs and chords
+  in equal circles, but they hold also for angles, arcs and chords in
+  the same circle.
+
+  The theorems are:--
+
+  Prop. 26. _In equal circles equal angles stand on equal arcs, whether
+  they be at the centres or circumferences_;
+
+  Prop. 27. (converse to Prop. 26). _In equal circles the angles which
+  stand on equal arcs are equal to one another, whether they be at the
+  centres or the circumferences_;
+
+  Prop. 28. _In equal circles equal straight lines_ (equal chords) _cut
+  off equal arcs, the greater equal to the greater, and the less equal
+  to the less_;
+
+  Prop. 29 (converse to Prop. 28). _In equal circles equal arcs are
+  subtended by equal straight lines._
+
+  S 35. Other important consequences of Props. 20-22 are:--
+
+  Prop. 31. _In a circle the angle in a semicircle is a right angle; but
+  the angle in a segment greater than a semicircle is less than a right
+  angle; and the angle in a segment less than a semicircle is greater
+  than a right angle_;
+
+  Prop. 32. _If a straight line touch a circle, and from the point of
+  contact a straight line be drawn cutting the circle, the angles which
+  this line makes with the line touching the circle shall be equal to
+  the angles which are in the alternate segments of the circle._
+
+  S 36. Propositions 30, 33, 34, contain problems which are solved by
+  aid of the propositions preceding them:--
+
+  Prop. 30. _To bisect a given arc, that is, to divide it into two equal
+  parts_;
+
+  Prop. 33. _On a given straight line to describe a segment of a circle
+  containing an angle equal to a given rectilineal angle_;
+
+  Prop. 34. _From a given circle to cut off a segment containing an
+  angle equal to a given rectilineal angle_.
+
+  S 37. If we draw chords through a point A within a circle, they will
+  each be divided by A into two segments. Between these segments the law
+  holds that the rectangle contained by them has the same area on
+  whatever chord through A the segments are taken. The value of this
+  rectangle changes, of course, with the position of A.
+
+  A similar theorem holds if the point A be taken without the circle. On
+  every straight line through A, which cuts the circle in two points B
+  and C, we have two segments AB and AC, and the rectangles contained by
+  them are again equal to one another, and equal to the square on a
+  tangent drawn from A to the circle.
+
+  The first of these theorems gives Prop. 35, and the second Prop. 36,
+  with its corollary, whilst Prop. 37, the last of Book III., gives the
+  converse to Prop. 36. The first two theorems may be combined in one:--
+
+  _If through a point A in the plane of a circle a straight line be
+  drawn cutting the circle in B and C, then the rectangle AB.AC has a
+  constant value so long as the point A be fixed; and if from A a
+  tangent AD can be drawn to the circle, touching at D, then the above
+  rectangle equals the square on AD._
+
+  Prop. 37 may be stated thus:--
+
+  _If from a point A without a circle a line be drawn cutting the circle
+  in B and C, and another line to a point D on the circle, and AB.AC =
+  AD^2, then the line AD touches the circle at D._
+
+  It is not difficult to prove also the converse to the general
+  proposition as above stated. This proposition and its converse may be
+  expressed as follows:--
+
+  _If four points ABCD be taken on the circumference of a circle, and if
+  the lines AB, CD, produced if necessary, meet at E, then_
+
+    EA.EB = EC.ED;
+
+  _and conversely, if this relation holds then the four points lie on a
+  circle, that is, the circle drawn through three of them passes through
+  the fourth._
+
+  That a circle may always be drawn through three points, provided that
+  they do not lie in a straight line, is proved only later on in Book
+  IV.
+
+
+  BOOK IV.
+
+  S 38. The fourth book contains only problems, all relating to the
+  construction of triangles and polygons inscribed in and circumscribed
+  about circles, and of circles inscribed in or circumscribed about
+  triangles and polygons. They are nearly all given for their own sake,
+  and not for future use in the construction of figures, as are most of
+  those in the former books. In seven definitions at the beginning of
+  the book it is explained what is understood by figures inscribed in or
+  described about other figures, with special reference to the case
+  where one figure is a circle. Instead, however, of saying that one
+  figure is described about another, it is now generally said that the
+  one figure is circumscribed about the other. We may then state the
+  definitions 3 or 4 thus:--
+
+  _Definition._--A polygon is said to be inscribed in a circle, and the
+  circle is said to be circumscribed about the polygon, if the vertices
+  of the polygon lie in the circumference of the circle.
+
+  And definitions 5 and 6 thus:--
+
+  _Definition._--A polygon is said to be circumscribed about a circle,
+  and a circle is said to be inscribed in a polygon, if the sides of the
+  polygon are tangents to the circle.
+
+  S 39. The first problem is merely constructive. It requires to draw in
+  a given circle a chord equal to a given straight line, which is not
+  greater than the diameter of the circle. The problem is not a
+  determinate one, inasmuch as the chord may be drawn from any point in
+  the circumference. This may be said of almost all problems in this
+  book, especially of the next two. They are:--
+
+  Prop. 2. _In a given circle to inscribe a triangle equiangular to a
+  given triangle;_
+
+  Prop. 3. _About a given circle to circumscribe a triangle equiangular
+  to a given triangle._
+
+  S 40. Of somewhat greater interest are the next problems, where the
+  triangles are given and the circles to be found.
+
+  Prop. 4. _To inscribe a circle in a given triangle._
+
+  The result is that the problem has always a solution, viz. the centre
+  of the circle is the point where the bisectors of two of the interior
+  angles of the triangle meet. The solution shows, though Euclid does
+  not state this, that the problem has but one solution; and also,
+
+  _The three bisectors of the interior angles of any triangle meet in a
+  point, and this is the centre of the circle inscribed in the
+  triangle._
+
+  The solutions of most of the other problems contain also theorems. Of
+  these we shall state those which are of special interest; Euclid does
+  not state any one of them.
+
+  S 41. Prop. 5. _To circumscribe a circle about a given triangle._
+
+  The one solution which always exists contains the following:--
+
+  _The three straight lines which bisect the sides of a triangle at
+  right angles meet in a point, and this point is the centre of the
+  circle circumscribed about the triangle._
+
+  Euclid adds in a corollary the following property:--
+
+  The centre of the circle circumscribed about a triangle lies within,
+  on a side of, or without the triangle, according as the triangle is
+  acute-angled, right-angled or obtuse-angled.
+
+  S 42. Whilst it is always possible to draw a circle which is inscribed
+  in or circumscribed about a given triangle, this is not the case with
+  quadrilaterals or polygons of more sides. Of those for which this is
+  possible the regular polygons, i.e. polygons which have all their
+  sides and angles equal, are the most interesting. In each of them a
+  circle may be inscribed, and another may be circumscribed about it.
+
+  Euclid does not use the word regular, but he describes the polygons in
+  question as _equiangular_ and _equilateral_. We shall use the name
+  regular polygon. The regular triangle is equilateral, the regular
+  quadrilateral is the square.
+
+  Euclid considers the regular polygons of 4, 5, 6 and 15 sides. For
+  each of the first three he solves the problems--(1) to inscribe such a
+  polygon in a given circle; (2) to circumscribe it about a given
+  circle; (3) to inscribe a circle in, and (4) to circumscribe a circle
+  about, such a polygon.
+
+  For the regular triangle the problems are not repeated, because more
+  general problems have been solved.
+
+  Props. 6, 7, 8 and 9 solve these problems for the square.
+
+  The general problem of inscribing in a given circle a regular polygon
+  of n sides depends upon the problem of dividing the circumference of a
+  circle into n equal parts, or what comes to the same thing, of drawing
+  from the centre of the circle n radii such that the angles between
+  consecutive radii are equal, that is, to divide the space about the
+  centre into n equal angles. Thus, if it is required to inscribe a
+  square in a circle, we have to draw four lines from the centre, making
+  the four angles equal. This is done by drawing two diameters at right
+  angles to one another. The ends of these diameters are the vertices of
+  the required square. If, on the other hand, tangents be drawn at these
+  ends, we obtain a square circumscribed about the circle.
+
+  S 43. To construct a _regular pentagon_, we find it convenient first
+  to construct a _regular decagon_. This requires to divide the space
+  about the centre into ten equal angles. Each will be 1/10th of a right
+  angle, or 1/5th of two right angles. If we suppose the decagon
+  constructed, and if we join the centre to the end of one side, we get
+  an isosceles triangle, where the angle at the centre equals 1/5th of
+  two right angles; hence each of the angles at the base will be 2/5ths
+  of two right angles, as all three angles together equal two right
+  angles. Thus we have to construct an isosceles triangle, having the
+  angle at the vertex equal to half an angle at the base. This is solved
+  in Prop. 10, by aid of the problem in Prop. 11 of the second book. If
+  we make the sides of this triangle equal to the radius of the given
+  circle, then the base will be the side of the regular decagon
+  inscribed in the circle. This side being known the decagon can be
+  constructed, and if the vertices are joined alternately, leaving out
+  half their number, we obtain the regular pentagon. (Prop. 11.)
+
+  Euclid does not proceed thus. He wants the pentagon before the
+  decagon. This, however, does not change the real nature of his
+  solution, nor does his solution become simpler by not mentioning the
+  decagon.
+
+  Once the regular pentagon is inscribed, it is easy to circumscribe
+  another by drawing tangents at the vertices of the inscribed pentagon.
+  This is shown in Prop. 12.
+
+  Props. 13 and 14 teach how a circle may be inscribed in or
+  circumscribed about any given regular pentagon.
+
+  S 44. The _regular hexagon_ is more easily constructed, as shown in
+  Prop. 15. The result is that the side of the regular hexagon inscribed
+  in a circle is equal to the radius of the circle.
+
+  For this polygon the other three problems mentioned are not solved.
+
+  S 45. The book closes with Prop. 16. To inscribe a regular quindecagon
+  in a given circle. If we inscribe a regular pentagon and a regular
+  hexagon in the circle, having one vertex in common, then the arc from
+  the common vertex to the next vertex of the pentagon is 1/5th of the
+  circumference, and to the next vertex of the hexagon is 1/6th of the
+  circumference. The difference between these arcs is, therefore, 1/5 -
+  1/6 = 1/30th of the circumference. The latter may, therefore, be
+  divided into thirty, and hence also in fifteen equal parts, and the
+  regular quindecagon be described.
+
+  S 46. We conclude with a few theorems about regular polygons which are
+  not given by Euclid.
+
+  _The straight lines perpendicular to and bisecting the sides of any
+  regular polygon meet in a point. The straight lines bisecting the
+  angles in the regular polygon meet in the same point. This point is
+  the centre of the circles circumscribed about and inscribed in the
+  regular polygon._
+
+  We can bisect any given arc (Prop. 30, III.). Hence we can divide a
+  circumference into 2n equal parts as soon as it has been divided into
+  n equal parts, or as soon as a regular polygon of n sides has been
+  constructed. Hence--
+
+  _If a regular polygon of n sides has been constructed, then a regular
+  polygon of 2n sides, of 4n, of 8n sides, &c., may also be
+  constructed._ Euclid shows how to construct regular polygons of 3, 4,
+  5 and 15 sides. It follows that we can construct regular polygons of
+
+     3,  6, 12,  24  sides
+     4,  8, 16,  32    "
+     5, 10, 20,  40    "
+    15, 30, 60, 120    "
+
+  The construction of any new regular polygon not included in one of
+  these series will give rise to a new series. Till the beginning of the
+  19th century nothing was added to the knowledge of regular polygons as
+  given by Euclid. Then Gauss, in his celebrated _Arithmetic_, proved
+  that every regular polygon of 2^n + 1 sides may be constructed if this
+  number 2^n + 1 be prime, and that no others except those with 2^m(2^n
+  + 1) sides can be constructed by elementary methods. This shows that
+  regular polygons of 7, 9, 13 sides cannot thus be constructed, but
+  that a regular polygon of 17 sides is possible; for 17 = 2^4 + 1. The
+  next polygon is one of 257 sides. The construction becomes already
+  rather complicated for 17 sides.
+
+
+  BOOK V.
+
+  S 47. The fifth book of the _Elements_ is not exclusively geometrical.
+  It contains the theory of ratios and proportion of quantities in
+  general. The treatment, as here given, is admirable, and in every
+  respect superior to the algebraical method by which Euclid's theory is
+  now generally replaced. We shall treat the subject in order to show
+  why the usual algebraical treatment of proportion is not really sound.
+  We begin by quoting those definitions at the beginning of Book V.
+  which are most important. These definitions have given rise to much
+  discussion.
+
+  The only definitions which are essential for the fifth book are Defs.
+  1, 2, 4, 5, 6 and 7. Of the remainder 3, 8 and 9 are more than
+  useless, and probably not Euclid's, but additions of later editors, of
+  whom Theon of Alexandria was the most prominent. Defs. 10 and 11
+  belong rather to the sixth book, whilst all the others are merely
+  nominal. The really important ones are 4, 5, 6 and 7.
+
+  S 48. To define a magnitude is not attempted by Euclid. The first two
+  definitions state what is meant by a "part," that is, a submultiple or
+  measure, and by a "multiple" of a given magnitude. The meaning of Def.
+  4 is that two given quantities can have a ratio to one another only in
+  case that they are comparable as to their magnitude, that is, if they
+  are of the same kind.
+
+  Def. 3, which is probably due to Theon, professes to define a ratio,
+  but is as meaningless as it is uncalled for, for all that is wanted is
+  given in Defs. 5 and 7.
+
+  In Def. 5 it is explained what is meant by saying that two magnitudes
+  have the same ratio to one another as two other magnitudes, and in
+  Def. 7 what we have to understand by a greater or a less ratio. The
+  6th definition is only nominal, explaining the meaning of the word
+  _proportional_.
+
+  Euclid represents magnitudes by lines, and often denotes them either
+  by single letters or, like lines, by two letters. We shall use only
+  single letters for the purpose. If a and b denote two magnitudes of
+  the same kind, their ratio will be denoted by a : b; if c and d are
+  two other magnitudes of the same kind, but possibly of a different
+  kind from a and b, then if c and d have the same ratio to one another
+  as a and b, this will be expressed by writing--
+
+    a : b :: c : d.
+
+  Further, if m is a (whole) number, ma shall denote the multiple of a
+  which is obtained by taking it m times.
+
+  S 49. The whole theory of ratios is based on Def. 5.
+
+  Def. 5. _The first of four magnitudes is said to have the same ratio
+  to the second that the third has to the fourth when, any equimultiples
+  whatever of the first and the third being taken, and any equimultiples
+  whatever of the second and the fourth, if the multiple of the first be
+  less than that of the second, the multiple of the third is also less
+  than that of the fourth; and if the multiple of the first is equal to
+  that of the second, the multiple of the third is also equal to that of
+  the fourth; and if the multiple of the first is greater than that of
+  the second, the multiple of the third is also greater than that of the
+  fourth._
+
+  It will be well to show at once in an example how this definition can
+  be used, by proving the first part of the first proposition in the
+  sixth book. _Triangles of the same altitude are to one another as
+  their bases_, or if a and b are the bases, and [alpha] and [beta] the
+  areas, of two triangles which have the same altitude, then a : b ::
+  [alpha] : [beta].
+
+  To prove this, we have, according to Definition 5, to show--
+
+    if ma > nb, then m[alpha] > n[beta],
+    if ma = nb, then m[alpha] = n[beta],
+    if ma < nb, then m[alpha] < n[beta].
+
+  That this is true is in our case easily seen. We may suppose that the
+  triangles have a common vertex, and their bases in the same line. We
+  set off the base a along the line containing the bases m times; we
+  then join the different parts of division to the vertex, and get m
+  triangles all equal to [alpha]. The triangle on ma as base equals,
+  therefore, m[alpha]. If we proceed in the same manner with the base b,
+  setting it off n times, we find that the area of the triangle on the
+  base nb equals n[beta], the vertex of all triangles being the same.
+  But if two triangles have the same altitude, then their areas are
+  equal if the bases are equal; hence m[alpha] = n[beta] if ma = nb, and
+  if their bases are unequal, then that has the greater area which is on
+  the greater base; in other words, m[alpha] is greater than, equal to,
+  or less than n[beta], according as ma is greater than, equal to, or
+  less than nb, which was to be proved.
+
+  S 50. It will be seen that even in this example it does not become
+  evident what a ratio really is. It is still an open question whether
+  ratios are magnitudes which we can compare. We do not know whether the
+  ratio of two lines is a magnitude of the same kind as the ratio of two
+  areas. Though we might say that Def. 5 defines _equal _ratios, still
+  we do not know whether they are equal in the sense of the axiom, that
+  two things which are equal to a third are equal to one another. That
+  this is the case requires a proof, and until this proof is given we
+  shall use the :: instead of the sign = , which, however, we shall
+  afterwards introduce.
+
+  As soon as it has been established that all ratios are like
+  magnitudes, it becomes easy to show that, in some cases at least, they
+  are numbers. This step was never made by Greek mathematicians. They
+  distinguished always most carefully between continuous magnitudes and
+  the discrete series of numbers. In modern times it has become the
+  custom to ignore this difference.
+
+  If, in determining the ratio of two lines, a common measure can be
+  found, which is contained m times in the first, and n times in the
+  second, then the ratio of the two lines equals the ratio of the two
+  numbers m : n. This is shown by Euclid in Prop. 5, X. But the ratio of
+  two numbers is, as a rule, a fraction, and the Greeks did not, as we
+  do, consider fractions as numbers. Far less had they any notion of
+  introducing irrational numbers, which are neither whole nor
+  fractional, as we are obliged to do if we wish to say that all ratios
+  are numbers. The incommensurable numbers which are thus introduced as
+  ratios of incommensurable quantities are nowadays as familiar to us as
+  fractions; but a proof is generally omitted that we may apply to them
+  the rules which have been established for rational numbers only.
+  Euclid's treatment of ratios avoids this difficulty. His definitions
+  hold for commensurable as well as for incommensurable quantities. Even
+  the notion of incommensurable quantities is avoided in Book V. But he
+  proves that the more elementary rules of algebra hold for ratios. We
+  shall state all his propositions in that algebraical form to which we
+  are now accustomed. This may, of course, be done without changing the
+  character of Euclid's method.
+
+  S. 51. Using the notation explained above we express the first
+  propositions as follows:--
+
+  Prop. 1. If   a = ma', b = mb', c = mc',
+  then          a + b + c = m(a' + b' + c').
+
+  Prop. 2. If   a = mb, and c = md,
+                e = nb, and f = nd,
+
+  then a + e is the same multiple of b as c + f is of d, viz.:--
+
+    a + e = (m + n)b, and c + f = (m + n)d.
+
+  Prop. 3. If a = mb, c = md, then is na the same multiple of b that nc
+  is of d, viz. na = nmb, nc = nmd.
+
+  Prop. 4. If   a : b :: c : d,
+  then          ma : nb :: mc : nd.
+
+  Prop. 5. If   a = mb, and c = md,
+  then          a - c = m(b - d).
+
+  Prop. 6. If   a = mb, c = md,
+
+  then are a - nb and c - nd either equal to, or equimultiples of, b and
+  d, viz. a - nb = (m - n)b and c - nd = (m - n)d, where m - n may be
+  unity.
+
+  All these propositions relate to _equimultiples_. Now follow
+  propositions about ratios which are compared as to their magnitude.
+
+  S 52. Prop. 7. If a = b, then a : c :: b : c and c : a :: c : b.
+
+  The proof is simply this. As a = b we know that ma = mb; therefore
+
+    if   ma > nc, then mb > nc,
+    if   ma = nc, then mb = nc,
+    if   ma < nc, then mb < nc,
+
+  therefore the first proportion holds by Definition 5.
+
+  Prop. 8. If a > b, then a : c > b : c,
+  and                     c : a < c : b.
+
+  The proof depends on Definition 7.
+
+  Prop. 9 (converse to Prop. 7). If
+             a : c :: b : c,
+  or if      c : a :: c : b, then a = b.
+
+  Prop. 10 (converse to Prop. 8). If
+             a : c > b : c, then a > b,
+  and if     c : a < c : b, then a < b.
+
+  Prop. 11. If  a : b :: c : d,
+  and           a : b :: e : f,
+  then          c : d :: e : f.
+
+  In words, _if too ratios are equal to a third, they are equal to one
+  another_. After these propositions have been proved, we have a right
+  to consider a ratio as a _magnitude_, for only now can we consider a
+  ratio as something for which the axiom about magnitudes holds: things
+  which are equal to a third are equal to one another.
+
+  We shall indicate this by writing in future the sign = instead of ::.
+  The remaining propositions, which explain themselves, may then be
+  stated as follows:
+
+  S 53. Prop. 12. If  a : b = c : d = e : f,
+  then                a + c + e : b + d + f = a : b.
+
+  Prop. 13. If    a : b = c : d and c : d > e : f,
+  then            a : b > e : f.
+
+  Prop. 14. If    a : b = c : d, and a > c, then b > d.
+
+  Prop. 15. Magnitudes have the same ratio to one another that their
+  equimultiples have--
+
+  ma : mb = a : b.
+
+  Prop. 16. If a, b, c, d are magnitudes of the same kind, and if
+                  a : b = c : d,
+  then            a : c = b : d.
+
+  Prop. 17. If    a + b : b = c + d : d,
+  then            a : b = c : d.
+
+  Prop. 18 (converse to 17). If
+                  a : b = c : d
+  then            a + b : b = c + d : d.
+
+  Prop. 19. If a, b, c, d are quantities of the same kind, and if
+                  a : b = c : d,
+  then            a - c : b - d = a : b.
+
+  S 54. Prop. 20. _If there be three magnitudes, and another three,
+  which have the same ratio, taken two and two, then if the first be
+  greater than the third, the fourth shall be greater than the sixth:
+  and if equal, equal; and if less, less._
+
+  If we understand by
+
+    a : b : c : d : e : ... = a' : b' : c' : d' : e' : ...
+
+  that the ratio of any two consecutive magnitudes on the first side
+  equals that of the corresponding magnitudes on the second side, we may
+  write this theorem in symbols, thus:--
+
+  If a, b, c be quantities of one, and d, e, f magnitudes of the same or
+  any other kind, such that
+
+           a : b : c = d : e : f,
+  and if   a > c, then d > f,
+  but if   a = c, then d = f,
+  and if   a < c, then d < f.
+
+  Prop. 21. If  a : b = e : f and b : c = d : e,
+  or if         a : b : c = 1/f : 1/e : 1/d,
+  and if        a > c, then d > f,
+  but if        a = c, then d = f,
+  and if        a < c, then d < f.
+
+  By aid of these two propositions the following two are proved.
+
+  S 55. Prop. 22. _If there be any number of magnitudes, and as many
+  others, which have the same ratio, taken two and two in order, the
+  first shall have to the last of the first magnitudes the same ratio
+  which the first of the others has to the last._
+
+  We may state it more generally, thus:
+
+  If   a : b : c : d : e: ... = a' : b' : c' : d' : e' : ... ,
+
+  then not only have two consecutive, but any two magnitudes on the
+  first side, the same ratio as the corresponding magnitudes on the
+  other. For instance--
+
+    a : c = a' : c'; b : e = b' : e', &c.
+
+  Prop. 23 we state only in symbols, viz.:--
+
+  If  a : b : c : d : e : ... = 1/a' : 1/b' : 1/c' : 1/d' : 1/e' ...,
+
+  then  a : c = c' : a',
+        b : e = e' : b',
+
+  and so on.
+
+  Prop. 24 comes to this: If a : b = c : d and e : b = f : d, then
+
+    a + e : b = c + f : d.
+
+  Some of the proportions which are considered in the above propositions
+  have special names. These we have omitted, as being of no use, since
+  algebra has enabled us to bring the different operations contained in
+  the propositions under a common point of view.
+
+  S 56. The last proposition in the fifth book is of a different
+  character.
+
+  Prop. 25. _If four magnitudes of the same kind be proportional, the
+  greatest and least of them together shall be greater than the other
+  two together._ In symbols--
+
+  If a, b, c, d be magnitudes of the same kind, and if a : b = c : d,
+  and if a is the greatest, hence d the least, then a + d > b + c.
+
+  S 57. We return once again to the question. What is a ratio? We have
+  seen that we may treat ratios as magnitudes, and that all ratios are
+  magnitudes of the same kind, for we may compare any two as to their
+  magnitude. It will presently be shown that ratios of lines may be
+  considered as _quotients_ of lines, so that a ratio appears as answer
+  to the question, How often is one line contained in another? But the
+  answer to this question is given by a number, at least in some cases,
+  and in all cases if we admit incommensurable numbers. Considered from
+  this point of view, we may say the fifth book of the _Elements_ shows
+  that some of the simpler algebraical operations hold for
+  incommensurable numbers. In the ordinary algebraical treatment of
+  numbers this proof is altogether omitted, or given by a process of
+  limits which does not seem to be natural to the subject.
+
+
+  BOOK VI.
+
+  S 58. The sixth book contains the theory of similar figures. After a
+  few definitions explaining terms, the first proposition gives the
+  first application of the theory of proportion.
+
+  Prop. 1. _Triangles and parallelograms of the same altitude are to one
+  another as their bases._
+
+  The proof has already been considered in S 49.
+
+  From this follows easily the important theorem
+
+  Prop. 2. _If a straight line be drawn parallel to one of the sides of
+  a triangle it shall cut the other sides, or those sides produced,
+  proportionally; and if the sides or the sides produced be cut
+  proportionally, the straight line which joins the points of section
+  shall be parallel to the remaining side of the triangle._
+
+  S 59. The next proposition, together with one added by Simson as Prop.
+  A, may be expressed more conveniently if we introduce a modern
+  phraseology, viz. if in a line AB we assume a point C between A and B,
+  we shall say that C divides AB internally in the ratio AC : CB; but if
+  C be taken in the line AB produced, we shall say that AB is divided
+  externally in the ratio AC : CB.
+
+  The two propositions then come to this:
+
+  Prop. 3. _The bisector of an angle in a triangle divides the opposite
+  side internally in a ratio equal to the ratio of the two sides
+  including that angle;_ and conversely, _if a line through the vertex
+  of a triangle divide the base internally in the ratio of the two other
+  sides, then that line bisects the angle at the vertex_.
+
+  Simson's Prop. A. _The line which bisects an exterior angle of a
+  triangle divides the opposite side externally in the ratio of the
+  other sides;_ and conversely, _if a line through the vertex of a
+  triangle divide the base externally in the ratio of the sides, then it
+  bisects an exterior angle at the vertex of the triangle_.
+
+  If we combine both we have--
+
+  _The two lines which bisect the interior and exterior angles at one
+  vertex of a triangle divide the opposite side internally and
+  externally in the same ratio, viz. in the ratio of the other two
+  sides._
+
+  S 60. The next four propositions contain the theory of similar
+  triangles, of which four cases are considered. They may be stated
+  together.
+
+  _Two triangles are similar_,--
+
+  1. (Prop. 4). _If the triangles are equiangular:_
+
+  2. (Prop. 5). _If the sides of the one are proportional to those of
+  the other_;
+
+  3. (Prop. 6). _If two sides in one are proportional to two sides in
+  the other, and if the angles contained by these sides are equal_;
+
+  4. (Prop. 7). _If two sides in one are proportional to two sides in
+  the other, if the angles opposite homologous sides are equal, and if
+  the angles opposite the other homologous sides are both acute, both
+  right or both obtuse; homologous sides being in each case those which
+  are opposite equal angles_.
+
+  An important application of these theorems is at once made to a
+  right-angled triangle, viz.:--
+
+  Prop. 8. _In a right-angled triangle, if a perpendicular be drawn from
+  the right angle to the base, the triangles on each side of it are
+  similar to the whole triangle, and to one another_.
+
+  _Corollary._--From this it is manifest that the perpendicular drawn
+  from the right angle of a right-angled triangle to the base is a mean
+  proportional between the segments of the base, and also that each of
+  the sides is a mean proportional between the base and the segment of
+  the base adjacent to that side.
+
+  S 61. There follow four propositions containing problems, in language
+  slightly different from Euclid's, viz.:--
+
+  Prop. 9. _To divide a straight line into a given number of equal
+  parts_.
+
+  Prop. 10. _To divide a straight line in a given ratio_.
+
+  Prop. 11. _To find a third proportional to two given straight lines_.
+
+  Prop. 12. _To find a fourth proportional to three given straight
+  lines_.
+
+  Prop. 13. _To find a mean proportional between two given straight
+  lines_.
+
+  The last three may be written as equations with one unknown
+  quantity--viz. if we call the given straight lines a, b, c, and the
+  required line x, we have to find a line x so that
+
+  Prop. 11.  a : b = b : x;
+
+  Prop. 12.  a : b = c : x;
+
+  Prop. 13.  a : x = x : b.
+
+  We shall see presently how these may be written without the signs of
+  ratios.
+
+  S 62. Euclid considers next proportions connected with parallelograms
+  and triangles which are equal in area.
+
+  Prop. 14. _Equal parallelograms which have one angle of the one equal
+  to one angle of the other have their sides about the equal angles
+  reciprocally proportional; and parallelograms which have one angle of
+  the one equal to one angle of the other, and their sides about the
+  equal angles reciprocally proportional, are equal to one another_.
+
+  Prop. 15. _Equal triangles which have one angle of the one equal to
+  one angle of the other, have their sides about the equal angles
+  reciprocally proportional; and triangles which have one angle of the
+  one equal to one angle of the other, and their sides about the equal
+  angles reciprocally proportional, are equal to one another_.
+
+  [Illustration]
+
+  The latter proposition is really the same as the former, for if, as in
+  the accompanying diagram, in the figure belonging to the former the
+  two equal parallelograms AB and BC be bisected by the lines DF and EG,
+  and if EF be drawn, we get the figure belonging to the latter.
+
+  It is worth noticing that the lines FE and DG are parallel. We may
+  state therefore the theorem--
+
+  _If two triangles are equal in area, and have one angle in the one
+  vertically opposite to one angle in the other, then the two straight
+  lines which join the remaining two vertices of the one to those of the
+  other triangle are parallel_.
+
+  S 63. A most important theorem is
+
+  _Prop. 16. If four straight lines be proportionals, the rectangle
+  contained by the extremes is equal to the rectangle contained by the
+  means; and if the rectangle contained by the extremes be equal to the
+  rectangle contained by the means, the four straight lines are
+  proportionals_.
+
+  In symbols, if a, b, c, d are the four lines, and
+  if                  a : b = c : d,
+  then                ad = bc;
+  and conversely, if  ad = bc,
+  then                a : b = c : d,
+
+  where ad and bc denote (as in S 20), the areas of the rectangles
+  contained by a and d and by b and c respectively.
+
+  This allows us to transform every proportion between four lines into
+  an equation between two products.
+
+  It shows further that the operation of forming a product of two lines,
+  and the operation of forming their ratio are each the inverse of the
+  other.
+
+  If we now define a quotient a/b of two lines as the _number_ which
+  multiplied into b gives a, so that
+
+    a
+    -- b = a,
+    b
+
+  we see that from the equality of two quotients
+
+    a    c
+    -- = --
+    b    d
+
+  follows, if we multiply both sides by bd,
+
+    a        c
+    -- b.d = -- d.b,
+    b        d
+
+    ad = cb.
+
+  But from this it follows, according to the last theorem, that
+
+    a : b = c : d.
+
+  Hence we conclude that the quotient a/b and the ratio a : b are
+  different forms of the same magnitude, only with this important
+  difference that the quotient a/b would have a meaning only if a and b
+  have a common measure, until we introduce incommensurable numbers,
+  while the ratio a : b has always a meaning, and thus gives rise to the
+  introduction of incommensurable numbers.
+
+  Thus it is really the theory of ratios in the fifth book which enables
+  us to extend the geometrical calculus given before in connexion with
+  Book II. It will also be seen that if we write the ratios in Book V.
+  as quotients, or rather as fractions, then most of the theorems state
+  properties of quotients or of fractions.
+
+  S 64. Prop. 17. _If three straight lines are proportional the
+  rectangle contained by the extremes is equal to the square on the
+  mean;_ and conversely, is only a special case of 16. After the
+  problem, Prop. 18, _On a given straight line to describe a rectilineal
+  figure similar and similarly situated to a given rectilineal figure_,
+  there follows another fundamental theorem:
+
+  Prop. 19. _Similar triangles are to one another in the duplicate ratio
+  of their homologous sides._ In other words, the areas of similar
+  triangles are to one another as the squares on homologous sides. This
+  is generalized in:
+
+  Prop. 20. _Similar polygons may be divided into the same number of
+  similar triangles, having the same ratio to one another that the
+  polygons have; and the polygons are to one another in the duplicate
+  ratio of their homologous sides._
+
+  S 65. Prop. 21. _Rectilineal figures which are similar to the same
+  rectilineal figure are also similar to each other_, is an immediate
+  consequence of the definition of similar figures. As similar figures
+  may be said to be equal in "shape" but not in "size," we may state it
+  also thus:
+
+  "Figures which are equal in shape to a third are equal in shape to
+  each other."
+
+  Prop. 22. _If four straight lines be proportionals, the similar
+  rectilineal figures similarly described on them shall also be
+  proportionals; and if the similar rectilineal figures similarly
+  described on four straight lines be proportionals, those straight
+  lines shall be proportionals._
+
+  This is essentially the same as the following:--
+
+  _If_    a : b = c : d,
+  _then_  a^2 : b^2 = c^2 : d^2.
+
+  S 66. Now follows a proposition which has been much discussed with
+  regard to Euclid's exact meaning in saying that a ratio is
+  _compounded_ of two other ratios, viz.:
+
+  Prop. 23. _Parallelograms which are equiangular to one another, have
+  to one another the ratio which is compounded of the ratios of their
+  sides._
+
+  The proof of the proposition makes its meaning clear. In symbols the
+  ratio a : c is compounded of the two ratios a : b and b : c, and if a
+  : b = a' : b', b : c = b" : c", then a : c is compounded of a' : b'
+  and b" : c".
+
+  If we consider the ratios as numbers, we may say that the one ratio is
+  the product of those of which it is compounded, or in symbols,
+
+    a    a    b    a'   b"     a    a'     b    b"
+    -- = -- . -- = -- . --, if -- = -- and -- = --.
+    c    b    c    b'   c"     b    b'     c    c"
+
+  The theorem in Prop. 23 is the foundation of all mensuration of areas.
+  From it we see at once that two rectangles have the ratio of their
+  areas compounded of the ratios of their sides.
+
+  If A is the area of a rectangle contained by a and b, and B that of a
+  rectangle contained by c and d, so that A = ab, B = cd, then A : B =
+  ab : cd, and this is, the theorem says, compounded of the ratios a : c
+  and b : d. In forms of quotients,
+
+    a    b    ab
+    -- . -- = --.
+    c    d    cd
+
+  This shows how to multiply quotients in our geometrical calculus.
+
+  Further, _Two triangles have the ratios of their areas compounded of
+  the ratios of their bases and their altitude._ For a triangle is equal
+  in area to half a parallelogram which has the same base and the same
+  altitude.
+
+  S 67. To bring these theorems to the form in which they are usually
+  given, we assume a straight line u as our unit of length (generally an
+  inch, a foot, a mile, &c.), and determine the number [alpha] which
+  expresses how often u is contained in a line a, so that [alpha]
+  denotes the ratio a : u whether commensurable or not, and that a =
+  [alpha]u. We call this number [alpha] the numerical value of a. If in
+  the same manner [beta] be the numerical value of a line b we have
+
+    a : b = [alpha] : [beta];
+
+  in words: _The ratio of two lines (and of two like quantities in
+  general) is equal to that of their numerical values._
+
+  This is easily proved by observing that a = [alpha]u, b = [beta]u,
+  therefore a : b = [alpha]u : [beta]u, and this may without difficulty
+  be shown to equal [alpha] : [beta].
+
+  If now a, b be base and altitude of one, a', b' those of another
+  parallelogram, [alpha], [beta] and [alpha]', [beta]' their numerical
+  values respectively, and A, A' their areas, then
+
+    A    a    b    [alpha]    [beta]    [alpha][beta]
+    -- = -- . -- = -------- . ------ = ---------------.
+    A'   a'   b'   [alpha]'   [beta]'  [alpha]'[beta]'
+
+  In words: _The areas of two parallelograms are to each other as the
+  products of the numerical values of their bases and altitudes._
+
+  If especially the second parallelogram is the unit square, i.e. a
+  square on the unit of length, then [alpha]' = [beta]' = 1, A' = u^2,
+  and we have
+
+    A
+    -- = [alpha][beta] or A = [alpha][beta] . u^2.
+    A'
+
+  This gives the theorem: The number of unit squares contained in a
+  parallelogram equals the product of the numerical values of base and
+  altitude, and similarly the number of unit squares contained in a
+  triangle equals half the product of the numerical values of base and
+  altitude.
+
+  This is often stated by saying that the area of a parallelogram is
+  equal to the product of the base and the altitude, meaning by this
+  product the product of the numerical values, and not the product as
+  defined above in S 20.
+
+  S 68. Propositions 24 and 26 relate to parallelograms about diagonals,
+  such as are considered in Book I., 43. They are--
+
+  Prop. 24. _Parallelograms about the diameter of any parallelogram are
+  similar to the whole parallelogram and to one another_; and its
+  converse (Prop. 26), _If two similar parallelograms have a common
+  angle, and be similarly situated, they are about the same diameter._
+
+  Between these is inserted a problem.
+
+  Prop. 25. _To describe a rectilineal figure which shall be similar to
+  one given rectilinear figure, and equal to another given rectilineal
+  figure_.
+
+  S 69. Prop. 27 contains a theorem relating to the theory of maxima and
+  minima. We may state it thus:
+
+  Prop. 27. _If a parallelogram be divided into two by a straight line
+  cutting the base, and if on half the base another parallelogram be
+  constructed similar to one of those parts, then this third
+  parallelogram is greater than the other part._
+
+  Of far greater interest than this general theorem is a special case of
+  it, where the parallelograms are changed into rectangles, and where
+  one of the parts into which the parallelogram is divided is made a
+  square; for then the theorem changes into one which is easily
+  recognized to be identical with the following:--
+
+  _Of all rectangles which have the same perimeter the square has the
+  greatest area._
+
+  This may also be stated thus:--
+
+  _Of all rectangles which have the same area the square has the least
+  perimeter._
+
+  S 70. The next three propositions contain problems which may be said
+  to be solutions of quadratic equations. The first two are, like the
+  last, involved in somewhat obscure language. We transcribe them as
+  follows:
+
+  _Problem_.--To describe on a given base a parallelogram, and to divide
+  it either internally (Prop. 28) or externally (Prop. 29) from a point
+  on the base into two parallelograms, of which the one has a given size
+  (is equal in area to a given figure), whilst the other has a given
+  shape (is similar to a given parallelogram).
+
+  If we express this again in symbols, calling the given base a, the one
+  part x, and the altitude y, we have to determine x and y in the first
+  case from the equations
+
+    (a - x)y = k^2,
+
+    x    p
+    -- = --,
+    y    q
+
+  k^2 being the given size of the first, and p and q the base and
+  altitude of the parallelogram which determine the shape of the second
+  of the required parallelograms.
+
+  If we substitute the value of y, we get
+
+               pk^2
+    (a - x)x = ----,
+                q
+
+  or,
+
+    ax - x^2 = b^2,
+
+  where a and b are known quantities, taking b^2 = pk^2/q.
+
+  The second case (Prop. 29) gives rise, in the same manner, to the
+  quadratic
+
+    ax + x^2 = b^2.
+
+  The next problem--
+
+  Prop. 30. _To cut a given straight line in extreme and mean ratio_,
+  leads to the equation
+
+    ax + x^2 = a^2.
+
+  This is, therefore, only a special case of the last, and is, besides,
+  an old acquaintance, being essentially the same problem as that
+  proposed in II. 11.
+
+  Prop. 30 may therefore be solved in two ways, either by aid of Prop.
+  29 or by aid of II. 11. Euclid gives both solutions.
+
+  S 71. Prop. 31 (Theorem). _In any right-angled triangle, any
+  rectilineal figure described on the side subtending the right angle is
+  equal to the similar and similarly-described figures on the sides
+  containing the right angle_,--is a pretty generalization of the
+  theorem of Pythagoras (I. 47).
+
+  Leaving out the next proposition, which is of little interest, we come
+  to the last in this book.
+
+  Prop. 33. _In equal circles angles, whether at the centres or the
+  circumferences, have the same ratio which the arcs on which they stand
+  have to one another; so also have the sectors_.
+
+  Of this, the part relating to angles at the centre is of special
+  importance; it enables us to measure angles by arcs.
+
+  With this closes that part of the _Elements_ which is devoted to the
+  study of figures in a plane.
+
+
+  BOOK XI.
+
+  S 72. In this book figures are considered which are not confined to a
+  plane, viz. first relations between lines and planes in space, and
+  afterwards properties of solids.
+
+  Of new definitions we mention those which relate to the
+  perpendicularity and the inclination of lines and planes.
+
+  Def. 3. _A straight line is perpendicular, or at right angles, to a
+  plane when it makes right angles with every straight line meeting it
+  in that plane_.
+
+  The definition of perpendicular planes (Def. 4) offers no difficulty.
+  Euclid defines the inclination of lines to planes and of planes to
+  planes (Defs. 5 and 6) by aid of plane angles, included by straight
+  lines, with which we have been made familiar in the first books.
+
+  The other important definitions are those of parallel planes, which
+  never meet (Def. 8), and of solid angles formed by three or more
+  planes meeting in a point (Def. 9).
+
+  To these we add the definition of a line parallel to a plane as a line
+  which does not meet the plane.
+
+  S 73. Before we investigate the contents of Book XI., it will be well
+  to recapitulate shortly what we know of planes and lines from the
+  definitions and axioms of the first book. There a plane has been
+  defined as a surface which has the property that every straight line
+  which joins two points in it lies altogether in it. This is equivalent
+  to saying that a straight line which has two points in a plane has all
+  points in the plane. Hence, a straight line which does not lie in the
+  plane cannot have more than one point in common with the plane. This
+  is virtually the same as Euclid's Prop. 1, viz.:--
+
+  Prop. 1. _One part of a straight line cannot be in a plane and another
+  part without it_.
+
+  It also follows, as was pointed out in S 3, in discussing the
+  definitions of Book I., that a plane is determined already by one
+  straight line and a point without it, viz. if all lines be drawn
+  through the point, and cutting the line, they will form a plane.
+
+  This may be stated thus:--
+
+  _A plane is determined_--
+
+  1st, _By a straight line and a point which does not lie on it;_
+
+  2nd, _By three points which do not lie in a straight line_; for if two
+  of these points be joined by a straight line we have case 1;
+
+  3rd, _By two intersecting straight lines_; for the point of
+  intersection and two other points, one in each line, give case 2;
+
+  4th, _By two parallel lines_ (Def. 35, I.).
+
+  The third case of this theorem is Euclid's
+
+  Prop. 2. _Two straight lines which cut one another are in one plane,
+  and three straight lines which meet one another are in one plane_.
+
+  And the fourth is Euclid's
+
+  Prop. 7. _If two straight lines be parallel, the straight line drawn
+  from any point in one to any point in the other is in the same plane
+  with the parallels_. From the definition of a plane further follows
+
+  Prop. 3. _If two planes cut one another, their common section is a
+  straight line_.
+
+  S 74. Whilst these propositions are virtually contained in the
+  definition of a plane, the next gives us a new and fundamental
+  property of space, showing at the same time that it is possible to
+  have a straight line perpendicular to a plane, according to Def. 3. It
+  states--
+
+  Prop. 4. _If a straight line is perpendicular to two straight lines in
+  a plane which it meets, then it is perpendicular to all lines in the
+  plane which it meets, and hence it is perpendicular to the plane_.
+
+  Def. 3 may be stated thus: If a straight line is perpendicular to a
+  plane, then it is perpendicular to every line in the plane which it
+  meets. The converse to this would be
+
+  _All straight lines which meet a given straight line in the same
+  point, and are perpendicular to it, lie in a plane which is
+  perpendicular to that line_.
+
+  This Euclid states thus:
+
+  Prop. 5. _If three straight lines meet all at one point, and a
+  straight line stands at right angles to each of them at that point,
+  the three straight lines shall be in one and the same plane_.
+
+  S 75. There follow theorems relating to the theory of parallel lines
+  in space, viz.:--
+
+  Prop. 6. _Any two lines which are perpendicular to the same plane are
+  parallel to each other;_ and conversely
+
+  Prop. 8. _If of two parallel straight lines one is perpendicular to a
+  plane, the other is so also._
+
+  Prop. 7. _If two straight lines are parallel, the straight line which
+  joins any point in one to any point in the other is in the same plane
+  as the parallels._ (See above, S 73.)
+
+  Prop. 9. _Two straight lines which are each of them parallel to the
+  same straight line, and not in the same plane with it, are parallel to
+  one another;_ where the words, "and not in the same plane with it,"
+  may be omitted, for they exclude the case of three parallels in a
+  plane, which has been proved before; and
+
+  Prop. 10. _If two angles in different planes have the two limits of
+  the one parallel to those of the other, then the angles are equal._
+  That their planes are parallel is shown later on in Prop. 15.
+
+  This theorem is not necessarily true, for the angles in question may
+  be supplementary; but then the one angle will be equal to that which
+  is adjacent and supplementary to the other, and this latter angle will
+  also have its limits parallel to those of the first.
+
+  From this theorem it follows that if we take any two straight lines in
+  space which do not meet, and if we draw through any point P in space
+  two lines parallel to them, then the angle included by these lines
+  will always be the same, whatever the position of the point P may be.
+  This angle has in modern times been called the angle between the given
+  lines:--
+
+  _By the angles between two not intersecting lines we understand the
+  angles which two intersecting lines include that are parallel
+  respectively to the two given lines._
+
+  S 76. It is now possible to solve the following two problems:--
+
+  _To draw a straight line perpendicular to a given plane from a given
+  point which lies_
+
+  1. _Not in the plane_ (Prop. 11).
+
+  2. _In the plane_ (Prop. 12).
+
+  The second case is easily reduced to the first--viz. if by aid of the
+  first we have drawn any perpendicular to the plane from some point
+  without it, we need only draw through the given point in the plane a
+  line parallel to it, in order to have the required perpendicular
+  given. The solution of the first part is of interest in itself. It
+  depends upon a construction which may be expressed as a theorem.
+
+  _If from a point A without a plane a perpendicular AB be drawn to the
+  plane, and if from the foot B of this perpendicular another
+  perpendicular BC be drawn to any straight line in the plane, then the
+  straight line joining A to the foot C of this second perpendicular
+  will also be perpendicular to the line in the plane._
+
+  The theory of perpendiculars to a plane is concluded by the theorem--
+
+  Prop. 13. _Through any point in space, whether in or without a plane,
+  only one straight line can be drawn perpendicular to the plane._
+
+  S 77. The next four propositions treat of parallel planes. It is shown
+  _that planes which have a common perpendicular are parallel_ (Prop.
+  14); _that two planes are parallel if two intersecting straight lines
+  in the one are parallel respectively to two straight lines in the
+  other plane_ (Prop. 15); _that parallel planes are cut by any plane in
+  parallel straight lines_ (Prop. 16); and lastly, _that any two
+  straight lines are cut proportionally by a series of parallel planes_
+  (Prop. 17).
+
+  This theory is made more complete by adding the following theorems,
+  which are easy deductions from the last: _Two parallel planes have
+  common perpendiculars_ (converse to 14); and _Two planes which are
+  parallel to a third plane are parallel to each other._
+
+  It will be noted that Prop. 15 at once allows of the solution of the
+  problem: "Through a given point to draw a plane parallel to a given
+  plane." And it is also easily proved that this problem allows always
+  of one, and only of one, solution.
+
+  S 78. We come now to planes which are perpendicular to one another.
+  Two theorems relate to them.
+
+  Prop. 18. _If a straight line be at right angles to a plane, every
+  plane which passes through it shall be at right angles to that plane._
+
+  Prop. 19. _If two planes which cut one another be each of them
+  perpendicular to a third plane, their common section shall be
+  perpendicular to the same plane._
+
+  S 79. If three planes pass through a common point, and if they bound
+  each other, a solid angle of three faces, or a _trihedral_ angle, is
+  formed, and similarly by more planes a solid angle of more faces, or a
+  _polyhedral_ angle. These have many properties which are quite
+  analogous to those of triangles and polygons in a plane. Euclid states
+  some, viz.:--
+
+  Prop. 20. _If a solid angle be contained by three plane angles, any
+  two of them are together greater than the third._
+
+  But the next--
+
+  Prop. 21. _Every solid angle is contained by plane angles, which are
+  together less than four right angles_--has no analogous theorem in the
+  plane.
+
+  We may mention, however, that the theorems about triangles contained
+  in the propositions of Book I., which do not depend upon the theory of
+  parallels (that is all up to Prop. 27), have their corresponding
+  theorems about trihedral angles. The latter are formed, if for "side
+  of a triangle" we write "plane angle" or "face" of trihedral angle,
+  and for "angle of triangle" we substitute "angle between two faces"
+  where the planes containing the solid angle are called its _faces_. We
+  get, for instance, from I. 4, the theorem, _If two trihedral angles
+  have the angles of two faces in the one equal to the angles of two
+  faces in the other, and have likewise the angles included by these
+  faces equal, then the angles in the remaining faces are equal, and the
+  angles between the other faces are equal each to each, viz. those
+  which are opposite equal faces._ The solid angles themselves are not
+  necessarily equal, for they may be only symmetrical like the right
+  hand and the left.
+
+  The connexion indicated between triangles and trihedral angles will
+  also be recognized in
+
+  Prop. 22. _If every two of three plane angles be greater than the
+  third, and if the straight lines which contain them be all equal, a
+  triangle may be made of the straight lines that join the extremities
+  of those equal straight lines._
+
+  And Prop. 23 solves the problem, _To construct a trihedral angle
+  having the angles of its faces equal to three given plane angles, any
+  two of them being greater than the third._ It is, of course, analogous
+  to the problem of constructing a triangle having its sides of given
+  length.
+
+  Two other theorems of this kind are added by Simson in his edition of
+  Euclid's _Elements_.
+
+  S 80. These are the principal properties of lines and planes in space,
+  but before we go on to their applications it will be well to define
+  the word _distance_. In geometry distance means always "shortest
+  distance"; viz. the distance of a point from a straight line, or from
+  a plane, is the length of the perpendicular from the point to the line
+  or plane. The distance between two non-intersecting lines is the
+  length of their common perpendicular, there being but one. The
+  distance between two parallel lines or between two parallel planes is
+  the length of the common perpendicular between the lines or the
+  planes.
+
+  S 81. _Parallelepipeds_.--The rest of the book is devoted to the study
+  of the parallelepiped. In Prop. 24 the possibility of such a solid is
+  proved, viz.:--
+
+  Prop. 24. _If a solid be contained by six planes two and two of which
+  are parallel, the opposite planes are similar and equal
+  parallelograms._
+
+  Euclid calls this solid henceforth a parallelepiped, though he never
+  defines the word. Either face of it may be taken as _base_, and its
+  distance from the opposite face as _altitude_.
+
+  Prop. 25. _If a solid parallelepiped be cut by a plane parallel to two
+  of its opposite planes, it divides the whole into two solids, the base
+  of one of which shall be to the base of the other as the one solid is
+  to the other_.
+
+  This theorem corresponds to the theorem (VI. 1) that parallelograms
+  between the same parallels are to one another as their bases. A
+  similar analogy is to be observed among a number of the remaining
+  propositions.
+
+  S 82. After solving a few problems we come to
+
+  Prop. 28. _If a solid parallelepiped be cut by a plane passing through
+  the diagonals of two of the opposite planes, it shall be cut in two
+  equal parts._
+
+  In the proof of this, as of several other propositions, Euclid
+  neglects the difference between solids which are symmetrical like the
+  right hand and the left.
+
+  Prop. 31. _Solid parallelepipeds, which are upon equal bases, and of
+  the same altitude, are equal to one another._
+
+  Props. 29 and 30 contain special cases of this theorem leading up to
+  the proof of the general theorem.
+
+  As consequences of this fundamental theorem we get
+
+  Prop. 32. _Solid parallelepipeds, which have the same altitude, are to
+  one another as their bases;_ and
+
+  Prop. 33. _Similar solid parallelepipeds are to one another in the
+  triplicate ratio of their homologous sides._
+
+  If we consider, as in S 67, the ratios of lines as numbers, we may
+  also say--
+
+  _The ratio of the volumes of similar parallelepipeds is equal to the
+  ratio of the third powers of homologous sides._
+
+  Parallelepipeds which are not similar but equal are compared by aid of
+  the theorem
+
+  Prop. 34. _The bases and altitudes of equal solid parallelepipeds are
+  reciprocally proportional; and if the bases and altitudes be
+  reciprocally proportional, the solid parallelepipeds are equal._
+
+  S 83. Of the following propositions the 37th and 40th are of special
+  interest.
+
+  Prop. 37. _If four straight lines be proportionals, the similar solid
+  parallelepipeds, similarly described from them, shall also be
+  proportionals; and if the similar parallelepipeds similarly described
+  from four straight lines be proportionals, the straight lines shall be
+  proportionals._
+
+  In symbols it says--
+
+    If a : b = c : d, then a^3 : b^3 = c^3 : d^3.
+
+  Prop. 40 teaches how to compare the volumes of triangular prisms with
+  those of parallelepipeds, by proving _that a triangular prism is equal
+  in volume to a parallelepiped, which has its altitude and its base
+  equal to the altitude and the base of the triangular prism._
+
+  S 84. From these propositions follow all results relating to the
+  mensuration of volumes. We shall state these as we did in the case of
+  areas. The starting-point is the "rectangular" parallelepiped, which
+  has every edge perpendicular to the planes it meets, and which takes
+  the place of the rectangle in the plane. If this has all its edges
+  equal we obtain the "cube."
+
+  If we take a certain line u as unit length, then the square on u is
+  the unit of area, and the cube on u the unit of volume, that is to
+  say, if we wish to measure a volume we have to determine how many unit
+  cubes it contains.
+
+  A rectangular parallelepiped has, as a rule, the three edges unequal,
+  which meet at a point. Every other edge is equal to one of them. If a,
+  b, c be the three edges meeting at a point, then we may take the
+  rectangle contained by two of them, say by b and c, as base and the
+  third as altitude. Let V be its volume, V' that of another rectangular
+  parallelepiped which has the edges a', b, c, hence the same base as
+  the first. It follows then easily, from Prop. 25 or 32, that V : V' =
+  a : a'; or in words,
+
+  _Rectangular parallelepipeds on equal bases are proportional to their
+  altitudes._
+
+  If we have two rectangular parallelepipeds, of which the first has the
+  volume V and the edges a, b, c, and the second, the volume V' and the
+  edges a', b', c', we may compare them by aid of two new ones which
+  have respectively the edges a', b, c and a', b', c, and the volumes V1
+  and V2. We then have
+
+    V : V1 = a : a'; V1 : V2 = b : b', V2 : V' = c : c'.
+
+  Compounding these, we have
+
+    V : V' = (a : a')(b : b')(c : c'),
+
+  or
+
+    V    a    b    c
+    -- = -- . -- . --.
+    V'   a'   b'   c'
+
+  Hence, as a special case, making V' equal to the unit cube U on u we
+  get
+
+    V    a    b    c
+    -- = -- . -- . -- = [alpha].[beta].[gamma],
+    U    u    u    u
+
+  where [alpha], [beta], [gamma] are the numerical values of a, b, c;
+  that is, _The number of unit cubes in a rectangular parallelepiped_ is
+  equal to the product of the numerical values of its three edges. This
+  is generally expressed by saying the volume of a rectangular
+  parallelepiped is measured by the product of its sides, or by the
+  product of its base into its altitude, which in this case is the same.
+
+  Prop. 31 allows us to extend this to any parallelepipeds, and Props.
+  28 or 40, to triangular prisms.
+
+  _The volume of any parallelepiped, or of any triangular prism, is
+  measured by the product of base and altitude._
+
+  The consideration that any polygonal prism may be divided into a
+  number of triangular prisms, which have the same altitude and the sum
+  of their bases equal to the base of the polygonal prism, shows further
+  that the same holds for any prism whatever.
+
+
+  BOOK XII.
+
+  S 85. In the last part of Book XI. we have learnt how to compare the
+  volumes of parallelepipeds and of prisms. In order to determine the
+  volume of any solid bounded by plane faces we must determine the
+  volume of pyramids, for every such solid may be decomposed into a
+  number of pyramids.
+
+  As every pyramid may again be decomposed into triangular pyramids, it
+  becomes only necessary to determine their volume. This is done by the
+
+  _Theorem._--Every triangular pyramid is equal in volume to one third
+  of a triangular prism having the same base and the same altitude as
+  the pyramid.
+
+  This is an immediate consequence of Euclid's
+
+  Prop. 7. _Every prism having a triangular base may be divided into
+  three pyramids that have triangular bases, and are equal to one
+  another._
+
+  The proof of this theorem is difficult, because the three triangular
+  pyramids into which the prism is divided are by no means equal in
+  shape, and cannot be made to coincide. It has first to be proved that
+  two triangular pyramids have equal volumes, if they have equal bases
+  and equal altitudes. This Euclid does in the following manner. He
+  first shows (Prop. 3) that a triangular pyramid may be divided into
+  four parts, of which two are equal triangular pyramids similar to the
+  whole pyramid, whilst the other two are equal triangular prisms, and
+  further, that these two prisms together are greater than the two
+  pyramids, hence more than half the given pyramid. He next shows (Prop.
+  4) that if two triangular pyramids are given, having equal bases and
+  equal altitudes, and if each be divided as above, then the two
+  triangular prisms in the one are equal to those in the other, and each
+  of the remaining pyramids in the one has its base and altitude equal
+  to the base and altitude of the remaining pyramids in the other. Hence
+  to these pyramids the same process is again applicable. We are thus
+  enabled to cut out of the two given pyramids equal parts, each greater
+  than half the original pyramid. Of the remainder we can again cut out
+  equal parts greater than half these remainders, and so on as far as we
+  like. This process may be continued till the last remainder is smaller
+  than any assignable quantity, however small. It follows, so we should
+  conclude at present, that the two volumes must be equal, for they
+  cannot differ by any assignable quantity.
+
+  To Greek mathematicians this conclusion offers far greater
+  difficulties. They prove elaborately, by a _reductio ad absurdum_,
+  that the volumes cannot be unequal. This proof must be read in the
+  _Elements._ We must, however, state that we have in the above not
+  proved Euclid's Prop. 5, but only a special case of it. Euclid does
+  not suppose that the bases of the two pyramids to be compared are
+  equal, and hence he proves that the volumes are as the bases. The
+  reasoning of the proof becomes clearer in the special case, from which
+  the general one may be easily deduced.
+
+  S 86. Prop. 6 extends the result to pyramids with polygonal bases.
+  From these results follow again the rules at present given for the
+  mensuration of solids, viz. a pyramid is the third part of a
+  triangular prism having the same base and the same altitude. But a
+  triangular prism is equal in volume to a parallelepiped which has the
+  same base and altitude. Hence if B is the base and h the altitude, we
+  have
+
+    Volume of prism   = Bh,
+    Volume of pyramid = 1/3Bh,
+
+  statements which have to be taken in the sense that B means the number
+  of square units in the base, h the number of units of length in the
+  altitude, or that B and h denote the numerical values of base and
+  altitude.
+
+  S 87. A method similar to that used in proving Prop. 5 leads to the
+  following results relating to solids bounded by simple curved
+  surfaces:--
+
+  Prop. 10. _Every cone is the third part of a cylinder which has the
+  same base, and is of an equal altitude with it._
+
+  Prop. 11. _Cones or cylinders of the same altitude are to one another
+  as their bases._
+
+  Prop. 12. _Similar cones or cylinders have to one another the
+  triplicate ratio of that which the diameters of their bases have._
+
+  Prop. 13. _If a cylinder be cut by a plane parallel to its opposite
+  planes or bases, it divides the cylinder into two cylinders, one of
+  which is to the other as the axis of the first to the axis of the
+  other;_ which may also be stated thus:--
+
+  _Cylinders on the same base are proportional to their altitudes._
+
+  Prop. 14. _Cones or cylinders upon equal bases are to one another as
+  their altitudes._
+
+  Prop. 15. _The bases and altitudes of equal cones or cylinders are
+  reciprocally proportional, and if the bases and altitudes be
+  reciprocally proportional, the cones or cylinders are equal to one
+  another._
+
+  These theorems again lead to formulae in mensuration, if we compare a
+  cylinder with a prism having its base and altitude equal to the base
+  and altitude of the cylinder. This may be done by the method of
+  exhaustion. We get, then, the result that their bases are equal, and
+  have, if B denotes the numerical value of the base, and h that of the
+  altitude,
+
+    Volume of cylinder = Bh,
+    Volume of cone     = 1/3Bh.
+
+  S 88. The remaining propositions relate to circles and spheres. Of the
+  sphere only one property is proved, viz.:--
+
+  Prop. 18. _Spheres have to one another the triplicate ratio of that
+  which their diameters have._ The mensuration of the sphere, like that
+  of the circle, the cylinder and the cone, had not been settled in the
+  time of Euclid. It was done by Archimedes.
+
+
+  BOOK XIII.
+
+  S 89. The 13th and last book of Euclid's _Elements_ is devoted to the
+  regular solids (see POLYHEDRON). It is shown that there are five of
+  them, viz.:--
+
+  1. The regular _tetrahedron_, with 4 triangular faces and 4 vertices;
+
+  2. The _cube_, with 8 vertices and 6 square faces;
+
+  3. The _octahedron_, with 6 vertices and 8 triangular faces;
+
+  4. The _dodecahedron_, with 12 pentagonal faces, 3 at each of the
+  20 vertices;
+
+  5. The _icosahedron_, with 20 triangular faces, 5 at each of the
+  12 vertices.
+
+  It is shown how to inscribe these solids in a given sphere, and how to
+  determine the lengths of their edges.
+
+  S 90. The 13th book, and therefore the _Elements_, conclude with the
+  scholium, "that no other regular solid exists besides the five ones
+  enumerated."
+
+  The proof is very simple. Each face is a regular polygon, hence the
+  angles of the faces at any vertex must be angles in equal regular
+  polygons, must be together less than four right angles (XI. 21), and
+  must be three or more in number. Each angle in a regular triangle
+  equals two-thirds of one right angle. Hence it is possible to form a
+  solid angle with three, four or five regular triangles or faces. These
+  give the solid angles of the tetrahedron, the octahedron and the
+  icosahedron. The angle in a square (the regular quadrilateral) equals
+  one right angle. Hence three will form a solid angle, that of the
+  cube, and four will not. The angle in the regular pentagon equals 6/5
+  of a right angle. Hence three of them equal 18/5 (i.e. less than 4)
+  right angles, and form the solid angle of the dodecahedron. Three
+  regular polygons of six or more sides cannot form a solid angle.
+  Therefore no other regular solids are possible.     (O. H.)
+
+
+II. PROJECTIVE GEOMETRY
+
+It is difficult, at the outset, to characterize projective geometry as
+compared with Euclidean. But a few examples will at least indicate the
+practical differences between the two.
+
+In Euclid's _Elements_ almost all propositions refer to the _magnitude_
+of lines, angles, areas or volumes, and therefore to measurement. The
+statement that an angle is right, or that two straight lines are
+parallel, refers to measurement. On the other hand, the fact that a
+straight line does or does not cut a circle is independent of
+measurement, it being dependent only upon the mutual "position" of the
+line and the circle. This difference becomes clearer if we project any
+figure from one plane to another (see PROJECTION). By this the length of
+lines, the magnitude of angles and areas, is altered, so that the
+projection, or shadow, of a square on a plane will not be a square; it
+will, however, be some quadrilateral. Again, the projection of a circle
+will not be a circle, but some other curve more or less resembling a
+circle. But one property may be stated at once--no straight line can cut
+the projection of a circle in more than two points, because no straight
+line can cut a circle in more than two points. There are, then, some
+properties of figures which do not alter by projection, whilst others
+do. To the latter belong nearly all properties relating to measurement,
+at least in the form in which they are generally given. The others are
+said to be projective properties, and their investigation forms the
+subject of projective geometry.
+
+Different as are the kinds of properties investigated in the old and the
+new sciences, the methods followed differ in a still greater degree. In
+Euclid each proposition stands by itself; its connexion with others is
+never indicated; the leading ideas contained in its proof are not
+stated; general principles do not exist. In the modern methods, on the
+other hand, the greatest importance is attached to the leading thoughts
+which pervade the whole; and general principles, which bring whole
+groups of theorems under one aspect, are given rather than separate
+propositions. The whole tendency is towards generalization. A straight
+line is considered as given in its entirety, extending both ways to
+infinity, while Euclid never admits anything but finite quantities. The
+treatment of the infinite is in fact another fundamental difference
+between the two methods: Euclid avoids it; in modern geometry it is
+systematically introduced.
+
+Of the different modern methods of geometry, we shall treat principally
+of the methods of projection and correspondence which have proved to be
+the most powerful. These have become independent of Euclidean Geometry,
+especially through the _Geometrie der Lage_ of V. Staudt and the
+_Ausdehnungslehre_ of Grassmann.
+
+For the sake of brevity we shall presuppose a knowledge of Euclid's
+_Elements_, although we shall use only a few of his propositions.
+
+  S 1. _Geometrical Elements._ We consider space as filled with points,
+  lines and planes, and these we call the elements out of which our
+  figures are to be formed, calling any combination of these elements a
+  "figure."
+
+  By a line we mean a straight line in its entirety, extending both ways
+  to infinity; and by a plane, a plane surface, extending in all
+  directions to infinity.
+
+  We accept the three-dimensional space of experience--the space assumed
+  by Euclid--which has for its properties (among others):--
+
+  Through any two points in space one and only one line may be drawn;
+
+  Through any three points which are not in a line, one and only one
+  plane may be placed;
+
+  The intersection of two planes is a line;
+
+  A line which has two points in common with a plane lies in the plane,
+  hence the intersection of a line and a plane is a single point; and
+
+  Three planes which do not meet in a line have one single point in
+  common.
+
+  These results may be stated differently in the following form:--
+
+  I. A plane is determined--           A point is determined--
+       1. By three points which do         1. By three planes which do
+           not lie in a line;                  not pass through a line;
+       2. By two intersecting lines;       2. By two intersecting lines;
+       3. By a line and a point            3. By a plane and a line
+           which does not lie in it.           which does not lie in it.
+  II. A line is determined--
+       1. By two points;                   2. By two planes.
+
+  It will be observed that not only are planes determined by points, but
+  also points by planes; that therefore the planes may be considered as
+  elements, like points; and also that in any one of the above
+  statements we may interchange the words point and plane, and we obtain
+  again a correct statement, provided that these statements themselves
+  are true. As they stand, we ought, in several cases, to add "if they
+  are not parallel," or some such words, parallel lines and planes being
+  evidently left altogether out of consideration. To correct this we
+  have to reconsider the theory of parallels.
+
+  [Illustration: FIG. 1.]
+
+  S 2. _Parallels. Point at Infinity._--Let us take in a plane a line p
+  (fig. 1), a point S not in this line, and a line q drawn through S.
+  Then this line q will meet the line p in a point A. If we turn the
+  line q about S towards q', its point of intersection with p will move
+  along p towards B, passing, on continued turning, to a greater and
+  greater distance, until it is moved out of our reach. If we turn q
+  still farther, its continuation will meet p, but now at the other side
+  of A. The point of intersection has disappeared to the right and
+  reappeared to the left. There is one intermediate position where q is
+  parallel to p--that is where it does not cut p. In every other
+  position it cuts p in some finite point. If, on the other hand, we
+  move the point A to an infinite distance in p, then the line q which
+  passes through A will be a line which does not cut p at any finite
+  point. Thus we are led to say: _Every_ line through S which joins it
+  to any point at an infinite distance in p is parallel to p. But by
+  Euclid's 12th axiom there is but one line parallel to p through S. The
+  difficulty in which we are thus involved is due to the fact that we
+  try to reason about infinity as if we, with our finite capabilities,
+  could comprehend the infinite. To overcome this difficulty, we may say
+  that all points at infinity in a line _appear_ to us as one, and may
+  be replaced by a single "ideal" point.
+
+  We may therefore now give the following definitions and axiom:--
+
+  _Definition._--Lines which meet at infinity are called parallel.
+
+  _Axiom._--All points at an infinite distance in a line may be
+  considered as one single point.
+
+  _Definition._--This ideal point is called the _point at infinity_ in
+  the line.
+
+  The axiom is equivalent to Euclid's Axiom 12, for it follows from
+  either that through any point only one line may be drawn parallel to a
+  given line.
+
+  This point at infinity in a line is reached whether we move a point in
+  the one or in the opposite direction of a line to infinity. A line
+  thus appears closed by this point, and we speak as if we could move a
+  point along the line from one position A to another B in two ways,
+  either through the point at infinity or through finite points only.
+
+  It must never be forgotten that this point at infinity is ideal; in
+  fact, the whole notion of "infinity" is only a mathematical
+  conception, and owes its introduction (as a method of research) to the
+  working generalizations which it permits.
+
+  S 3. _Line and Plane at Infinity._--Having arrived at the notion of
+  replacing all points at infinity in a line by one ideal point, there
+  is no difficulty in replacing all points at infinity in a plane by one
+  ideal line.
+
+  To make this clear, let us suppose that a line p, which cuts two fixed
+  lines a and b in the points A and B, moves parallel to itself to a
+  greater and greater distance. It will at last cut both a and b at
+  their points at infinity, so that a line which joins the two points at
+  infinity in two intersecting lines lies altogether at infinity. Every
+  other line in the plane will meet it therefore at infinity, and thus
+  it contains all points at infinity in the plane.
+
+  _All points at infinity in a plane lie in a line, which is called the_
+  line at infinity _in the plane._
+
+  It follows that parallel planes must be considered as planes having a
+  common line at infinity, for any other plane cuts them in parallel
+  lines which have a point at infinity in common.
+
+  If we next take two intersecting planes, then the point at infinity in
+  their line of intersection lies in both planes, so that their lines at
+  infinity meet. Hence every line at infinity meets every other line at
+  infinity, and they are therefore all in one plane.
+
+  _All points at infinity in space may be considered as lying in one
+  ideal plane, which is called the_ plane at infinity.
+
+  S 4. _Parallelism._--We have now the following definitions:--
+
+  Parallel lines are lines which meet at infinity;
+
+  Parallel planes are planes which meet at infinity;
+
+  A line is parallel to a plane if it meets it at infinity.
+
+  Theorems like this--Lines (or planes) which are parallel to a third
+  are parallel to each other--follow at once.
+
+  This view of parallels leads therefore to no contradiction of Euclid's
+  _Elements._
+
+  As immediate consequences we get the propositions:--
+
+  Every line meets a plane in one point, or it lies in it;
+
+  Every plane meets every other plane in a line;
+
+  Any two lines in the same plane meet.
+
+  S 5. _Aggregates of Geometrical Elements._--We have called points,
+  lines and planes the elements of geometrical figures. We also say that
+  an element of one kind contains one of the other if it lies in it or
+  passes through it.
+
+  All the elements of one kind which are contained in one or two
+  elements of a different kind form aggregates which have to be
+  enumerated. They are the following:--
+
+  I. Of one dimension.
+
+     1. The _row_, or range, _of points_ formed by all points in a line,
+          which is called its base.
+
+     2. The _flat pencil_ formed by all the lines through a point in a
+          plane. Its base is the point in the plane.
+
+     3. The _axial pencil_ formed by all planes through a line which is
+          called its base or axis.
+
+  II. Of two dimensions.
+
+     1. The field of points and lines--that is, a plane with all its
+          points and all its lines.
+
+     2. The pencil of lines and planes--that is, a point in space with
+          all lines and all planes through it.
+
+  III. Of three dimensions.
+
+     The space of points--that is, all points in space.
+
+     The space of planes--that is, all planes in space.
+
+  IV. Of four dimensions.
+
+     The space of lines, or all lines in space.
+
+  S 6. _Meaning of "Dimensions."_--The word dimension in the above needs
+  explanation. If in a plane we take a row p and a pencil with centre Q,
+  then through every point in p one line in the pencil will pass, and
+  every ray in Q will cut p in one point, so that we are entitled to say
+  a row contains as many points as a flat pencil lines, and, we may add,
+  as an axial pencil planes, because an axial pencil is cut by a plane
+  in a flat pencil.
+
+  The number of elements in the row, in the flat pencil, and in the
+  axial pencil is, of course, infinite and indefinite too, but the same
+  in all. This number may be denoted by [infinity]. Then a plane
+  contains [infinity]^2 points and as many lines. To see this, take a
+  flat pencil in a plane. It contains [infinity] lines, and each line
+  contains [infinity] points, whilst each point in the plane lies on one
+  of these lines. Similarly, in a plane each line cuts a fixed line in a
+  point. But this line is cut at each point by [infinity] lines and
+  contains [infinity] points; hence there are [infinity]^2 lines in a
+  plane.
+
+  A pencil in space contains as many lines as a plane contains points
+  and as many planes as a plane contains lines, for any plane cuts the
+  pencil in a field of points and lines. Hence a pencil contains
+  [infinity]^2 lines and [infinity]^2 planes. _The field and the pencil
+  are of two dimensions._
+
+  To count the number of points in space we observe that each point lies
+  on some line in a pencil. But the pencil contains [infinity]^2 lines,
+  and each line [infinity] points; hence space contains [infinity]^3
+  points. Each plane cuts any fixed plane in a line. But a plane
+  contains [infinity]^2 lines, and through each pass [infinity] planes;
+  therefore space contains [infinity]^3 planes.
+
+  Hence space contains as many planes as points, but it contains an
+  infinite number of times more lines than points or planes. To count
+  them, notice that every line cuts a fixed plane in one point. But
+  [infinity]^2 lines pass through each point, and there are [infinity]^2
+  points in the plane. Hence there are [infinity]^4 lines in space. _The
+  space of points and planes is of three dimensions, but the space of
+  lines is of four dimensions._
+
+  A field of points or lines contains an infinite number of rows and
+  flat pencils; a pencil contains an infinite number of flat pencils and
+  of axial pencils; space contains a triple infinite number of pencils
+  and of fields, [infinity]^4 rows and axial pencils and [infinity]^5
+  flat pencils--or, in other words, each point is a centre of
+  [infinity]^2 flat pencils.
+
+  S 7. The above enumeration allows a classification of figures. Figures
+  in a row consist of groups of points only, and figures in the flat or
+  axial pencil consist of groups of lines or planes. In the plane we may
+  draw polygons; and in the pencil or in the point, solid angles, and so
+  on.
+
+  We may also distinguish the different measurements We have--
+
+    In the row, length of segment;
+    In the flat pencil, angles;
+    In the axial pencil, dihedral angles between two planes;
+    In the plane, areas;
+    In the pencil, solid angles;
+    In the space of points or planes, volumes.
+
+
+  SEGMENTS OF A LINE
+
+  S 8. Any two points A and B in space determine on the line through
+  them a finite part, which may be considered as being described by a
+  point moving from A to B. This we shall denote by AB, and distinguish
+  it from BA, which is supposed as being described by a point moving
+  from B to A, and hence in a direction or in a "sense" opposite to AB.
+  Such a finite line, which has a definite sense, we shall call a
+  "segment," so that AB and BA denote different segments, which are said
+  to be equal in length but of opposite sense. The one sense is often
+  called positive and the other negative.
+
+  In introducing the word "sense" for direction in a line, we have the
+  word direction reserved for direction of the line itself, so that
+  different lines have different directions, unless they be parallel,
+  whilst in each line we have a positive and negative sense.
+
+  We may also say, with Clifford, that AB denotes the "step" of going
+  from A to B.
+
+  [Illustration: FIG. 2.]
+
+  S 9. If we have three points A, B, C in a line (fig. 2), the step AB
+  will bring us from A to B, and the step BC from B to C. Hence both
+  steps are equivalent to the one step AC. This is expressed by saying
+  that AC is the "sum" of AB and BC; in symbols--
+
+    AB + BC = AC,
+
+  where account is to be taken of the sense.
+
+  This equation is true whatever be the position of the three points on
+  the line. As a special case we have
+
+    AB + BA = 0, (1)
+
+  and similarly
+
+    AB + BC + CA = 0, (2)
+
+  which again is true for any three points in a line.
+
+  We further write
+
+    AB = -BA.
+
+  where - denotes negative sense.
+
+  We can then, just as in algebra, change subtraction of segments into
+  addition by changing the sense, so that AB - CB is the same as AB +
+  (-CB) or AB + BC. A figure will at once show the truth of this. The
+  sense is, in fact, in every respect equivalent to the "sign" of a
+  number in algebra.
+
+  S 10. Of the many formulae which exist between points in a line we
+  shall have to use only one more, which connects the segments between
+  any four points A, B, C, D in a line. We have
+
+    BC = BD + DC, CA = CD + DA, AB = AD + DB;
+
+  or multiplying these by AD, BD, CD respectively, we get
+
+    BC.AD = BD.AD + DC.AD = BD.AD - CD.AD
+
+    CA.BD = CD.BD + DA.BD = CD.BD - AD.BD
+
+    AB.CD = AD.CD + DB.CD = AD.CD - BD.CD.
+
+  It will be seen that the sum of the right-hand sides vanishes, hence
+  that
+
+    BC.AD + CA.BD + AB.CD = 0 (3)
+
+  for any four points on a line.
+
+  [Illustration: FIG. 3.]
+
+  S 11. If C is any point in the line AB, then we say that C divides the
+  segment AB in the ratio AC/CB, account being taken of the sense of the
+  two segments AC and CB. If C lies between A and B the ratio is
+  positive, as AC and CB have the same sense. But if C lies without the
+  segment AB, i.e. if C divides AB externally, then the ratio is
+  negative. To see how the value of this ratio changes with C, we will
+  move C along the whole line (fig. 3), whilst A and B remain fixed. If
+  C lies at the point A, then AC = 0, hence the ratio AC : CB vanishes.
+  As C moves towards B, AC increases and CB decreases, so that our ratio
+  increases. At the middle point M of AB it assumes the value +1, and
+  then increases till it reaches an infinitely large value, when C
+  arrives at B. On passing beyond B the ratio becomes negative. If C is
+  at P we have AC = AP = AB + BP, hence
+
+    AC   AB   BP     AB
+    -- = -- + -- = - -- - 1.
+    CB   PB   PB     BP
+
+  In the last expression the ratio AB : BP is positive, has its greatest
+  value [infinity] when C coincides with B, and vanishes when BC becomes
+  infinite. Hence, as C moves from B to the right to the point at
+  infinity, the ratio AC : CB varies from -[infinity] to -1.
+
+  If, on the other hand, C is to the left of A, say at Q, we have AC =
+  AQ = AB + BQ = AB - QB, hence AC/CB = AB/QB - 1.
+
+  Here AB < QB, hence the ratio AB : QB is positive and always less than
+  one, so that the whole is negative and < 1. If C is at the point at
+  infinity it is -1, and then increases as C moves to the right, till
+  for C at A we get the ratio = 0. Hence--
+
+  "As C moves along the line from an infinite distance to the left to an
+  infinite distance at the right, the ratio always increases; it starts
+  with the value -1, reaches 0 at A, +1 at M, [infinity] at B, now
+  changes sign to -[infinity], and increases till at an infinite
+  distance it reaches again the value -1. _It assumes therefore all
+  possible values from -[infinity] to +[infinity], and each value only
+  once, so that not only does every position of C determine a definite
+  value of the ratio AC : CB, but also, conversely, to every positive or
+  negative value of this ratio belongs one single point in the line AB._
+
+  [Relations between segments of lines are interesting as showing an
+  application of algebra to geometry. The genesis of such relations
+  from algebraic identities is very simple. For example, if a, b, c, x
+  be any four quantities, then
+
+              a                       b
+    --------------------- + --------------------- +
+    (a - b)(a - c)(x - a)   (b - c)(b - a)(x - b)
+
+                c                       x
+      --------------------- = ---------------------;
+      (c - a)(c - b)(x - c)   (x - a)(x - b)(x - c)
+
+  this may be proved, cumbrously, by multiplying up, or, simply, by
+  decomposing the right-hand member of the identity into partial
+  fractions. Now take a line ABCDX, and let AB = a, AC = b, AD = c, AX =
+  x. Then obviously (a - b) = AB - AC = -BC, paying regard to signs; (a
+  - c) = AB - AD = DB, and so on. Substituting these values in the
+  identity we obtain the following relation connecting the segments
+  formed by five points on a line:--
+
+       AB         AC         AD         AX
+    -------- + -------- + -------- = --------.
+    BC.BD.BX   CD.CB.CX   DB.DC.DX   BX.CX.DX
+
+  Conversely, if a metrical relation be given, its validity may be
+  tested by reducing to an algebraic equation, which is an identity if
+  the relation be true. For example, if ABCDX be five collinear points,
+  prove
+
+    AD.AX   BD.BX   CD.CX
+    ----- + ----- + ----- = 1.
+    AB.AC   BC.BA   CA.CB
+
+  Clearing of fractions by multiplying throughout by AB.BC.CA, we have
+  to prove
+
+    -AD.AX.BC - BD.BX.CA - CD.CX.AB = AB.BC.CA.
+
+  Take A as origin and let AB = a, AC = b, AD = c, AX = x. Substituting
+  for the segments in terms of a, b, c, x, we obtain on simplification
+
+    a^2b - ab^2 = -ab^2 + a^2b, an obvious identity.
+
+  An alternative method of testing a relation is illustrated in the
+  following example:-- If A, B, C, D, E, F be six collinear points, then
+
+      AE.AF      BE.BF      CE.CF      DE.DF
+    -------- + -------- + -------- + -------- = 0.
+    AB.AC.AD   BC.BD.BA   CD.CA.CB   DA.DB.DC
+
+  Clearing of fractions by multiplying throughout by AB.BC.CD.DA, and
+  reducing to a common origin O (calling OA = a, OB = b, &c.), an
+  equation containing the second and lower powers of OA (= a), &c., is
+  obtained. Calling OA = x, it is found that x = b, x = c, x = d are
+  solutions. Hence the quadratic has three roots; consequently it is an
+  identity.
+
+  The relations connecting five points which we have instanced above may
+  be readily deduced from the six-point relation; the first by taking D
+  at infinity, and the second by taking F at infinity, and then making
+  the obvious permutations of the points.]
+
+
+  PROJECTION AND CROSS-RATIOS
+
+  S 12. If we join a point A to a point S, then the point where the line
+  SA cuts a fixed plane [pi] is called the projection of A on the plane
+  [pi] from S as centre of projection. If we have two planes [pi] and
+  [pi]' and a point S, we may project every point A in [pi] to the other
+  plane. If A' is the projection of A, then A is also the projection of
+  A', so that the relations are reciprocal. To every figure in [pi] we
+  get as its projection a corresponding figure in [pi]'.
+
+  We shall determine such properties of figures as remain true for the
+  projection, and which are called projective properties. For this
+  purpose it will be sufficient to consider at first only constructions
+  in one plane.
+
+  [Illustration: FIG. 4.]
+
+  [Illustration: FIG. 5.]
+
+  Let us suppose we have given in a plane two lines p and p' and a
+  centre S (fig. 4); we may then project the points in p from S to p'.
+  Let A', B' ... be the projections of A, B ..., the point at infinity
+  in p which we shall denote by I will be projected into a finite point
+  I' in p', viz. into the point where the parallel to p through S cuts
+  p'. Similarly one point J in p will be projected into the point J' at
+  infinity in p'. This point J is of course the point where the parallel
+  to p' through S cuts p. We thus see that every point in p is projected
+  into a single point in p'.
+
+  Fig. 5 shows that a segment AB will be projected into a segment A'B'
+  which is not equal to it, at least not as a rule; and also that the
+  ratio AC : CB is not equal to the ratio A'C' : C'B' formed by the
+  projections. These ratios will become equal only if p and p' are
+  parallel, for in this case the triangle SAB is similar to the triangle
+  SA'B'. Between three points in a line and their projections there
+  exists therefore in general no relation. But between four points a
+  relation does exist.
+
+  S 13. Let A, B, C, D be four points in p, A', B', C, D' their
+  projections in p', then the ratio of the two ratios AC : CB and AD :
+  DB into which C and D divide the segment AB is equal to the
+  corresponding expression between A', B', C', D'. In symbols we have
+
+    AC   AD   A'C'   A'D'
+    -- : -- = ---- : ----.
+    CB   DB   C'B'   D'B'
+
+  This is easily proved by aid of similar triangles.
+
+  [Illustration: FIG. 6.]
+
+  Through the points A and B on p draw parallels to p', which cut the
+  projecting rays in C2, D2, B2 and A1, C1, D1, as indicated in fig. 6.
+  The two triangles ACC2 and BCC1 will be similar, as will also be the
+  triangles ADD2 and BDD1.
+
+  The proof is left to the reader.
+
+  This result is of fundamental importance.
+
+  The expression AC/CB : AD/DB has been called by Chasles the
+  "anharmonic ratio of the four points A, B, C, D." Professor Clifford
+  proposed the shorter name of "cross-ratio." We shall adopt the latter.
+  We have then the
+
+  FUNDAMENTAL THEOREM.--_The cross-ratio of four points in a line is
+  equal to the cross-ratio of their projections on any other line which
+  lies in the same plane with it._
+
+  S 14. Before we draw conclusions from this result, we must investigate
+  the meaning of a cross-ratio somewhat more fully.
+
+  If four points A, B, C, D are given, and we wish to form their
+  cross-ratio, we have first to divide them into two groups of two, the
+  points in each group being taken in a definite order. Thus, let A, B
+  be the first, C, D the second pair, A and C being the first points in
+  each pair. The cross-ratio is then the ratio AC : CB divided by AD :
+  DB. This will be denoted by (AB, CD), so that
+
+               AC   AD
+    (AB, CD) = -- : --.
+               CB   DB
+
+  This is easily remembered. In order to write it out, make first the
+  two lines for the fractions, and put above and below these the letters
+  A and B in their places, thus, A*/B : A*/B; and then fill up,
+  crosswise, the first by C and the other by D.
+
+  S 15. If we take the points in a different order, the value of the
+  cross-ratio will change. We can do this in twenty-four different ways
+  by forming all permutations of the letters. But of these twenty-four
+  cross-ratios groups of four are equal, so that there are really only
+  six different ones, and these six are reciprocals in pairs.
+
+  We have the following rules:--
+
+  I. If in a cross-ratio the two groups be interchanged, its value
+  remains unaltered, i.e.
+
+    (AB, CD) = (CD, AB) = (BA, DC) = (DC, BA).
+
+  II. If in a cross-ratio the two points belonging to one of the two
+  groups be interchanged, the cross-ratio changes into its reciprocal,
+  i.e.
+
+    (AB, CD) = 1/(AB, DC) = 1/(BA, CD) = 1/(CD, BA) = 1/(DC, AB).
+
+  From I. and II. we see that eight cross-ratios are associated with
+  (AB, CD).
+
+  III. If in a cross-ratio the two middle letters be interchanged, the
+  cross-ratio [alpha] changes into its complement 1 - [alpha], i.e. (AB,
+  CD) = 1 - (AC, BD).
+
+  [S 16. If [lambda] = (AB, CD), [mu] = (AC, DB), [nu] = (AD, BC), then
+  [lambda], [mu], [nu] and their reciprocals 1/[lambda], 1/[mu], 1/[nu]
+  are the values of the total number of twenty-four cross-ratios.
+  Moreover, [lambda], [mu], [nu] are connected by the relations
+
+    [lambda] + 1/[mu] = [mu] + 1/[nu] = [nu] + 1/[lambda] = -[lambda][mu][nu] = 1;
+
+  this proposition may be proved by substituting for [lambda], [mu],
+  [nu] and reducing to a common origin. There are therefore four
+  equations between three unknowns; hence if one cross-ratio be given,
+  the remaining twenty-three are determinate. Moreover, two of the
+  quantities [lambda], [mu], [nu] are positive, and the remaining one
+  negative.
+
+  The following scheme shows the twenty-four cross-ratios expressed in
+  terms of [lambda], [mu], [nu].]
+
+    +---------+-----------------------+---------------+---------------+
+    |(AB, CD) |                       |               |               |
+    |(BA, DC) |       [lambda]        |    1 - [mu]   |  1/(1 - [nu]) |
+    |(CD, AB) |                       |               |               |
+    |(DC, BA) |                       |               |               |
+    +---------+-----------------------+---------------+---------------+
+    |(AC, DB) |                       |               |               |
+    |(BD, CA) |   1/(1 - [lambda])    |     1/[mu]    |([nu] - 1)/[nu]|
+    |(CA, BD) |                       |               |               |
+    |(DB, AC) |                       |               |               |
+    +---------+-----------------------+---------------+---------------+
+    |(AB, DC) |                       |               |               |
+    |(BA, CD) |      1/[lambda]       |  1/(1 - [mu]) |   1 - [nu]    |
+    |(CD, BA) |                       |               |               |
+    |(DC, AB) |                       |               |               |
+    +---------+-----------------------+---------------+---------------+
+    |(AD, BC) |                       |               |               |
+    |(BC, AD) |([lambda] - 1)/[lambda]|[mu]/([mu] - 1)|     [nu]      |
+    |(CB, DA) |                       |               |               |
+    |(DA, CB) |                       |               |               |
+    +---------+-----------------------+---------------+---------------+
+    |(AC, BD) |                       |               |               |
+    |(BD, AC) |     1 - [lambda]      |     [mu]      |[nu]/([nu] - 1)|
+    |(CA, DB) |                       |               |               |
+    |(DB, CA) |                       |               |               |
+    +---------+-----------------------+---------------+---------------+
+    |(AD, CB) |                       |               |               |
+    |(BC, DA) |[lambda]/([lambda] - 1)|([mu] - 1)/[mu]|    1/[nu]     |
+    |(CB, AD) |                       |               |               |
+    |(DA, BC) |                       |               |               |
+    +---------+-----------------------+---------------+---------------+
+
+  S 17. If one of the points of which a cross-ratio is formed is the
+  point at infinity in the line, the cross-ratio changes into a simple
+  ratio. It is convenient to let the point at infinity occupy the last
+  place in the symbolic expression for the cross-ratio. Thus if I is a
+  point at infinity, we have (AB, CI) = -AC/CB, because AI : IB = -1.
+
+  Every common ratio of three points in a line may thus be expressed as
+  a cross-ratio, by adding the point at infinity to the group of points.
+
+
+  HARMONIC RANGES
+
+  S 18. If the points have special positions, the cross-ratios may have
+  such a value that, of the six different ones, two and two become
+  equal. If the first two shall be equal, we get [lambda] = 1/[lambda],
+  or [lambda]^2 = 1, [lambda] = [+-]1.
+
+  If we take [lambda] = +1, we have (AB, CD) = 1, or AC/CB = AD/DB; that
+  is, the points C and D coincide, provided that A and B are different.
+
+  If we take [lambda] = -1, so that (AB, CD) = -1, we have AC/CB =
+  -AD/DB. _Hence C and D divide AB internally and externally in the same
+  ratio._
+
+  The four points are in this case said to be _harmonic points_, and _C
+  and D are said to be harmonic conjugates with regard to A and B._
+
+  But we have also (CD, AB) = -1, so that A and B are harmonic
+  conjugates with regard to C and D.
+
+  The principal property of harmonic points is that their cross-ratio
+  remains unaltered if we interchange the two points belonging to one
+  pair, viz.
+
+    (AB, CD) = (AB, DC) = (BA, CD).
+
+  For four harmonic points the six cross-ratios become equal two and
+  two:
+
+                                       [lambda]
+    [lambda] = -1, 1 - [lambda] = 2, ------------ = 1/2,
+                                     [lambda] - 1
+
+           1                1              [lambda] - 1
+      = -------- = -1, ------------ = 1/2, ------------ = 2.
+        [lambda]       1 - [lambda]          [lambda]
+
+  Hence if we get four points whose cross-ratio is 2 or 1/2, then they
+  are harmonic, but not arranged so that conjugates are paired. If this
+  is the case the cross-ratio = -1.
+
+  S 19. If we equate any two of the above six values of the
+  cross-ratios, we get either [lambda] = 1, 0, [infinity], or [lambda] =
+  -1, 2, 1/2, or else [lambda] becomes a root of the equation [lambda]^2
+  - [lambda] + 1 = 0, that is, an imaginary cube root of -1. In this
+  case the six values become three and three equal, so that only two
+  different values remain. This case, though important in the theory of
+  cubic curves, is for our purposes of no interest, whilst harmonic
+  points are all-important.
+
+  S 20. From the definition of harmonic points, and by aid of S 11, the
+  following properties are easily deduced.
+
+  If C and D are harmonic conjugates with regard to A and B, then one of
+  them lies in, the other without AB; it is impossible to move from A to
+  B without passing either through C or through D; the one blocks the
+  finite way, the other the way through infinity. This is expressed by
+  saying A and B are "separated" by C and D.
+
+  For every position of C there will be one and only one point D which
+  is its harmonic conjugate with regard to any point pair A, B.
+
+  If A and B are different points, and if C coincides with A or B, D
+  does. But if A and B coincide, one of the points C or D, lying between
+  them, coincides with them, and the other may be anywhere in the line.
+  It follows that, "_if of four harmonic conjugates two coincide, then a
+  third coincides with them, and the fourth may be any point in the
+  line_."
+
+  If C is the middle point between A and B, then D is the point at
+  infinity; for AC : CB = +1, hence AD : DB must be equal to -1. _The
+  harmonic conjugate of the point at infinity in a line with regard to
+  two points A, B is the middle point of AB._
+
+  This important property gives a first example how metric properties
+  are connected with projective ones.
+
+  [S 21. _Harmonic properties of the complete quadrilateral and
+  quadrangle._
+
+  [Illustration: FIG. 7.]
+
+  [Illustration: FIG. 8.]
+
+  A figure formed by four lines in a plane is called a _complete
+  quadrilateral_, or, shorter, a _four-side_. The four sides meet in six
+  points, named the "vertices," which may be joined by three lines
+  (other than the sides), named the "diagonals" or "harmonic lines." The
+  diagonals enclose the "harmonic triangle of the quadrilateral." In
+  fig. 7, A'B'C', B'AC, C'AB, CBA' are the sides, A, A', B, B', C, C'
+  the vertices, AA', BB', CC' the harmonic lines, and
+  [alpha][beta][gamma] the harmonic triangle of the quadrilateral. A
+  figure formed by four coplanar points is named a _complete
+  quadrangle_, or, shorter, a _four-point_. The four points may be
+  joined by six lines, named the "sides," which intersect in three other
+  points, termed the "diagonal or harmonic points." The harmonic points
+  are the vertices of the "harmonic triangle of the complete
+  quadrangle." In fig. 8, AA', BB' are the points, AA', BB', A'B', B'A,
+  AB, BA' are the sides, L, M, N are the diagonal points, and LMN is the
+  harmonic triangle of the quadrangle.
+
+  The harmonic property of the complete quadrilateral is: Any diagonal
+  or harmonic line is harmonically divided by the other two; and of a
+  complete quadrangle: The angle at any harmonic point is divided
+  harmonically by the joins to the other harmonic points. To prove the
+  first theorem, we have to prove (AA', [beta][gamma]), (BB',
+  [gamma][alpha]), (CC', [beta][alpha]) are harmonic. Consider the
+  cross-ratio (CC', [alpha][beta]). Then projecting from A on BB' we
+  have A(CC', [alpha][beta]) = A(B'B, [alpha][gamma]). Projecting from
+  A' on BB', A'(CC', [alpha][beta]) = A'(BB', [alpha][gamma]). Hence
+  (B'B, [alpha][gamma]) = (BB', [alpha][gamma]), i.e. the cross-ratio
+  (BB', [alpha][gamma]) equals that of its reciprocal; hence the range
+  is harmonic.
+
+  The second theorem states that the pencils L(BA, NM), M(B'A, LN),
+  N(BA, LM) are harmonic. Deferring the subject of harmonic pencils to
+  the next section, it will suffice to state here that any transversal
+  intersects an harmonic pencil in an harmonic range. Consider the
+  pencil L(BA, NM), then it is sufficient to prove (BA', NM') is
+  harmonic. This follows from the previous theorem by considering A'B as
+  a diagonal of the quadrilateral ALB'M.]
+
+  This property of the complete quadrilateral allows the solution of the
+  problem:
+
+  _To construct the harmonic conjugate D to a point C with regard to two
+  given points A and B._
+
+  Through A draw any two lines, and through C one cutting the former two
+  in G and H. Join these points to B, cutting the former two lines in E
+  and F. The point D where EF cuts AB will be the harmonic conjugate
+  required.
+
+  This remarkable construction requires nothing but the drawing of
+  lines, and is therefore independent of measurement. In a similar
+  manner the harmonic conjugate of the line VA for two lines VC, VD is
+  constructed with the aid of the property of the complete quadrangle.
+
+  S 22. _Harmonic Pencils._--The theory of cross-ratios may be extended
+  from points in a row to lines in a flat pencil and to planes in an
+  axial pencil. We have seen (S 13) that if the lines which join four
+  points A, B, C, D to any point S be cut by any other line in A', B',
+  C', D', then (AB, CD) = (A'B', C'D'). In other words, four lines in a
+  flat pencil are cut by every other line in four points whose
+  cross-ratio is constant.
+
+  _Definition._--By the cross-ratio of four rays in a flat pencil is
+  meant the cross-ratio of the four points in which the rays are cut by
+  any line. If a, b, c, d be the lines, then this cross-ratio is denoted
+  by (ab, cd).
+
+  _Definition._--By the cross-ratio of four planes in an axial pencil is
+  understood the cross-ratio of the four points in which any line cuts
+  the planes, or, what is the same thing, the cross-ratio of the four
+  rays in which any plane cuts the four planes.
+
+  In order that this definition may have a meaning, it has to be proved
+  that all lines cut the pencil in points which have the same
+  cross-ratio. This is seen at once for two intersecting lines, as their
+  plane cuts the axial pencil in a flat pencil, which is itself cut by
+  the two lines. The cross-ratio of the four points on one line is
+  therefore equal to that on the other, and equal to that of the four
+  rays in the flat pencil.
+
+  If two non-intersecting lines p and q cut the four planes in A, B, C,
+  D and A', B', C', D', draw a line r to meet both p and q, and let this
+  line cut the planes in A", B", C", D". Then (AB, CD) = (A'B', C'D'),
+  for each is equal to (A"B", C"D").
+
+  S 23. We may now also extend the notion of harmonic elements, viz.
+
+  _Definition._--Four rays in a flat pencil and four planes in an axial
+  pencil are said to be harmonic if their cross-ratio equals -1, that
+  is, if they are cut by a line in four harmonic points.
+
+  If we understand by a "median line" of a triangle a line which joins a
+  vertex to the middle point of the opposite side, and by a "median
+  line" of a parallelogram a line joining middle points of opposite
+  sides, we get as special cases of the last theorem:
+
+  _The diagonals and median lines of a parallelogram form an harmonic
+  pencil_; and
+
+  _At a vertex of any triangle, the two sides, the median line, and the
+  line parallel to the base form an harmonic pencil._
+
+  Taking the parallelogram a rectangle, or the triangle isosceles, we
+  get:
+
+  _Any two lines and the bisections of their angles form an harmonic
+  pencil._ Or:
+
+  _In an harmonic pencil, if two conjugate rays are perpendicular, then
+  the other two are equally inclined to them_; and, conversely, _if one
+  ray bisects the angle between conjugate rays, it is perpendicular to
+  its conjugate_.
+
+  This connects perpendicularity and bisection of angles with projective
+  properties.
+
+  S 24. We add a few theorems and problems which are easily proved or
+  solved by aid of harmonics.
+
+  An harmonic pencil is cut by a line parallel to one of its rays in
+  three equidistant points.
+
+  Through a given point to draw a line such that the segment determined
+  on it by a given angle is bisected at that point.
+
+  Having given two parallel lines, to bisect on either any given segment
+  without using a pair of compasses.
+
+  Having given in a line a segment and its middle point, to draw through
+  any given point in the plane a line parallel to the given line.
+
+  To draw a line which joins a given point to the intersection of two
+  given lines which meet off the drawing paper (by aid of S 21).
+
+
+  CORRESPONDENCE. HOMOGRAPHIC AND PERSPECTIVE RANGES
+
+  S 25. Two rows, p and p', which are one the projection of the other
+  (as in fig. 5), stand in a definite relation to each other,
+  characterized by the following properties.
+
+  1. _To each point in either corresponds one point in the other_; that
+  is, those points are said to correspond which are projections of one
+  another.
+
+  2. _The cross-ratio of any four points in one equals that of the
+  corresponding points in the other._
+
+  3. _The lines joining corresponding points all pass through the same
+  point._
+
+  If we suppose corresponding points marked, and the rows brought into
+  any other position, then the lines joining corresponding points will
+  no longer meet in a common point, and hence the third of the above
+  properties will not hold any longer; but we have still a
+  correspondence between the points in the two rows possessing the first
+  two properties. Such a correspondence has been called a _one-one
+  correspondence_, whilst the two rows between which such correspondence
+  has been established are said to be _projective_ or _homographic_. Two
+  rows which are each the projection of the other are therefore
+  _projective_. We shall presently see, also, that any two projective
+  rows may always be placed in such a position that one appears as the
+  projection of the other. If they are in such a position the rows are
+  said to be in _perspective position_, or simply to be in
+  _perspective_.
+
+  S 26. The notion of a one-one correspondence between rows may be
+  extended to flat and axial pencils, viz. a flat pencil will be said to
+  be projective to a flat pencil if to each ray in the first corresponds
+  one ray in the second, and if the cross-ratio of four rays in one
+  equals that of the corresponding rays in the second.
+
+  Similarly an axial pencil may be projective to an axial pencil. But a
+  flat pencil may also be projective to an axial pencil, or either
+  pencil may be projective to a row. The definition is the same in each
+  case: there is a one-one correspondence between the elements, and four
+  elements have the same cross-ratio as the corresponding ones.
+
+  S 27. There is also in each case a special position which is called
+  _perspective_, viz.
+
+  1. Two projective rows are perspective if they lie in the same plane,
+  and if the one row is a projection of the other.
+
+  2. Two projective flat pencils are perspective--(1) if they lie in the
+  same plane, and have a row as a common section; (2) if they lie in the
+  same pencil (in space), and are both sections of the same axial
+  pencil; (3) if they are in space and have a row as common section, or
+  are both sections of the same axial pencil, one of the conditions
+  involving the other.
+
+  3. Two projective axial pencils, if their axes meet, and if they have
+  a flat pencil as a common section.
+
+  4. A row and a projective flat pencil, if the row is a section of the
+  pencil, each point lying in its corresponding line.
+
+  5. A row and a projective axial pencil, if the row is a section of the
+  pencil, each point lying in its corresponding line.
+
+  6. A flat and a projective axial pencil, if the former is a section of
+  the other, each ray lying in its corresponding plane.
+
+  That in each case the correspondence established by the position
+  indicated is such as has been called projective follows at once from
+  the definition. It is not so evident that the perspective position may
+  always be obtained. We shall show in S 30 this for the first three
+  cases. First, however, we shall give a few theorems which relate to
+  the general correspondence, not to the perspective position.
+
+  S 28. _Two rows or pencils, flat or axial, which are projective to a
+  third are projective to each other_; this follows at once from the
+  definitions.
+
+  S 29. _If two rows, or two pencils, either flat or axial, or a row and
+  a pencil, be projective, we may assume to any three elements in the
+  one the three corresponding elements in the other, and then the
+  correspondence is uniquely determined._
+
+  For if in two projective rows we assume that the points A, B, C in the
+  first correspond to the given points A', B', C' in the second, then to
+  any fourth point D in the first will correspond a point D' in the
+  second, so that
+
+    (AB, CD) = (A'B', C'D').
+
+  But there is only one point, D', which makes the cross-ratio (A'B',
+  C'D') equal to the given number (AB, CD).
+
+  The same reasoning holds in the other cases.
+
+  S 30. If two rows are perspective, then the lines joining
+  corresponding points all meet in a point, the centre of projection;
+  and the point in which the two bases of the rows intersect as a point
+  in the first row coincides with its corresponding point in the second.
+
+  This follows from the definition. The converse also holds, viz.
+
+  _If two projective rows have such a position that one point in the one
+  coincides with its corresponding point in the other, then they are
+  perspective, that is, the lines joining corresponding points all pass
+  through a common point, and form a flat pencil._
+
+  For let A, B, C, D ... be points in the one, and A', B', C', D' ...
+  the corresponding points in the other row, and let A be made to
+  coincide with its corresponding point A'. Let S be the point where the
+  lines BB' and CC' meet, and let us join S to the point D in the first
+  row. This line will cut the second row in a point D", so that A, B, C,
+  D are projected from S into the points A, B', C', D". The cross-ratio
+  (AB, CD) is therefore equal to (AB', C'D"), and by hypothesis it is
+  equal to (A'B', C'D'). Hence (A'B', C'D") = (A'B', C'D'), that is, D"
+  is the same point as D'.
+
+  S 31. If two projected flat pencils in the same plane are in
+  perspective, then the intersections of corresponding lines form a row,
+  and the line joining the two centres as a line in the first pencil
+  corresponds to the same line as a line in the second. And conversely,
+
+  _If two projective pencils in the same plane, but with different
+  centres, have one line in the one coincident with its corresponding
+  line in the other, then the two pencils are perspective, that is, the
+  intersection of corresponding lines lie in a line._
+
+  The proof is the same as in S 30.
+
+  S 32. If two projective flat pencils in the same point (pencil in
+  space), but not in the same plane, are perspective, then the planes
+  joining corresponding rays all pass through a line (they form an axial
+  pencil), and the line common to the two pencils (in which their planes
+  intersect) corresponds to itself. And conversely:--
+
+  If two flat pencils which have a common centre, but do not lie in a
+  common plane, are placed so that one ray in the one coincides with its
+  corresponding ray in the other, then they are perspective, that is,
+  the planes joining corresponding lines all pass through a line.
+
+  S 33. If two projective axial pencils are perspective, then the
+  intersection of corresponding planes lie in a plane, and the plane
+  common to the two pencils (in which the two axes lie) corresponds to
+  itself. And conversely:--
+
+  If two projective axial pencils are placed in such a position that a
+  plane in the one coincides with its corresponding plane, then the two
+  pencils are perspective, that is, corresponding planes meet in lines
+  which lie in a plane.
+
+  The proof again is the same as in S 30.
+
+  S 34. These theorems relating to perspective position become illusory
+  if the projective rows of pencils have a common base. We then have:--
+
+  In two projective rows on the same line--and also in two projective
+  and concentric flat pencils in the same plane, or in two projective
+  axial pencils with a common axis--every element in the one coincides
+  with its corresponding element in the other as soon as three elements
+  in the one coincide with their corresponding elements in the other.
+
+  _Proof_ (in case of two rows).--Between four elements A, B, C, D and
+  their corresponding elements A', B', C', D' exists the relation (ABCD)
+  = (A'B'C'D'). If now A', B', C' coincide respectively with A, B, C, we
+  get (AB, CD) = (AB, CD'), hence D and D' coincide.
+
+  The last theorem may also be stated thus:--
+
+  In two projective rows or pencils, which have a common base but are
+  not identical, not more than two elements in the one can coincide with
+  their corresponding elements in the other.
+
+  Thus two projective rows on the same line cannot have more than two
+  pairs of coincident points unless every point coincides with its
+  corresponding point.
+
+  It is easy to construct two projective rows on the same line, which
+  have two pairs of corresponding points coincident. Let the points A,
+  B, C as points belonging to the one row correspond to A, B, and C' as
+  points in the second. Then A and B coincide with their corresponding
+  points, but C does not. It is, however, not necessary that two such
+  rows have twice a point coincident with its corresponding point; it is
+  possible that this happens only once or not at all. Of this we shall
+  see examples later.
+
+  [Illustration: FIG. 9.]
+
+  S 35. If two projective rows or pencils are in perspective position,
+  we know at once which element in one corresponds to any given element
+  in the other. If p and q (fig. 9) are two projective rows, so that K
+  corresponds to itself, and if we know that to A and B in p correspond
+  A' and B' in q, then the point S, where AA' meets BB', is the centre
+  of projection, and hence, in order to find the point C' corresponding
+  to C, we have only to join C to S; the point C', where this line cuts
+  q, is the point required.
+
+  [Illustration: FIG. 10.]
+
+  If two flat pencils, S1 and S2, in a plane are perspective (fig. 10),
+  we need only to know two pairs, a, a' and b, b', of corresponding rays
+  in order to find the axis s of projection. This being known, a ray c'
+  in S2, corresponding to a given ray c in S1, is found by joining S2 to
+  the point where c cuts the axis s.
+
+  A similar construction holds in the other cases of perspective
+  figures.
+
+  On this depends the solution of the following general problem.
+
+  S 36. Three pairs of corresponding elements in two projective rows or
+  pencils being given, to determine for any element in one the
+  corresponding element in the other.
+
+  We solve this in the two cases of two projective rows and of two
+  projective flat pencils in a plane.
+
+  _Problem_ I.--Let A, B, C be          _Problem_ II.--Let a, b, c be
+  three points in a row s, A', B',      three rays in a pencil S, a',
+  C' the corresponding points in a      b', c' the corresponding rays in
+  projective row s', both being in      a projective pencil S', both
+  a plane; it is required to find       being in the same plane; it is
+  for any point D in s the              required to find for any ray d
+  corresponding point D' in s'.         in S the corresponding ray d' in
+                                        S'.
+
+  The solution is made to depend on the construction of an auxiliary row
+  or pencil which is perspective to both the given ones. This is found
+  as follows:--
+
+  [Illustration: FIG. 11.]
+
+  _Solution of Problem_ I.--On the line joining two corresponding
+  points, say AA' (fig. 11), take any two points, S and S', as centres
+  of auxiliary pencils. Join the intersection B1 of SB and S'B' to the
+  intersection C1 of SC and S'C' by the line s1. Then a row on s1 will
+  be perspective to s with S as centre of projection, and to s' with S'
+  as centre. To find now the point D' on s' corresponding to a point D
+  on s we have only to determine the point D1, where the line SD cuts
+  s1, and to draw S'D1; the point where this line cuts s' will be the
+  required point D'.
+
+  _Proof._--The rows s and s' are both perspective to the row s1, hence
+  they are projective to one another. To A, B, C, D on s correspond A1,
+  B1, C1, D1 on s1, and to these correspond A', B', C', D' on s'; so
+  that D and D' are corresponding points as required.
+
+  [Illustration: FIG. 12.]
+
+  _Solution of Problem_ II.--Through the intersection A of two
+  corresponding rays a and a' (fig. 12), take two lines, s and s', as
+  bases of auxiliary rows. Let S1 be the point where the line b1, which
+  joins B and B', cuts the line c1, which joins C and C'. Then a pencil
+  S1 will be perspective to S with s as axis of projection. To find the
+  ray d' in S' corresponding to a given ray d in S, cut d by s at D;
+  project this point from S1 to D' on s' and join D' to S'. This will be
+  the required ray.
+
+  _Proof._--That the pencil S1 is perspective to S and also to S'
+  follows from construction. To the lines a1, b1, c1, d1 in S1
+  correspond the lines a, b, c, d in S and the lines a', b', c', d' in
+  S', so that d and d' are corresponding rays.
+
+  In the first solution the two centres, S, S', are _any_ two points on
+  a line joining any two corresponding points, so that the solution of
+  the problem allows of a great many different constructions. _But
+  whatever construction be used, the point D', corresponding to D, must
+  be always the same_, according to the theorem in S 29. This gives rise
+  to a number of theorems, into which, however, we shall not enter. The
+  same remarks hold for the second problem.
+
+  S 37. _Homological Triangles._--As a further application of the
+  theorems about perspective rows and pencils we shall prove the
+  following important theorem.
+
+  _Theorem._--If ABC and A'B'C' (fig. 13) be two triangles, such that
+  the lines AA', BB', CC' meet in a point S, then the intersections of
+  BC and B'C', of CA and C'A', and of AB and A'B' will lie in a line.
+  Such triangles are said to be homological, or in perspective. The
+  triangles are "co-axial" in virtue of the property that the meets of
+  corresponding sides are collinear and copolar, since the lines joining
+  corresponding vertices are concurrent.
+
+  _Proof._--Let a, b, c denote the lines AA', BB', CC', which meet at S.
+  Then these may be taken as bases of projective rows, so that A, A', S
+  on a correspond to B, B', S on b, and to C, C', S on c. As the point S
+  is common to all, any two of these rows will be perspective.
+
+    If S1 be the centre of projection of rows b and c,
+       S2      "          "          "        c and a,
+       S3      "          "          "        a and b,
+
+  and if the line S1S2 cuts a in A1, and b in B1, and c in C1, then A1,
+  B1 will be corresponding points in a and b, both corresponding to C1
+  in c. But a and b are perspective, therefore the line A1B1, that is
+  S1S2, joining corresponding points must pass through the centre of
+  projection S3 of a and b. In other words, S1, S2, S3 lie in a line.
+  This is Desargues' celebrated theorem if we state it thus:--
+
+  [Illustration: FIG. 13.]
+
+  _Theorem of Desargues._--If each of two triangles has one vertex on
+  each of three concurrent lines, then the intersections of
+  corresponding sides lie in a line, those sides being called
+  corresponding which are opposite to vertices on the same line.
+
+  The converse theorem holds also, viz.
+
+  _Theorem._--If the sides of one triangle meet those of another in
+  three points which lie in a line, then the vertices lie on three lines
+  which meet in a point.
+
+  The proof is almost the same as before.
+
+  S 38. _Metrical Relations between Projective Rows._--Every row
+  contains one point which is distinguished from all others, viz. the
+  point at infinity. In two projective rows, to the point I at infinity
+  in one corresponds a point I' in the other, and to the point J' at
+  infinity in the second corresponds a point J in the first. The points
+  I' and J are in general finite. If now A and B are any two points in
+  the one, A', B' the corresponding points in the other row, then
+
+    (AB, JI) = (A'B', J'I'),
+
+  or
+
+    AJ/JB : AI/IB = A'J'/J'B' : A'I'/I'B'.
+
+  But, by S 17,
+
+    AI/IB = A'J'/J'B' = -1;
+
+  therefore the last equation changes into
+
+    AJ.A'I' = BJ.B'I',
+
+  that is to say--
+
+  _Theorem._--The product of the distances of any two corresponding
+  points in two projective rows from the points which correspond to the
+  points at infinity in the other is constant, viz. AJ.A'I' = k.
+  Steiner has called this number k the _Power of the correspondence_.
+
+  [The relation AJ . A'I' = k shows that if J, I' be given then the
+  point A' corresponding to a specified point A is readily found; hence
+  A, A' generate homographic ranges of which I and J' correspond to the
+  points at infinity on the ranges. If we take any two origins O, O', on
+  the ranges and reduce the expression AJ . A'I' = k to its algebraic
+  equivalent, we derive an equation of the form [alpha]xx' + [beta]x +
+  [gamma]x' + [delta] = 0. Conversely, if a relation of this nature
+  holds, then points corresponding to solutions in x, x' form
+  homographic ranges.]
+
+  S 39. _Similar Rows._--If the points at infinity in two projective
+  rows correspond so that I' and J are at infinity, this result loses
+  its meaning. But if A, B, C be any three points in one, A', B', C' the
+  corresponding ones on the other row, we have
+
+    (AB, CI) = (A'B', C'I'),
+
+  which reduces to
+
+    AC/CB = A'C'/C'B' or AC/A'C' = BC/B'C',
+
+  that is, corresponding segments are proportional. Conversely, if
+  corresponding segments are proportional, then to the point at infinity
+  in one corresponds the point at infinity in the other. If we call such
+  rows _similar_, we may state the result thus--
+
+  _Theorem._--Two projective rows are similar if to the point at
+  infinity in one corresponds the point at infinity in the other, and
+  conversely, if two rows are similar then they are projective, and the
+  points at infinity are corresponding points.
+
+  From this the well-known propositions follow:--
+
+  Two lines are cut proportionally (in similar rows) by a series of
+  parallels. The rows are perspective, with centre of projection at
+  infinity.
+
+  If two similar rows are placed parallel, then the lines joining
+  homologous points pass through a common point.
+
+  S 40. If two flat pencils be projective, then there exists in either,
+  one single pair of lines at right angles to one another, such that the
+  corresponding lines in the other pencil are again at right angles.
+
+  [Illustration: FIG. 14.]
+
+  To prove this, we place the pencils in perspective position (fig. 14)
+  by making one ray coincident with its corresponding ray. Corresponding
+  rays meet then on a line p. And now we draw the circle which has its
+  centre O on p, and which passes through the centres S and S' of the
+  two pencils. This circle cuts p in two points H and K. The two pairs
+  of rays, h, k, and h', k', joining these points to S and S' will be
+  pairs of corresponding rays at right angles. The construction gives in
+  general but one circle, but if the line p is the perpendicular
+  bisector of SS', there exists an infinite number, and _to every right
+  angle in the one pencil corresponds a right angle in the other_.
+
+
+  PRINCIPLE OF DUALITY
+
+  S 41. It has been stated in S 1 that not only points, but also planes
+  and lines, are taken as elements out of which figures are built up. We
+  shall now see that the construction of one figure which possesses
+  certain properties gives rise in many cases to the construction of
+  another figure, by replacing, according to definite rules, elements of
+  one kind by those of another. The new figure thus obtained will then
+  possess properties which may be stated as soon as those of the
+  original figure are known.
+
+  We obtain thus a principle, known as the _principle of duality_ or of
+  _reciprocity_, which enables us to construct to any figure not
+  containing any measurement in its construction a _reciprocal_ figure,
+  as it is called, and to deduce from any theorem a _reciprocal_
+  theorem, for which no further proof is needed.
+
+  It is convenient to print reciprocal propositions on opposite sides of
+  a page broken into two columns, and this plan will occasionally be
+  adopted.
+
+  We begin by repeating in this form a few of our former statements:--
+
+  Two points determine a line.        Two planes determine a line.
+
+  Three points which are not in a     Three planes which do not pass
+  line determine a plane.             through a line determine a point.
+
+  A line and a point without it       A line and a plane not through
+  determine a plane.                  it determine a point.
+
+  Two lines in a plane determine      Two lines through a point
+  a point.                            determine a plane.
+
+  These propositions show that it will be possible, when any figure is
+  given, to construct a second figure by taking planes instead of
+  points, and points instead of planes, but lines where we had lines.
+
+  For instance, if in the first figure we take a plane and three points
+  in it, we have to take in the second figure a point and three planes
+  through it. The three points in the first, together with the three
+  lines joining them two and two, form a triangle; the three planes in
+  the second and their three lines of intersection form a trihedral
+  angle. A triangle and a trihedral angle are therefore reciprocal
+  figures.
+
+  Similarly, to any figure in a plane consisting of points and lines
+  will correspond a figure consisting of planes and lines passing
+  through a point S, and hence belonging to the pencil which has S as
+  centre.
+
+  The figure reciprocal to four points in space which do not lie in a
+  plane will consist of four planes which do not meet in a point. In
+  this case each figure forms a tetrahedron.
+
+  S 42. As other examples we have the following:--
+
+  To a row                   is reciprocal  an axial pencil,
+
+  " a flat pencil                  "        a flat pencil,
+
+  " a field of points and lines    "        a pencil of planes and lines,
+
+  " the space of points            "        the space of planes.
+
+  For the row consists of a line and all the points in it, reciprocal to
+  it therefore will be a line with all planes through it, that is, an
+  axial pencil; and so for the other cases.
+
+  This correspondence of reciprocity breaks down, however, if we take
+  figures which contain measurement in their construction. For instance,
+  there is no figure reciprocal to two planes at _right angles_, because
+  there is no segment in a row which has a magnitude as definite as a
+  right angle.
+
+  We add a few examples of reciprocal propositions which are easily
+  proved.
+
+  _Theorem._--If A, B, C, D are        _Theorem._--If [alpha], [beta],
+  any four points in space, and if     [gamma], [delta] are four planes
+  the lines AB and CD meet, then       in space, and if the lines
+  all four points lie in a plane,      [alpha][beta] and [gamma][delta]
+  hence also AC and BD, as well        meet, then all four planes lie
+  as AD and BC, meet.                  in a point (pencil), hence also
+                                       [alpha][gamma] and [beta][delta],
+                                       well as [alpha][delta] and
+                                       as [beta][gamma], meet.
+
+  Theorem.--_If of any number of lines every one meets every other,
+  whilst all do not_
+
+  _lie in a point, then all lie in     _lie in a plane, then all lie in
+  a plane._                            a point (pencil)._
+
+  S 43. Reciprocal figures as explained lie both in space of three
+  dimensions. If the one is confined to a plane (is formed of elements
+  which lie in a plane), then the reciprocal figure is confined to a
+  pencil (is formed of elements which pass through a point).
+
+  But there is also a more special principle of duality, according to
+  which figures are reciprocal which lie both in a plane or both in a
+  pencil. In the plane we take points and lines as reciprocal elements,
+  for they have this fundamental property in common, that two elements
+  of one kind determine one of the other. In the pencil, on the other
+  hand, lines and planes have to be taken as reciprocal, and here it
+  holds again that two lines or planes determine one plane or line.
+
+  Thus, to one plane figure we can construct one reciprocal figure in
+  the plane, and to each one reciprocal figure in a pencil. We mention a
+  few of these. At first we explain a few names:--
+
+  A figure consisting of n points     A figure consisting of n lines
+  in a plane will be called an        in a plane will be called an
+  n-point.                            n-side.
+
+  A figure consisting of n planes     A figure consisting of n lines
+  in a pencil will be called an       in a pencil will be called an
+  n-flat.                             n-edge.
+
+  It will be understood that an n-side is different from a polygon of n
+  sides. The latter has sides of finite length and n vertices, the
+  former has sides all of infinite extension, and every point where two
+  of the sides meet will be a vertex. A similar difference exists
+  between a solid angle and an n-edge or an n-flat. We notice
+  particularly--
+
+  A four-point has six sides, of      A four-side has six vertices, of
+  which two and two are opposite,     which two and two are opposite,
+  and three diagonal points, which    and three diagonals, which join
+  are intersections of opposite       opposite vertices.
+  sides.
+
+  A four-flat has six edges, of       A four-edge has six faces, of
+  which two and two are opposite,     which two and two are opposite,
+  and three diagonal planes, which    and three diagonal edges, which
+  pass through opposite edges.        are intersections of opposite
+                                      faces.
+
+  A four-side is usually called a complete quadrilateral, and a
+  four-point a complete quadrangle. The above notation, however, seems
+  better adapted for the statement of reciprocal propositions.
+
+  S 44.
+
+  If a point moves in a plane it      If a line moves in a plane it
+  describes a plane curve.            envelopes a plane curve (fig. 15).
+
+  If a plane moves in a pencil it     If a line moves in a pencil it
+  envelopes a cone.                   describes a cone.
+
+  A curve thus appears as generated either by points, and then we call
+  it a "locus," or by lines, and then we call it an "envelope." In the
+  same manner a cone, which means here a surface, appears either as the
+  locus of lines passing through a fixed point, the "vertex" of the
+  cone, or as the envelope of planes passing through the same point.
+
+  [Illustration: FIG. 15.]
+
+  To a surface as locus of points corresponds, in the same manner, a
+  surface as envelope of planes; and to a curve in space as locus of
+  points corresponds a developable surface as envelope of planes.
+
+  It will be seen from the above that we may, by aid of the principle of
+  duality, construct for every figure a reciprocal figure, and that to
+  any property of the one a reciprocal property of the other will exist,
+  as long as we consider only properties which depend upon nothing but
+  the positions and intersections of the different elements and not upon
+  measurement.
+
+  For such propositions it will therefore be unnecessary to prove more
+  than one of two reciprocal theorems.
+
+
+  GENERATION OF CURVES AND CONES OF SECOND ORDER OR SECOND CLASS
+
+  S 45. _Conics._--If we have two projective pencils in a plane,
+  corresponding rays will meet, and their point of intersection will
+  constitute some locus which we have to investigate. Reciprocally, if
+  two projective rows in a plane are given, then the lines which join
+  corresponding points will envelope some curve. We prove first:--
+
+  _Theorem._--If two projective       _Theorem._--If two projective
+  flat pencils lie in a plane, but    rows lie in a plane, but are
+  are neither in perspective nor      neither in perspective nor on a
+  concentric, then the locus of       common base, then the envelope
+  intersections of corresponding      of lines joining corresponding
+  rays is a curve of the second       points is a curve of the second
+  order, that is, no line contains    class, that is, through no point
+  more than two points of the         pass more than two of the
+  locus.                              enveloping lines.
+
+  Proof.--We draw any line t.         _Proof._--We take any point T
+  This cuts each of the pencils in    and join it to all points in each
+  a row, so that we have on t two     row. This gives two concentric
+  rows, and these are projective      pencils, which are projective
+  because the pencils are             because the rows are projective.
+  projective. If corresponding rays   If a line joining corresponding
+  of the two pencils meet on the      points in the two rows passes
+  line t, their intersection will     through T, it will be a line in
+  be a point in the one row which     the one pencil which coincides
+  coincides with its corresponding    with its corresponding line in
+  point in the other. But two         the other. But two projective
+  projective rows on the same base    concentric flat pencils in the
+  cannot have more than two           same plane cannot have more than
+  points of one coincident with       two lines of one coincident with
+  their corresponding points in       their corresponding line in the
+  the other (S 34).                   other (S 34).
+
+  It will be seen that the proofs are reciprocal, so that the one may be
+  copied from the other by simply interchanging the words point and
+  line, locus and envelope, row and pencil, and so on. We shall
+  therefore in future prove seldom more than one of two reciprocal
+  theorems, and often state one theorem only, the reader being
+  recommended to go through the reciprocal proof by himself, and to
+  supply the reciprocal theorems when not given.
+
+  S 46. We state the theorems in the pencil reciprocal to the last,
+  without proving them:--
+
+  _Theorem._--If two projective       _Theorem._--If two projective
+  flat pencils are concentric, but    axial pencils lie in the same
+  are neither perspective nor         pencil (their axes meet in a
+  coplanar, then the envelope of      point), but are neither perspective
+  the planes joining corresponding    nor co-axial, then the locus
+  rays is a cone of the second        of lines joining corresponding
+  class; that is, no line through     planes is a cone of the second
+  the common centre contains more     order; that is, no plane in the
+  than two of the enveloping          pencil contains more than two
+  planes.
+
+  S 47. Of theorems about cones of second order and cones of second
+  class we shall state only very few. We point out, however, the
+  following connexion between the curves and cones under consideration:
+
+  The lines which join any point      Every plane section of a cone
+  in space to the points on a curve   of the second order is a curve of
+  of the second order form a cone     the second order.
+  of the second order.
+
+  The planes which join any           Every plane section of a cone
+  point in space to the lines         of the second class is a curve of
+  enveloping a curve of the           the second class.
+  second class envelope themselves
+  a cone of the second class.
+
+  By its aid, or by the principle of duality, it will be easy to obtain
+  theorems about them from the theorems about the curves.
+
+  We prove the first. A curve of the second order is generated by two
+  projective pencils. These pencils, when joined to the point in space,
+  give rise to two projective axial pencils, which generate the cone in
+  question as the locus of the lines where corresponding planes meet.
+
+  S48.
+
+  _Theorem._--The curve of second    _Theorem._--The envelope of
+  order which is generated by two    second class which is generated
+  projective flat pencils passes     by two projective rows contains
+  through the centres of the two     the bases of these rows as
+  pencils.                           enveloping lines or tangents.
+
+  _Proof._--If S and S' are the      _Proof._--If s and s' are the
+  two pencils, then to the ray SS'   two rows, then to the point ss'
+  or p' in the pencil S'             or P' as a point in s'
+  corresponds in the pencil S a      corresponds in s a point P,
+  ray p, which is different from     which is not coincident with P',
+  p', for the pencils are not        for the rows are not
+  perspective. But p and p' meet     perspective. But P and P' are
+  at S, so that S is a point on      joined by s, so that s is one of
+  the curve, and similarly S'.       the enveloping lines, and
+                                     similarly s'.
+
+  It follows that every line in one of the two pencils cuts the curve in
+  two points, viz. once at the centre S of the pencil, and once where it
+  cuts its corresponding ray in the other pencil. These two points,
+  however, coincide, if the line is cut by its corresponding line at S
+  itself. The line p in S, which corresponds to the line SS' in S', is
+  therefore the only line through S which has but one point in common
+  with the curve, or which cuts the curve in two coincident points. Such
+  a line is called a _tangent_ to the curve, touching the latter at the
+  point S, which is called the "point of contact."
+
+  In the same manner we get in the reciprocal investigation the result
+  that through every point in one of the rows, say in s, two tangents
+  may be drawn to the curve, the one being s, the other the line joining
+  the point to its corresponding point in s'. There is, however, one
+  point P in s for which these two lines coincide. Such a point in one
+  of the tangents is called the "point of contact" of the tangent. We
+  thus get--
+
+  _Theorem._--To the line joining    _Theorem._--To the point of
+  the centres of the projective      intersection of the bases of two
+  pencils as a line in one pencil    projective rows as a point in
+  corresponds in the other the       one row corresponds in the other
+  tangent at its centre.             the _point of contact_ of its
+                                     base.
+
+  S 49. Two projective pencils are determined if three pairs of
+  corresponding lines are given. Hence if a1, b1, c1 are three lines in
+  a pencil S1, and a2, b2, c2 the corresponding lines in a projective
+  pencil S2, the correspondence and therefore the curve of the second
+  order generated by the points of intersection of corresponding rays is
+  determined. Of this curve we know the two centres S1 and S2, and the
+  three points a1a2, b1b2, c1c2, hence five points in all. This and the
+  reciprocal considerations enable us to solve the following two
+  problems:
+
+  _Problem._--To construct a curve   _Problem._--To construct a curve
+  of the second order, of which      of the second class, of which
+  five points S1, S2, A, B, C are    five tangents u1, u2, a, b, c
+  given.                             are given.
+
+  In order to solve the left-hand problem, we take two of the given
+  points, say S1 and S2, as centres of pencils. These we make projective
+  by taking the rays a1, b1, c1, which join S1 to A, B, C respectively,
+  as corresponding to the rays a2, b2, c2, which join S2 to A, B, C
+  respectively, so that three rays meet their corresponding rays at the
+  given points A, B, C. This determines the correspondence of the
+  pencils which will generate a curve of the second order passing
+  through A, B, C and through the centres S1 and S2, hence through the
+  five given points. To find more points on the curve we have to
+  construct for any ray in S1 the corresponding ray in S2. This has been
+  done in S 36. But we repeat the construction in order to deduce
+  further properties from it. We also solve the right-hand problem. Here
+  we select two, viz. u1, u2 of the five given lines, u1, u2, a, b, c,
+  as bases of two rows, and the points A1, B1, C1 where a, b, c cut u1
+  as corresponding to the points A2, B2, C2 where a, b, c cut u2.
+
+  We get then the following solutions of the two problems:
+
+  _Solution._--Through the point A   _Solution._--In the line a take
+  draw any two lines, u1 and u2      any two points S1 and S2 as
+  (fig. 16), the first u1 to cut     centres of pencils (fig. 17),
+  the pencil S1 in a row AB1C1,      the first S1 (A1B1C1) to project
+  the other u2 to cut the pencil     the row u1, the other S2
+  S2 in a row AB2C2. These two       (A2B2C2) to project the row u2.
+  rows will be perspective, as the   These two pencils will be
+  point A corresponds to itself,     perspective, the line S1A1 being
+  and the centre of projection       the same as the corresponding
+  will be the point S, where the     line S2A2, and the axis of
+  lines B1B2 and C1C2 meet. To       projection will be the line u,
+  find now for any ray d1 in S1      which joins the intersection B
+  its corresponding ray d2 in S2,    of S1B1 and S2B2 to the
+  we determine the point D1 where    intersection C of S1C1 and S2C2.
+  d1 cuts u1, project this point     To find now for any point D1 in
+  from S to D2 on u2 and join S2     u1 the corresponding point D2 in
+  to D2. This will be the required   u2, we draw S1D1 and project the
+  ray d2 which cuts d1 at some       point D where this line cuts u
+  point D on the curve.              from S2 to u2. This will give
+                                     the required point D2, and the
+                                     line d joining D1 to D2 will be
+                                     a new tangent to the curve.
+
+  S 50. These constructions prove, when rightly interpreted, very
+  important properties of the curves in question.
+
+  [Illustration: FIG. 16.]
+
+  If in fig. 16 we draw in the pencil S1 the ray k1 which passes through
+  the auxiliary centre S, it will be found that the corresponding ray k2
+  cuts it on u2. Hence--
+
+  _Theorem._--In the above           _Theorem._--In the above
+  construction the bases of the      construction (fig. 17) the
+  auxiliary rows u1 and u2 cut the   tangents to the curve from the
+  curve where they cut the rays      centres of the auxiliary pencils
+  S2S and S1S respectively.          S1 and S2 are the lines which
+                                     pass through u2u and u1u
+                                     respectively.
+
+  As A is any given point on the curve, and u1 any line through it, we
+  have solved the problems:
+
+  _Problem._--To find the second     _Problem._--To find the second
+  point in which any line through    tangent which can be drawn from
+  a known point on the curve cuts    any point in a given tangent to
+  the curve.                         the curve.
+
+  If we determine in S1 (fig. 16) the ray corresponding to the ray S2S1
+  in S2, we get the tangent at S1. Similarly, we can determine the point
+  of contact of the tangents u1 or u2 in fig. 17.
+
+  [Illustration: FIG. 17.]
+
+  S 51. If five points are given, of which not three are in a line, then
+  we can, as has just been shown, always draw a curve of the second
+  order through them; we select two of the points as centres of
+  projective pencils, and then one such curve is determined. It will be
+  presently shown that we get always the same curve if two other points
+  are taken as centres of pencils, that therefore five points
+  _determine_ one curve of the second order, and reciprocally, that five
+  tangents determine one curve of the second class. Six points taken at
+  random will therefore not lie on a curve of the second order. In order
+  that this may be the case a certain condition has to be satisfied, and
+  this condition is easily obtained from the construction in S 49, fig.
+  16. If we consider the conic determined by the five points A, S1, S2,
+  K, L, then the point D will be on the curve if, and only if, the
+  points on D1, S, D2 be in a line.
+
+  [Illustration: FIG. 18.]
+
+  This may be stated differently if we take AKS1DS2L (figs. 16 and 18)
+  as a hexagon inscribed in the conic, then AK and DS2 will be opposite
+  sides, so will be KS1 and S2L, as well as S1D and LA. The first two
+  meet in D2, the others in S and D1 respectively. We may therefore
+  state the required condition, together with the reciprocal one, as
+  follows:--
+
+  _Pascal's Theorem._--If a hexagon  _Brianchon's Theorem._--If a
+  be inscribed in a curve of the     hexagon be circumscribed about
+  second order, then the             a curve of the second class, then
+  intersectionsof opposite sides     the lines joining opposite vertices
+  are three points in a line.        are three lines meeting in a point.
+
+  These celebrated theorems, which are known by the names of their
+  discoverers, are perhaps the most fruitful in the whole theory of
+  conics. Before we go over to their applications we have to show that
+  we obtain the same curve if we take, instead of S1, S2, any two other
+  points on the curve as centres of projective pencils.
+
+  S 52. We know that the curve depends only upon the correspondence
+  between the pencils S1 and S2, and not upon the special construction
+  used for finding new points on the curve. The point A (fig. 16 or 18),
+  through which the two auxiliary rows u1, u2 were drawn, may therefore
+  be changed to any other point on the curve. Let us now suppose the
+  curve drawn, and keep the points S1, S2, K, L and D, and hence also
+  the point S fixed, whilst we move A along the curve. Then the line AL
+  will describe a pencil about L as centre, and the point D1 a row on
+  S1D perspective to the pencil L. At the same time AK describes a
+  pencil about K and D2 a row perspective to it on S2D. But by Pascal's
+  theorem D1 and D2 will always lie in a line with S, so that the rows
+  described by D1 and D2 are perspective. It follows that the pencils K
+  and L will themselves be projective, corresponding rays meeting on the
+  curve. This proves that we get the same curve whatever pair of the
+  five given points we take as centres of projective pencils. Hence--
+
+  Only one curve of the second       Only one curve of the second
+  order can be drawn which passes    class can be drawn which touches
+  through five given points.         five given lines.
+
+  We have seen that if on a curve of the second order two points
+  coincide at A, the line joining them becomes the tangent at A. If,
+  therefore, a point on the curve and its tangent are given, this will
+  be equivalent to having given two points on the curve. Similarly, if
+  on the curve of second class a tangent and its point of contact are
+  given, this will be equivalent to two given tangents.
+
+  We may therefore extend the last theorem:
+
+  Only one curve of the second       Only one curve of the second
+  order can be drawn, of which       class can be drawn, of which four
+  four points and the tangent at     tangents and the point of contact
+  oneof them, or three points        at one of them, or three tangents
+  and the tangents at two of         and the points of contact at two
+  them, are given.                   of them, are given.
+
+  S 53. At the same time it has been proved:
+
+  If all points on a curve of the    All tangents to a curve of second
+  second order be joined to any      class are cut by any two of
+  two of them, then the two          them in projective rows, those
+  pencils thus formed are            being corresponding points which
+  projective, those rays being       lie on the same tangent. Hence--
+  corresponding which meet on the
+  curve. Hence--
+
+  The cross-ratio of four rays       The cross-ratio of the four
+  joining a point S on a curve of    points in which any tangent u is
+  second order to four fixed         cut by four fixed tangents a, b, c,
+  points A, B, C, D in the curve     d is independent of the position of
+  is independent of the position     u, and is called the cross-ratio of
+  of S, and is called the cross-     the four tangents a, b, c, d.
+  ratio of the four points A, B,
+  C, D.
+
+  If this cross-ratio equals -1     If this cross-ratio equals -1
+  the four points are said to be    the four tangents are said to be
+  four harmonic points.             four harmonic tangents.
+
+  We have seen that a curve of second order, as generated by projective
+  pencils, has at the centre of each pencil one tangent; and further,
+  that any point on the curve may be taken as centre of such pencil.
+  Hence--
+
+  A curve of second order has       A curve of second class has on
+  at every point one tangent.       every tangent a point of contact.
+
+  S 54. We return to Pascal's and Brianchon's theorems and their
+  applications, and shall, as before, state the results both for curves
+  of the second order and curves of the second class, but prove them
+  only for the former.
+
+  Pascal's theorem may be used when five points are given to find more
+  points on the curve, viz. it enables us to find the point where any
+  line through one of the given points cuts the curve again. It is
+  convenient, in making use of Pascal's theorem, to number the points,
+  to indicate the order in which they are to be taken in forming a
+  hexagon, which, by the way, may be done in 60 different ways. It will
+  be seen that 1 2 (leaving out 3) 4 5 are opposite sides, so are 2 3
+  and (leaving out 4) 5 6, and also 3 4 and (leaving out 5) 6 1.
+
+  If the points 1 2 3 4 5 are given, and we want a 6th point on a line
+  drawn through 1, we know all the sides of the hexagon with the
+  exception of 5 6, and this is found by Pascal's theorem.
+
+  If this line should happen to pass through 1, then 6 and 1 coincide,
+  or the line 6 1 is the tangent at 1. And always if two consecutive
+  vertices of the hexagon approach nearer and nearer, then the side
+  joining them will ultimately become a tangent.
+
+  We may therefore consider a pentagon inscribed in a curve of second
+  order and the tangent at one of its vertices as a hexagon, and thus
+  get the theorem:
+
+  Every pentagon inscribed in a      Every pentagon circumscribed
+  curve of second order has the      about a curve of the second class
+  property that the intersections    has the property that the lines
+  of two pairs of non-consecutive    which join two pairs of non-
+  sides lie in a line with the       consecutive vertices meet on that
+  point where the fifth side cuts    line which joins the fifth vertex
+  the tangent at the opposite        to the point of contact of the
+  vertex.                            opposite side.
+
+  This enables us also to solve the following problems.
+
+  Given five points on a curve of    Given five tangents to a curve
+  second order to construct the      of second class to construct the
+  tangent at any one of them.        point of contact of any one of
+                                     them.
+
+  [Illustration: FIG. 19.]
+
+  If two pairs of adjacent vertices coincide, the hexagon becomes a
+  quadrilateral, with tangents at two vertices. These we take to be
+  opposite, and get the following theorems:
+
+  If a quadrilateral be inscribed    If a quadrilateral be circumscribed
+  in a curve of second order, the    about a curve of second
+  intersections of opposite sides,   class, the lines joining opposite
+  and also the intersections of      vertices, and also the lines joining
+  the tangents at opposite           points of contact of opposite
+  vertices, lie in a line (fig.      sides, meet in a point.
+  19).
+
+  [Illustration: FIG. 20.]
+
+  If we consider the hexagon made up of a triangle and the tangents at
+  its vertices, we get--
+
+  If a triangle is inscribed in a    If a triangle be circumscribed
+  curve of the second order, the     about a curve of second class,
+  points in which the sides are      the lines which join the vertices
+  cut by the tangents at the         to the points of contact of the
+  opposite vertices meet in a        opposite sides meet in a point
+  point.                             (fig. 20).
+
+  S 55. Of these theorems, those about the quadrilateral give rise to a
+  number of others. Four points A, B, C, D may in three different ways
+  be formed into a quadrilateral, for we may take them in the order
+  ABCD, or ACBD, or ACDB, so that either of the points B, C, D may be
+  taken as the vertex opposite to A. Accordingly we may apply the
+  theorem in three different ways.
+
+  Let A, B, C, D be four points on a curve of second order (fig. 21),
+  and let us take them as forming a quadrilateral by taking the points
+  in the order ABCD, so that A, C and also B, D are pairs of opposite
+  vertices. Then P, Q will be the points where opposite sides meet, and
+  E, F the intersections of tangents at opposite vertices. The four
+  points P, Q, E, F lie therefore in a line. The quadrilateral ACBD
+  gives us in the same way the four points Q, R, G, H in a line, and the
+  quadrilateral ABDC a line containing the four points R, P, I, K. These
+  three lines form a triangle PQR.
+
+  The relation between the points and lines in this figure may be
+  expressed more clearly if we consider ABCD as a four-point inscribed
+  in a conic, and the tangents at these points as a four-side
+  circumscribed about it,--viz. it will be seen that P, Q, R are the
+  diagonal points of the four-point ABCD, whilst the sides of the
+  triangle PQR are the diagonals of the circumscribing four-side. Hence
+  the theorem--
+
+  _Any four-point on a curve of the second order and the four-side
+  formed by the tangents at these points stand in this relation that the
+  diagonal points of the four-point lie in the diagonals of the
+  four-side._ And conversely,
+
+  _If a four-point and a circumscribed four-side stand in the above
+  relation, then a curve of the second order may be described which
+  passes through the four points and touches there the four sides of
+  these figures._
+
+  That the last part of the theorem is true follows from the fact that
+  the four points A, B, C, D and the line a, as tangent at A, determine
+  a curve of the second order, and the tangents to this curve at the
+  other points B, C, D are given by the construction which leads to fig.
+  21.
+
+  [Illustration: FIG. 21.]
+
+  The theorem reciprocal to the last is--
+
+  _Any four-side circumscribed about a curve of second class and the
+  four-point formed by the points of contact stand in this relation that
+  the diagonals of the four-side pass through the diagonal points of the
+  four-point._ And conversely,
+
+  _If a four-side and an inscribed four-point stand in the above
+  relation, then a curve of the second class may be described which
+  touches the sides of the four-side at the points of the four-point._
+
+  S 56. The four-point and the four-side in the two reciprocal theorems
+  are alike. Hence if we have a four-point ABCD and a four-side abcd
+  related in the manner described, then not only may a curve of the
+  second order be drawn, but also a curve of the second class, which
+  both touch the lines a, b, c, d at the points A, B, C, D.
+
+  The curve of second order is already more than determined by the
+  points A, B, C and the tangents a, b, c at A, B and C. The point D may
+  therefore be _any_ point on this curve, and d any tangent to the
+  curve. On the other hand the curve of the second class is more than
+  determined by the three tangents a, b, c and their points of contact
+  A, B, C, so that d is any tangent to this curve. It follows that every
+  tangent to the curve of second order is a tangent of a curve of the
+  second class having the same point of contact. In other words, the
+  curve of second order is a curve of second class, and _vice versa_.
+  Hence the important theorems--
+
+  _Every curve of second order is    _Every curve of second class is a
+  a curve of second class._          curve of second order._
+
+  The curves of second order and of second class, having thus been
+  proved to be identical, shall henceforth be called by the common name
+  of _Conics_.
+
+  For these curves hold, therefore, all properties which have been
+  proved for curves of second order or of second class. We may therefore
+  now state Pascal's and Brianchon's theorem thus--
+
+  _Pascal's Theorem._--If a hexagon be inscribed in a conic, then the
+  intersections of opposite sides lie in a line.
+
+  _Brianchon's Theorem._--If a hexagon be circumscribed about a conic,
+  then the diagonals forming opposite centres meet in a point.
+
+  S 57. If we suppose in fig. 21 that the point D together with the
+  tangent d moves along the curve, whilst A, B, C and their tangents a,
+  b, c remain fixed, then the ray DA will describe a pencil about A, the
+  point Q a projective row on the fixed line BC, the point F the row b,
+  and the ray EF a pencil about E. But EF passes always through Q. Hence
+  the pencil described by AD is projective to the pencil described by
+  EF, and therefore to the row described by F on b. At the same time the
+  line BD describes a pencil about B projective to that described by AD
+  (S 53). Therefore the pencil BD and the row F on b are projective.
+  Hence--
+
+  _If on a conic a point A be taken and the tangent a at this point,
+  then the cross-ratio of the four rays which join A to any four points
+  on the curve is equal to the cross-ratio of the points in which the
+  tangents at these points cut the tangent at A._
+
+  S 58. There are theorems about cones of second order and second class
+  in a pencil which are reciprocal to the above, according to S 43. We
+  mention only a few of the more important ones.
+
+  The locus of intersections of corresponding planes in two projective
+  axial pencils whose axes meet is a cone of the second order.
+
+  The envelope of planes which join corresponding lines in two
+  projective flat pencils, not in the same plane, is a cone of the
+  second class.
+
+  Cones of second order and cones of second class are identical.
+
+  Every plane cuts a cone of the second order in a conic.
+
+  _A cone of second order is uniquely determined by five of its edges or
+  by five of its tangent planes, or by four edges and the tangent plane
+  at one of them, &c. &c._
+
+  _Pascal's Theorem._--If a solid angle of six faces be inscribed in a
+  cone of the second order, then the intersections of opposite faces are
+  three lines in a plane.
+
+  _Brianchon's Theorem._--If a solid angle of six edges be circumscribed
+  about a cone of the second order, then the planes through opposite
+  edges meet in a line.
+
+  Each of the other theorems about conics may be stated for cones of the
+  second order.
+
+  S 59. _Projective Definitions of the Conics._--We now consider the
+  shape of the conics. We know that any line in the plane of the conic,
+  and hence that the line at infinity, either has no point in common
+  with the curve, or one (counting for two coincident points) or two
+  distinct points. If the line at infinity has no point on the curve the
+  latter is altogether finite, and is called an _Ellipse_ (fig. 21). If
+  the line at infinity has only one point in common with the conic, the
+  latter extends to infinity, and has the line at infinity a tangent. It
+  is called a _Parabola_ (fig. 22). If, lastly, the line at infinity
+  cuts the curve in two points, it consists of two separate parts which
+  each extend in two branches to the points at infinity where they meet.
+  The curve is in this case called an _Hyperbola_ (see fig. 20). The
+  tangents at the two points at infinity are finite because the line at
+  infinity is not a tangent. They are called _Asymptotes_. The branches
+  of the hyperbola approach these lines indefinitely as a point on the
+  curves moves to infinity.
+
+  [Illustration: FIG. 22.]
+
+  S 60. That the circle belongs to the curves of the second order is
+  seen at once if we state in a slightly different form the theorem that
+  in a circle all angles at the circumference standing upon the same arc
+  are equal. If two points S1, S2 on a circle be joined to any other two
+  points A and B on the circle, then the angle included by the rays S1A
+  and S1B is equal to that between the rays S2A and S2B, so that as A
+  moves along the circumference the rays S1A and S2A describe equal and
+  therefore projective pencils. The circle can thus be generated by two
+  projective pencils, and is a curve of the second order.
+
+  If we join a point in space to all points on a circle, we get a
+  (circular) cone of the second order (S 43). Every plane section of
+  this cone is a conic. This conic will be an ellipse, a parabola, or an
+  hyperbola, according as the line at infinity in the plane has no, one
+  or two points in common with the conic in which the plane at infinity
+  cuts the cone. It follows that our curves of second order may be
+  obtained as sections of a circular cone, and that they are identical
+  with the "Conic Sections" of the Greek mathematicians.
+
+  S 61. Any two tangents to a parabola are cut by all others in
+  projective rows; but the line at infinity being one of the tangents,
+  the points at infinity on the rows are corresponding points, and the
+  rows therefore similar. Hence the theorem--
+
+  _The tangents to a parabola cut each other proportionally._
+
+
+  POLE AND POLAR
+
+  S 62. We return once again to fig. 21, which we obtained in S 55.
+
+  If a four-side be circumscribed about and a four-point inscribed in a
+  conic, so that the vertices of the second are the points of contact of
+  the sides of the first, then the triangle formed by the diagonals of
+  the first is the same as that formed by the diagonal points of the
+  other.
+
+  Such a triangle will be called a _polar-triangle_ of the conic, so
+  that PQR in fig. 21 is a polar-triangle. It has the property that on
+  the side p opposite P meet the tangents at A and B, and also those at
+  C and D. From the harmonic properties of four-points and four-sides it
+  follows further that the points L, M, where it cuts the lines AB and
+  CD, are harmonic conjugates with regard to AB and CD respectively.
+
+  If the point P is given, and we draw a line through it, cutting the
+  conic in A and B, then the point Q harmonic conjugate to P with regard
+  to AB, and the point H where the tangents at A and B meet, are
+  determined. But they lie both on p, and therefore this line is
+  determined. If we now draw a second line through P, cutting the conic
+  in C and D, then the point M harmonic conjugate to P with regard to
+  CD, and the point G where the tangents at C and D meet, must also lie
+  on p. As the first line through P already determines p, the second may
+  be any line through P. Now every two lines through P determine a
+  four-point ABCD on the conic, and therefore a polar-triangle which has
+  one vertex at P and its opposite side at p. This result, together with
+  its reciprocal, gives the theorems--
+
+  _All polar-triangles which have one vertex in common have also the
+  opposite side in common._
+
+  _All polar-triangles which have one side in common have also the
+  opposite vertex in common._
+
+  S 63. To any point P in the plane of, but not on, a conic corresponds
+  thus one line p as the side opposite to P in all polar-triangles which
+  have one vertex at P, and reciprocally to every line p corresponds one
+  point P as the vertex opposite to p in all triangles which have p as
+  one side.
+
+  We call the line p the _polar_ of P, and the point P the _pole_ of the
+  line p with regard to the conic.
+
+  If a point lies on the conic, we call the tangent at that point its
+  polar; and reciprocally we call the point of contact the pole of
+  tangent.
+
+  S 64. From these definitions and former results follow--
+
+  The polar of any point P not       The pole of any line p not a
+  on the conic is a line p, which    tangent to the conic is a point
+  has the following properties:--    P, which has the following
+                                     properties:--
+
+  1. On every line through P         1. Of all lines through a point
+  which cuts the conic, the polar    on p from which two tangents
+  of P contains the harmonic         may be drawn to the conic, the
+  conjugate  of P with regard to     pole P contains the line which is
+  those points on the conic.         harmonic conjugate to p, with
+                                     regard to the two tangents.
+
+  2. If tangents can be drawn        2. If p cuts the conic, the
+  from P, their points of contact    tangents at the intersections
+  lie on p.                          meet at P.
+
+  3. Tangents drawn at the           3. The point of contact of
+  points where any line through P    tangents drawn from any point
+  cuts the conic meet on p; and      on p to the conic lie in a line
+  conversely,                        with P; and conversely,
+
+  4. If from any point on p,         4. Tangents drawn at points
+  tangents be drawn, their points    where any line through P cuts the
+  of contact will lie in a line      conic meet on p.
+  with P.
+
+  5. Any four-point on the conic     5. Any four-side circumscribed
+  which has one diagonal point at    about a conic which has one
+  P has the other two lying on p.    diagonal on p has the other two
+                                     meeting at P.
+
+  The truth of 2 follows from 1. If T be a point where p cuts the conic,
+  then one of the points where PT cuts the conic, and which are harmonic
+  conjugates with regard to PT, coincides with T; hence the other
+  does--that is, PT touches the curve at T.
+
+  That 4 is true follows thus: If we draw from a point H on the polar
+  one tangent a to the conic, join its point of contact A to the pole P,
+  determine the second point of intersection B of this line with the
+  conic, and draw the tangent at B, it will pass through H, and will
+  therefore be the second tangent which may be drawn from H to the
+  curve.
+
+  S 65. The second property of the polar or pole gives rise to the
+  theorem--
+
+  From a point in the plane of a     A line in the plane of a conic
+  conic, two, one or no tangents     has two, one or no points in
+  may be drawn to the conic,         common with the conic, according
+  as its polar has two,              as two, one or no tangents
+  one, or no points in common        can be drawn from its pole to the
+  with the curve.                    conic.
+
+  Of any point in the plane of a conic we say that it was _without_, on
+  or _within_ the curve according as two, one or no tangents to the
+  curve pass through it. The points on the conic separate those within
+  the conic from those without. That this is true for a circle is known
+  from elementary geometry. That it also holds for other conics follows
+  from the fact that every conic may be considered as the projection of
+  a circle, which will be proved later on.
+
+  The fifth property of pole and polar stated in S 64 shows how to find
+  the polar of any point and the pole of any line by aid of the
+  straight-edge only. Practically it is often convenient to draw three
+  secants through the pole, and to determine only one of the diagonal
+  points for two of the four-points formed by pairs of these lines and
+  the conic (fig. 22).
+
+  These constructions also solve the problem--
+
+  From a point without a conic, to draw the two tangents to the conic by
+  aid of the straight-edge only.
+
+  For we need only draw the polar of the point in order to find the
+  points of contact.
+
+  S 66. The property of a polar-triangle may now be stated thus--
+
+  In a polar-triangle each side is the polar of the opposite vertex, and
+  each vertex is the pole of the opposite side.
+
+  [Illustration: FIG. 23.]
+
+  If P is one vertex of a polar-triangle, then the other vertices, Q and
+  R, lie on the polar p of P. One of these vertices we may choose
+  arbitrarily. For if from any point Q on the polar a secant be drawn
+  cutting the conic in A and D (fig. 23), and if the lines joining these
+  points to P cut the conic again at B and C, then the line BC will pass
+  through Q. Hence P and Q are two of the vertices on the polar-triangle
+  which is determined by the four-point ABCD. The third vertex R lies
+  also on the line p. It follows, therefore, also--
+
+  _If Q is a point on the polar of P, then P is a point on the polar of
+  Q_; and reciprocally,
+
+  _If q is a line through the pole of p, then p is a line through the
+  pole of q._
+
+  This is a very important theorem. It may also be stated thus--
+
+  _If a point moves along a line describing a row, its polar turns about
+  the pole of the line describing a pencil._
+
+  _This pencil is projective to the row, so that the cross-ratio of four
+  poles in a row equals the cross-ratio of its four polars, which pass
+  through the pole of the row._
+
+  To prove the last part, let us suppose that P, A and B in fig. 23
+  remain fixed, whilst Q moves along the polar p of P. This will make CD
+  turn about P and move R along p, whilst QD and RD describe projective
+  pencils about A and B. Hence Q and R describe projective rows, and
+  hence PR, which is the polar of Q, describes a pencil projective to
+  either.
+
+  S 67. Two points, of which one, and therefore each, lies on the polar
+  of the other, are said to be _conjugate with regard to the conic_; and
+  two lines, of which one, and therefore each, passes through the pole
+  of the other, are said to be _conjugate with regard to the conic_.
+  Hence all points conjugate to a point P lie on the polar of P; all
+  lines conjugate to a line p pass through the pole of p.
+
+  If the line joining two conjugate poles cuts the conic, then the poles
+  are harmonic conjugates with regard to the points of intersection;
+  hence one lies within the other without the conic, and all points
+  conjugate to a point within a conic lie without it.
+
+  Of a polar-triangle any two vertices are conjugate poles, any two
+  sides conjugate lines. If, therefore, one side cuts a conic, then one
+  of the two vertices which lie on this side is within and the other
+  without the conic. The vertex opposite this side lies also without,
+  for it is the pole of a line which cuts the curve. In this case
+  therefore one vertex lies within, the other two without. If, on the
+  other hand, we begin with a side which does not cut the conic, then
+  its pole lies within and the other vertices without. Hence--
+
+  Every polar-triangle has one and only one vertex within the conic.
+
+  We add, without a proof, the theorem--
+
+  The four points in which a conic is cut by two conjugate polars are
+  four harmonic points in the conic.
+
+  S 68. If two conics intersect in four points (they cannot have more
+  points in common, S 52), there exists one and only one four-point
+  which is inscribed in both, and therefore one polar-triangle common to
+  both.
+
+  _Theorem._--Two conics which intersect in four points have always one
+  and only one common polar-triangle; and reciprocally,
+
+  Two conics which have four common tangents have always one and only
+  one common polar-triangle.
+
+
+  DIAMETERS AND AXES OF CONICS
+
+  S 69. _Diameters._--The theorems about the harmonic properties of
+  poles and polars contain, as special cases, a number of important
+  metrical properties of conics. These are obtained if either the pole
+  or the polar is moved to infinity,--it being remembered that the
+  harmonic conjugate to a point at infinity, with regard to two points
+  A, B, is the middle point of the segment AB. The most important
+  properties are stated in the following theorems:--
+
+  _The middle points of parallel chords of a conic lie in a line--viz.
+  on the polar to the point at infinity on the parallel chords._
+
+  This line is called a _diameter_.
+
+  _The polar of every point at infinity is a diameter._
+
+  _The tangents at the end points of a diameter are parallel, and are
+  parallel to the chords bisected by the diameter._
+
+  _All diameters pass through a common point, the pole of the line at
+  infinity._
+
+  _All diameters of a parabola are parallel_, the pole to the line at
+  infinity being the point where the curve touches the line at
+  infinity.
+
+  In case of the ellipse and hyperbola, the pole to the line at infinity
+  is a finite point called the _centre_ of the curve.
+
+  _A centre of a conic bisects every chord through it._
+
+  _The centre of an ellipse is within the curve_, for the line at
+  infinity does not cut the ellipse.
+
+  _The centre of an hyperbola is without the curve_, because the line at
+  infinity cuts the curve. Hence also--
+
+  _From the centre of an hyperbola two tangents can be drawn to the
+  curve which have their point of contact at infinity._ These are called
+  _Asymptotes_ (S 59).
+
+  _To construct a diameter_ of a conic, draw two parallel chords and
+  join their middle points.
+
+  _To find the centre_ of a conic, draw two diameters; their
+  intersection will be the centre.
+
+  S 70. _Conjugate Diameters._--A polar-triangle with one vertex at the
+  centre will have the opposite side at infinity. The other two sides
+  pass through the centre, and are called _conjugate diameters_, each
+  being the polar of the point at infinity on the other.
+
+  _Of two conjugate diameters each bisects the chords parallel to the
+  other, and if one cuts the curve, the tangents at its ends are
+  parallel to the other diameter._
+
+  Further--
+
+  _Every parallelogram inscribed in a conic has its sides parallel to
+  two conjugate diameters_; and
+
+  _Every parallelogram circumscribed about a conic has as diagonals two
+  conjugate diameters._
+
+  This will be seen by considering the parallelogram in the first case
+  as an inscribed four-point, in the other as a circumscribed four-side,
+  and determining in each case the corresponding polar-triangle. The
+  first may also be enunciated thus--
+
+  _The lines which join any point on an ellipse or an hyperbola to the
+  ends of a diameter are parallel to two conjugate diameters._
+
+  S 71. _If every diameter is perpendicular to its conjugate the conic
+  is a circle._
+
+  For the lines which join the ends of a diameter to any point on the
+  curve include a right angle.
+
+  _A conic which has more than one pair of conjugate diameters at right
+  angles to each other is a circle._
+
+  [Illustration: FIG. 24.]
+
+  Let AA' and BB' (fig. 24) be one pair of conjugate diameters at right
+  angles to each other, CC and DD' a second pair. If we draw through the
+  end point A of one diameter a chord AP parallel to DD', and join P to
+  A', then PA and PA' are, according to S 70, parallel to two conjugate
+  diameters. But PA is parallel to DD', hence PA' is parallel to CC, and
+  therefore PA and PA' are perpendicular. If we further draw the
+  tangents to the conic at A and A', these will be perpendicular to AA',
+  they being parallel to the conjugate diameter BB'. We know thus five
+  points on the conic, viz. the points A and A' with their tangents, and
+  the point P. Through these a circle may be drawn having AA' as
+  diameter; and as through five points one conic only can be drawn, this
+  circle must coincide with the given conic.
+
+  S 72. _Axes._--Conjugate diameters perpendicular to each other are
+  called _axes_, and the points where they cut the curve _vertices_ of
+  the conic.
+
+  In a circle every diameter is an axis, every point on it is a vertex;
+  and any two lines at right angles to each other may be taken as a pair
+  of axes of any circle which has its centre at their intersection.
+
+  [Illustration: FIG. 25.]
+
+  If we describe on a diameter AB of an ellipse or hyperbola a circle
+  concentric to the conic, it will cut the latter in A and B (fig. 25).
+  Each of the semicircles in which it is divided by AB will be partly
+  within, partly without the curve, and must cut the latter therefore
+  again in a point. The circle and the conic have thus four points A, B,
+  C, D, and therefore one polar-triangle, in common (S 68). Of this the
+  centre is one vertex, for the line at infinity is the polar to this
+  point, both with regard to the circle and the other conic. The other
+  two sides are conjugate diameters of both, hence perpendicular to each
+  other. This gives--
+
+  An ellipse as well as an hyperbola has one pair of axes.
+
+  This reasoning shows at the same time _how to construct the axis of an
+  ellipse or of an hyperbola_.
+
+  _A parabola has one axis_, if we define an axis as a diameter
+  perpendicular to the chords which it bisects. It is easily
+  constructed. The line which bisects any two parallel chords is a
+  diameter. Chords perpendicular to it will be bisected by a parallel
+  diameter, and this is the axis.
+
+  S 73. The first part of the right-hand theorem in S 64 may be stated
+  thus: any two conjugate lines through a point P without a conic are
+  harmonic conjugates with regard to the two tangents that may be drawn
+  from P to the conic.
+
+  If we take instead of P the centre C of an hyperbola, then the
+  conjugate lines become conjugate diameters, and the tangents
+  asymptotes. Hence--
+
+  _Any two conjugate diameters of an hyperbola are harmonic conjugates
+  with regard to the asymptotes._
+
+  As the axes are conjugate diameters at right angles to one another, it
+  follows (S 23)--
+
+  _The axes of an hyperbola bisect the angles between the asymptotes._
+
+  [Illustration: FIG. 26.]
+
+  Let O be the centre of the hyperbola (fig. 26), t any secant which
+  cuts the hyperbola in C, D and the asymptotes in E, F, then the line
+  OM which bisects the chord CD is a diameter conjugate to the diameter
+  OK which is parallel to the secant t, so that OK and OM are harmonic
+  with regard to the asymptotes. The point M therefore bisects EF. But
+  by construction M bisects CD. It follows that DF = EC, and ED = CF; or
+
+  _On any secant of an hyperbola the segments between the curve and the
+  asymptotes are equal._
+
+  If the chord is changed into a tangent, this gives--
+
+  _The segment between the asymptotes on any tangent to an hyperbola is
+  bisected by the point of contact._
+
+  The first part allows a simple solution of the problem to find any
+  number of points on an hyperbola, of which the asymptotes and one
+  point are given. This is equivalent to three points and the tangents
+  at two of them. This construction requires measurement.
+
+  S 74. For the parabola, too, follow some metrical properties. A
+  diameter PM (fig. 27) bisects every chord conjugate to it, and the
+  pole P of such a chord BC lies on the diameter. But a diameter cuts
+  the parabola once at infinity. Hence--
+
+  _The segment PM which joins the middle point M of a chord of a
+  parabola to the pole P of the chord is bisected by the parabola at A._
+
+  S 75. Two asymptotes and any two tangents to an hyperbola may be
+  considered as a quadrilateral circumscribed about the hyperbola. But
+  in such a quadrilateral the intersections of the diagonals and the
+  points of contact of opposite sides lie in a line (S 54). If therefore
+  DEFG (fig. 28) is such a quadrilateral, then the diagonals DF and GE
+  will meet on the line which joins the points of contact of the
+  asymptotes, that is, on the line at infinity; hence they are parallel.
+  From this the following theorem is a simple deduction:
+
+  _All triangles formed by a tangent and the asymptotes of an hyperbola
+  are equal in area._
+
+  [Illustration: FIG. 27.]
+
+  [Illustration: FIG. 28.]
+
+  If we draw at a point P (fig. 28) on an hyperbola a tangent, the part
+  HK between the asymptotes is bisected at P. The parallelogram PQOQ'
+  formed by the asymptotes and lines parallel to them through P will be
+  half the triangle OHK, and will therefore be constant. If we now take
+  the asymptotes OX and OY as oblique axes of co-ordinates, the lines OQ
+  and QP will be the co-ordinates of P, and will satisfy the equation xy
+  = const. = a^2.
+
+  _For the asymptotes as axes of co-ordinates the equation of the
+  hyperbola is xy = const._
+
+
+  INVOLUTION
+
+  [Illustration: FIG. 29.]
+
+  S 76. If we have two projective rows, ABC on u and A'B'C' on u', and
+  place their bases on the same line, then each point in this line
+  counts twice, once as a point in the row u and once as a point in the
+  row u'. In fig. 29 we denote the points as points in the one row by
+  letters above the line A, B, C ..., and as points in the second row by
+  A', B', C' ... below the line. Let now A and B' be the same point,
+  then to A will correspond a point A' in the second, and to B' a point
+  B in the first row. In general these points A' and B will be
+  different. It may, however, happen that they coincide. Then the
+  correspondence is a peculiar one, as the following theorem shows:
+
+  _If two projective rows lie on the same base, and if it happens that
+  to one point in the base the same point corresponds, whether we
+  consider the point as belonging to the first or to the second row,
+  then the same will happen for every point in the base--that is to say,
+  to every point in the line corresponds the same point in the first as
+  in the second row._
+
+  [Illustration: FIG. 30.]
+
+  In order to determine the correspondence, we may assume three pairs of
+  corresponding points in two projective rows. Let then A', B', C', in
+  fig. 30, correspond to A, B, C, so that A and B', and also B and A',
+  denote the same point. Let us further denote the point C' when
+  considered as a point in the first row by D; then it is to be proved
+  that the point D', which corresponds to D, is the same point as C. We
+  know that the cross-ratio of four points is equal to that of the
+  corresponding row. Hence
+
+    (AB, CD) = (A'B', C'D')
+
+  but replacing the dashed letters by those undashed ones which denote
+  the same points, the second cross-ratio equals (BA, DD'), which,
+  according to S 15, equals (AB, D'D); so that the equation becomes
+
+    (AB, CD) = (AB, D'D).
+
+  This requires that C and D' coincide.
+
+  S 77. Two projective rows on the same base, which have the above
+  property, that to every point, whether it be considered as a point in
+  the one or in the other row, corresponds the same point, are said to
+  be in _involution_, or to form an _involution_ of points on the line.
+
+  We mention, but without proving it, that any two projective rows may
+  be placed so as to form an involution.
+
+  An involution may be said to consist of a row of pairs of points, to
+  every point A corresponding a point A', and to A' again the point A.
+  These points are said to be conjugate, or, better, one point is termed
+  the "mate" of the other.
+
+  From the definition, according to which an involution may be
+  considered as made up of two projective rows, follow at once the
+  following important properties:
+
+  1. The cross-ratio of four points equals that of the four conjugate
+  points.
+
+  2. If we call a point which coincides with its mate a "focus" or
+  "double point" of the involution, we may say: An involution has either
+  two foci, or one, or none, and is called respectively a hyperbolic,
+  parabolic or elliptic involution (S 34).
+
+  3. In an hyperbolic involution any two conjugate points are harmonic
+  conjugates with regard to the two foci.
+
+  For if A, A' be two conjugate points, F1, F2 the two foci, then to the
+  points F1, F2, A, A' in the one row correspond the points F1, F2, A',
+  A in the other, each focus corresponding to itself. Hence (F1F2, AA')
+  = (F1F2, A'A)--that is, we may interchange the two points AA' without
+  altering the value of the cross-ratio, which is the characteristic
+  property of harmonic conjugates (S 18).
+
+  4. The point conjugate to the point at infinity is called the "centre"
+  of the involution. Every involution has a centre, unless the point at
+  infinity be a focus, in which case we may say that the centre is at
+  infinity.
+
+  In an hyperbolic involution the centre is the middle point between the
+  foci.
+
+  5. The product of the distances of two conjugate points A, A' from the
+  centre O is constant: OA . OA' = c.
+
+  For let A, A' and B, B' be two pairs of conjugate points, the centre,
+  I the point at infinity, then
+
+    (AB, OI) = (A'B', IO),
+
+  or
+
+    OA . OA' = OB . OB'.
+
+  In order to determine the distances of the foci from the centre, we
+  write F for A and A' and get
+
+  OF^2 = c; OF = [+-][root]c.
+
+  Hence if c is positive OF is real, and has two values, equal and
+  opposite. The involution is hyperbolic.
+
+  If c = 0, OF = 0, and the two foci both coincide with the centre. If c
+  is negative, [root]c becomes imaginary, and there are no foci. Hence
+  we may write--
+
+    In an hyperbolic involution, OA.OA' = k^2,
+    In a parabolic involution, OA.OA' = 0,
+    In an elliptic involution, OA.OA' = -k^2.
+
+  From these expressions it follows that conjugate points A, A' in an
+  hyperbolic involution lie on the same side of the centre, and in an
+  elliptic involution on opposite sides of the centre, and that in a
+  parabolic involution one coincides with the centre.
+
+  In the first case, for instance, OA.OA' is positive; hence OA and OA'
+  have the same sign.
+
+  It also follows that two segments, AA' and BB', between pairs of
+  conjugate points have the following positions: in an hyperbolic
+  involution they lie either one altogether within or altogether without
+  each other; in a parabolic involution they have one point in common;
+  and in an elliptic involution they overlap, each being partly within
+  and partly without the other.
+
+  _Proof._--We have OA.OA' = OB.OB' = k^2 in case of an hyperbolic
+  involution. Let A and B be the points in each pair which are nearer to
+  the centre O. If now A, A' and B, B' lie on the same side of O, and if
+  B is nearer to O than A, so that OB < OA, then OB' > OA'; hence B' lies
+  farther away from O than A', or the segment AA' lies within BB'. And so
+  on for the other cases.
+
+  6. An involution is determined--
+
+    ([alpha]) By two pairs of conjugate points. Hence also
+    ([beta]) By one pair of conjugate points and the centre;
+    ([gamma]) By the two foci;
+    ([delta]) By one focus and one pair of conjugate points;
+    ([epsilon]) By one focus and the centre.
+
+  7. The condition that A, B, C and A', B', C' may form an involution
+  may be written in one of the forms--
+
+     (AB, CC') = (A'B', C'C),
+
+  or (AB, CA') = (A'B', C'A),
+
+  or (AB, C'A') = (A'B', CA),
+
+  for each expresses that in the two projective rows in which A, B, C
+  and A', B', C' are conjugate points two conjugate elements may be
+  interchanged.
+
+  8. Any three pairs. A, A', B, B', C, C', of conjugate points are
+  connected by the relations:
+
+    AB'.BC'.CA'   AB'.BC.C'A'   AB.B'C'.CA'   AB.B'C.C'A'
+    ----------- = ----------- = ----------- = ----------- = -1.
+    A'B.B'C.C'A   A'B.B'C'.CA   A'B'.BC.C'A   A'B'.BC'.CA
+
+  These relations readily follow by working out the relations in (7)
+  (above).
+
+  S 78. _Involution of a quadrangle.--The sides of any four-point are
+  cut by any line in six points in involution, opposite sides being cut
+  in conjugate points._
+
+  Let A1B1C1D1 (fig. 31) be the four-point. If its sides be cut by the
+  line p in the points A, A', B, B', C, C', if further, C1D1 cuts the
+  line A1B1 in C2, and if we project the row A1B1C2C to p once from D1
+  and once from C1, we get (A'B', C'C) = (BA, C'C).
+
+  Interchanging in the last cross-ratio the letters in each pair we get
+  (A'B', C'C) = (AB, CC'). Hence by S 77 (7) the points are in
+  involution.
+
+  The theorem may also be stated thus:
+
+  _The three points in which any line cuts the sides of a triangle and
+  the projections, from any point in the plane, of the vertices of the
+  triangle on to the same line are six points in involution._
+
+  [Illustration: FIG. 31.]
+
+  Or again--
+
+  The projections from any point on to any line of the six vertices of a
+  four-side are six points in involution, the projections of opposite
+  vertices being conjugate points.
+
+  This property gives a simple means to construct, by aid of the
+  straight edge only, in an involution of which two pairs of conjugate
+  points are given, to any point its conjugate.
+
+  S 79. _Pencils in Involution._--The theory of involution may at once
+  be extended from the row to the flat and the axial pencil--viz. we say
+  that there is an involution in a flat or in an axial pencil if any
+  line cuts the pencil in an involution of points. An involution in a
+  pencil consists of pairs of conjugate rays or planes; it has two, one
+  or no _focal rays_ (double lines) or _planes_, but nothing
+  corresponding to a centre.
+
+  An involution in a flat pencil contains always one, and in general
+  only one, pair of conjugate rays which are perpendicular to one
+  another. For in two projective flat pencils exist always two
+  corresponding right angles (S 40).
+
+  Each involution in an axial pencil contains in the same manner one
+  pair of conjugate planes at right angles to one another.
+
+  As a rule, there exists but one pair of conjugate lines or planes at
+  right angles to each other. But it is possible that there are more,
+  and then there is an infinite number of such pairs. An involution in a
+  flat pencil, in which every ray is perpendicular to its conjugate ray,
+  is said to be _circular_. That such involution is possible is easily
+  seen thus: if in two concentric flat pencils each ray on one is made
+  to correspond to that ray on the other which is perpendicular to it,
+  then the two pencils are projective, for if we turn the one pencil
+  through a right angle each ray in one coincides with its corresponding
+  ray in the other. But these two projective pencils are in involution.
+
+  A circular involution has no focal rays, because no ray in a pencil
+  coincides with the ray perpendicular to it.
+
+  S 80. _Every elliptical involution in a row may be considered as a
+  section of a circular involution._
+
+  In an elliptical involution any two segments AA' and BB' lie partly
+  within and partly without each other (fig. 32). Hence two circles
+  described on AA' and BB' as diameters will intersect in two points E
+  and E'. The line EE' cuts the base of the involution at a point O,
+  which has the property that OA.OA' = OB . OB', for each is equal to
+  OE . OE'. The point O is therefore the centre of the involution. If we
+  wish to construct to any point C the conjugate point C', we may draw
+  the circle through CEE'. This will cut the base in the required point
+  C' for OC.OC' = OA.OA'. But EC and EC' are at right angles. Hence the
+  involution which is obtained by joining E or E' to the points in the
+  given involution is circular. This may also be expressed thus:
+
+  [Illustration: FIG. 32.]
+
+  _Every elliptical involution has the property that there are two
+  definite points in the plane from which any two conjugate points are
+  seen under a right angle._
+
+  At the same time the following problem has been solved:
+
+  To determine the centre and also the point corresponding to any given
+  point in an elliptical involution of which two pairs of conjugate
+  points are given.
+
+  S 81. _Involution Range on a Conic._--By the aid of S 53, the points
+  on a conic may be made to correspond to those on a line, so that the
+  row of points on the conic is projective to a row of points on a line.
+  We may also have two projective rows on the same conic, and these will
+  be in involution as soon as one point on the conic has the same point
+  corresponding to it all the same to whatever row it belongs. An
+  involution of points on a conic will have the property (as follows
+  from its definition, and from S 53) that the lines which join
+  conjugate points of the involution to any point on the conic are
+  conjugate lines of an involution in a pencil, and that a fixed tangent
+  is cut by the tangents at conjugate points on the conic in points
+  which are again conjugate points of an involution on the fixed
+  tangent. For such involution on a conic the following theorem holds:
+
+  _The lines which join corresponding points in an involution on a conic
+  all pass through a fixed point; and reciprocally, the points of
+  intersection of conjugate lines in an involution among tangents to a
+  conic lie on a line._
+
+  [Illustration: FIG. 33]
+
+  We prove the first part only. The involution is determined by two
+  pairs of conjugate points, say by A, A' and B, B' (fig. 33). Let AA'
+  and BB' meet in P. If we join the points in involution to any point on
+  the conic, and the conjugate points to another point on the conic, we
+  obtain two projective pencils. We take A and A' as centres of these
+  pencils, so that the pencils A(A'BB') and A'(AB'B) are projective, and
+  in perspective position, because AA' corresponds to A'A. Hence
+  corresponding rays meet in a line, of which two points are found by
+  joining AB' to A'B and AB to A'B'. It follows that the _axis_ of
+  perspective is the polar of the point P, where AA' and BB' meet. If we
+  now wish to construct to any other point C on the conic the
+  corresponding point C', we join C to A' and the point where this line
+  cuts p to A. The latter line cuts the conic again in C'. But we know
+  from the theory of pole and polar that the line CC' passes through P.
+  The point of concurrence is called the "pole of the involution," and
+  the line of collinearity of the meets is called the "axis of the
+  involution."
+
+
+  INVOLUTION DETERMINED BY A CONIC ON A LINE.--FOCI
+
+  S 82. The polars, with regard to a conic, of points in a row p form a
+  pencil P projective to the row (S 66). This pencil cuts the base of
+  the row p in a projective row.
+
+  If A is a point in the given row, A' the point where the polar of A
+  cuts p, then A and A' will be corresponding points. If we take A' a
+  point in the first row, then the polar of A' will pass through A, so
+  that A corresponds to A'--in other words, the rows are in involution.
+  The conjugate points in this involution are conjugate points with
+  regard to the conic. Conjugate points coincide only if the polar of a
+  point A passes through A--that is, if A lies on the conic. Hence--
+
+  _A conic determines on every line in its plane an involution, in which
+  those points are conjugate which are also conjugate with regard to the
+  conic._
+
+  _If the line cuts the conic the involution is hyperbolic, the points
+  of intersection being the foci._
+
+  _If the line touches the conic the involution is parabolic, the two
+  foci coinciding at the point of contact._
+
+  _If the line does not cut the conic the involution is elliptic, having
+  no foci._
+
+  If, on the other hand, we take a point P in the plane of a conic, we
+  get to each line a through P one conjugate line which joins P to the
+  pole of a. These pairs of conjugate lines through P form an involution
+  in the pencil at P. The focal rays of this involution are the tangents
+  drawn from P to the conic. This gives the theorem reciprocal to the
+  last, viz:--
+
+  _A conic determines in every pencil in its plane an involution,
+  corresponding lines being conjugate lines with regard to the conic._
+
+  _If the point is without the conic the involution is hyperbolic, the
+  tangents from the points being the focal rays._
+
+  _If the point lies on the conic the involution is parabolic, the
+  tangent at the point counting for coincident focal rays._
+
+  _If the point is within the conic the involution is elliptic, having
+  no focal rays._
+
+  It will further be seen that the involution determined by a conic on
+  any line p is a section of the involution, which is determined by the
+  conic at the pole P of p.
+
+  S 83. _Foci._--The centre of a pencil in which the conic determines a
+  circular involution is called a "focus" of the conic.
+
+  In other words, a focus is such a point that every line through it is
+  perpendicular to its conjugate line. The polar to a focus is called a
+  _directrix_ of the conic.
+
+  From the definition it follows that _every focus lies on an axis_, for
+  the line joining a focus to the centre of the conic is a diameter to
+  which the conjugate lines are perpendicular; and _every line joining
+  two foci is an axis_, for the perpendiculars to this line through the
+  foci are conjugate to it. These conjugate lines pass through the pole
+  of the line, the pole lies therefore at infinity, and the line is a
+  diameter, hence by the last property an axis.
+
+  It follows that all _foci lie on one axis_, for no line joining a
+  point in one axis to a point in the other can be an axis.
+
+  As the conic determines in the pencil which has its centre at a focus
+  a circular involution, no tangents can be drawn from the focus to the
+  conic. Hence _each focus lies within a conic_; and _a directrix does
+  not cut the conic_.
+
+  Further properties are found by the following considerations:
+
+  S 84. Through a point P one line p can be drawn, which is with regard
+  to a given conic conjugate to a given line q, viz. that line which
+  joins the point P to the pole of the line q. If the line q is made to
+  describe a pencil about a point Q, then the line p will describe a
+  pencil about P. These two pencils will be projective, for the line p
+  passes through the pole of q, and whilst q describes the pencil Q, its
+  pole describes a projective row, and this row is perspective to the
+  pencil P.
+
+  We now take the point P on an axis of the conic, draw any line p
+  through it, and from the pole of p draw a perpendicular q to p. Let q
+  cut the axis in Q. Then, in the pencils of conjugate lines, which have
+  their centres at P and Q, the lines p and q are conjugate lines at
+  right angles to one another. Besides, to the axis as a ray in either
+  pencil will correspond in the other the perpendicular to the axis (S
+  72). The conic generated by the intersection of corresponding lines in
+  the two pencils is therefore the circle on PQ as diameter, _so that
+  every line in P is perpendicular to its corresponding line in Q_.
+
+  To every point P on an axis of a conic corresponds thus a point Q,
+  such that conjugate lines through P and Q are perpendicular.
+
+  We shall show that these _point-pairs_ P, Q _form an involution_. To
+  do this let us move P along the axis, and with it the line p, keeping
+  the latter parallel to itself. Then P describes a row, p a perspective
+  pencil (of parallels), and the pole of p a projective row. At the same
+  time the line q describes a pencil of parallels perpendicular to p,
+  and perspective to the row formed by the pole of p. The point Q,
+  therefore, where q cuts the axis, describes a row projective to the
+  row of points P. The two points P and Q describe thus two projective
+  rows on the axis; and not only does P as a point in the first row
+  correspond to Q, but also Q as a point in the first corresponds to P.
+  The two rows therefore form an involution. _The centre of this
+  involution, it is easily seen, is the centre of the conic._
+
+  _A focus of this involution has the property that any two conjugate
+  lines through it are perpendicular; hence, it is a focus to the
+  conic._
+
+  Such involution exists on each axis. But only one of these can have
+  foci, because all foci lie on the same axis. The involution on one of
+  the axes is elliptic, and appears (S 80) therefore as the section of
+  two circular involutions in two pencils whose centres lie in the other
+  axis. These centres are foci, hence the one axis contains two foci,
+  the other axis none; _or every central conic has two foci which lie on
+  one axis equidistant from the centre_.
+
+  The axis which contains the foci is called the _principal axis_; in
+  case of an hyperbola it is the axis which cuts the curve, because the
+  foci lie within the conic.
+
+  In case of the parabola there is but one axis. The involution on this
+  axis has its centre at infinity. One focus is therefore at infinity,
+  the one focus only is finite. _A parabola has only one focus._
+
+  [Illustration: FIG. 34.]
+
+  S 85. If through any point P (fig. 34) on a conic the tangent PT and
+  the normal PN (i.e. the perpendicular to the tangent through the point
+  of contact) be drawn, these will be conjugate lines with regard to the
+  conic, and at right angles to each other. They will therefore cut the
+  principal axis in two points, which are conjugate in the involution
+  considered in S 84; hence they are harmonic conjugates with regard to
+  the foci. If therefore the two foci F1 and F2 be joined to P, these
+  lines will be harmonic with regard to the tangent and normal. As the
+  latter are perpendicular, they will bisect the angles between the
+  other pair. Hence--
+
+  _The lines joining any point on a conic to the two foci are equally
+  inclined to the tangent and normal at that point._
+
+  In case of the parabola this becomes--
+
+  _The line joining any point on a parabola to the focus and the
+  diameter through the point, are equally inclined to the tangent and
+  normal at that point._
+
+  From the definition of a focus it follows that--
+
+  _The segment of a tangent between the directrix and the point of
+  contact is seen from the focus belonging to the directrix under a
+  right angle_, because the lines joining the focus to the ends of this
+  segment are conjugate with regard to the conic, and therefore
+  perpendicular.
+
+  With equal ease the following theorem is proved:
+
+  _The two lines which join the points of contact of two tangents each
+  to one focus, but not both to the same, are seen from the intersection
+  of the tangents under equal angles._
+
+  S 86. Other focal properties of a conic are obtained by the following
+  considerations:
+
+  [Illustration: FIG. 35.]
+
+  Let F (fig. 35) be a focus to a conic, f the corresponding directrix,
+  A and B the points of contact of two tangents meeting at T, and P the
+  point where the line AB cuts the directrix. Then TF will be the polar
+  of P (because polars of F and T meet at P). Hence TF and PF are
+  conjugate lines through a focus, and therefore perpendicular. They are
+  further harmonic conjugates with regard to FA and FB (SS 64 and 13),
+  so that they bisect the angles formed by these lines. This by the way
+  proves--
+
+  _The segments between the point of intersection of two tangents to a
+  conic and their points of contact are seen from a focus under equal
+  angles._
+
+  If we next draw through A and B lines parallel to TF, then the points
+  A1, B1 where these cut the directrix will be harmonic conjugates with
+  regard to P and the point where FT cuts the directrix. The lines FT
+  and FP bisect therefore also the angles between FA1 and FB1. From this
+  it follows easily that the triangles FAA1 and FBB1 are equiangular,
+  and therefore similar, so that FA : AA1 = FB : BB1.
+
+  The triangles AA1A2 and BB1B2 formed by drawing perpendiculars from A
+  and B to the directrix are also similar, so that AA1 : AA2 = = BB1 :
+  BB2. This, combined with the above proportion, gives FA : AA2 = FB :
+  BB2. Hence the theorem:
+
+  _The ratio of the distances of any point on a conic from a focus and
+  the corresponding directrix is constant._
+
+  To determine this ratio we consider its value for a vertex on the
+  principal axis. In an ellipse the focus lies between the two vertices
+  on this axis, hence the focus is nearer to a vertex than to the
+  corresponding directrix. Similarly, in an hyperbola a vertex is nearer
+  to the directrix than to the focus. In a parabola the vertex lies
+  halfway between directrix and focus.
+
+  It follows in an ellipse the ratio between the distance of a point
+  from the focus to that from the directrix is less than unity, in the
+  parabola it equals unity, and in the hyperbola it is greater than
+  unity.
+
+  It is here the same which focus we take, because the two foci lie
+  symmetrical to the axis of the conic. If now P is any point on the
+  conic having the distances r1 and r2 from the foci and the distances
+  d1 and d2 from the corresponding directrices, then r1/d1 = r2/d2 =
+  e, where e is constant. Hence also r1 [+-] r2 / d1 [+-] d2 = e.
+
+  In the ellipse, which lies between the directrices, d1 + d2 is
+  constant, therefore also r1 +r2. In the hyperbola on the other hand d1
+  - d2 is constant, equal to the distance between the directrices,
+  therefore in this case r1 - r2 is constant.
+
+  If we call the distances of a point on a conic from the focus its
+  focal distances we have the theorem:
+
+  _In an ellipse the sum of the focal distances is constant; and in an
+  hyperbola the difference of the focal distances is constant._
+
+  _This constant sum or difference equals in both cases the length of
+  the principal axis._
+
+
+  PENCIL OF CONICS
+
+  S 87. Through four points A, B, C, D in a plane, of which no three lie
+  in a line, an infinite number of conics may be drawn, viz. through
+  these four points and any fifth one single conic. This system of
+  conics is called a pencil of conics. Similarly, all conics touching
+  four fixed lines form a system such that any fifth tangent determines
+  one and only one conic. We have here the theorems:
+
+  The pairs of points in which        The pairs of tangents which
+  any line is cut by a system of      can be drawn from a point to
+  conics through four fixed points    a system of conics touching four
+  are in involution.                  fixed lines are in involution.
+
+  [Illustration: FIG. 36.]
+
+  We prove the first theorem only. Let ABCD (fig. 36) be the four-point,
+  then any line t will cut two opposite sides AC, BD in the points E,
+  E', the pair AD, BC in points F, F', and any conic of the system in M,
+  N, and we have A(CD, MN) = B(CD, MN).
+
+  If we cut these pencils by t we get
+
+     (EF, MN) = (F'E', MN)
+
+  or (EF, MN) = (E'F', NM).
+
+  But this is, according to S 77 (7), the condition that M, N are
+  corresponding points in the involution determined by the point pairs
+  E, E', F, F' in which the line t cuts pairs of opposite sides of the
+  four-point ABCD. This involution is independent of the particular
+  conic chosen.
+
+  S 88. There follow several important theorems:
+
+  _Through four points two, one, or no conics may be drawn which touch
+  any given line, according as the involution determined by the given
+  four-point on the line has real, coincident or imaginary foci._
+
+  _Two, one, or no conics may be drawn which touch four given lines and
+  pass through a given point, according as the involution determined by
+  the given four-side at the point has real, coincident or imaginary
+  focal rays._
+
+  For the conic through four points which touches a given line has its
+  point of contact at a focus of the involution determined by the
+  four-point on the line.
+
+  As a special case we get, by taking the line at infinity:
+
+  _Through four points of which none is at infinity either two or no
+  parabolas may be drawn._
+
+  The problem of drawing a conic through four points and touching a
+  given line is solved by determining the points of contact on the line,
+  that is, by determining the foci of the involution in which the line
+  cuts the sides of the four-point. The corresponding remark holds for
+  the problem of drawing the conics which touch four lines and pass
+  through a given point.
+
+
+  RULED QUADRIC SURFACES
+
+  S 89. We have considered hitherto projective rows which lie in the
+  same plane, in which case lines joining corresponding points envelop a
+  conic. We shall now consider projective rows whose bases do not meet.
+  In this case, corresponding points will be joined by lines which do
+  not lie in a plane, but on some surface, which like every surface
+  generated by lines is called a _ruled_ surface. This surface clearly
+  contains the bases of the two rows.
+
+  If the points in either row be joined to the base of the other, we
+  obtain two axial pencils which are also projective, those planes being
+  corresponding which pass through corresponding points in the given
+  rows. If A', A be two corresponding points, [alpha], [alpha]' the
+  planes in the axial pencils passing through them, then AA' will be the
+  line of intersection of the corresponding planes [alpha], [alpha]' and
+  also the line joining corresponding points in the rows.
+
+  If we cut the whole figure by a plane this will cut the axial pencils
+  in two projective flat pencils, and the curve of the second order
+  generated by these will be the curve in which the plane cuts the
+  surface. Hence
+
+  _The locus of lines joining corresponding points in two projective
+  rows which do not lie in the same plane is a surface which contains
+  the bases of the rows, and which can also be generated by the lines of
+  intersection of corresponding planes in two projective axial pencils.
+  This surface is cut by every plane in a curve of the second order,
+  hence either in a conic or in a line-pair. No line which does not lie
+  altogether on the surface can have more than two points in common with
+  the surface, which is therefore said to be of the second order or is
+  called a ruled quadric surface._
+
+  That no line which does not lie on the surface can cut the surface in
+  more than two points is seen at once if a plane be drawn through the
+  line, for this will cut the surface in a conic. It follows also that a
+  line which contains more than two points of the surface lies
+  altogether on the surface.
+
+  S 90. Through any point in space one line can always be drawn cutting
+  two given lines which do not themselves meet.
+
+  If therefore three lines in space be given of which no two meet, then
+  through every point in either one line may be drawn cutting the other
+  two.
+
+  _If a line moves so that it always cuts three given lines of which no
+  two meet, then it generates a ruled quadric surface._
+
+  Let a, b, c be the given lines, and p, q, r ... lines cutting them in
+  the points A, A', A" ...; B, B', B" ...; C, C', C" ... respectively;
+  then the planes through a containing p, q, r, and the planes through b
+  containing the same lines, may be taken as corresponding planes in two
+  axial pencils which are projective, because both pencils cut the line
+  c in the same row, C, C', C" ...; the surface can therefore be
+  generated by projective axial pencils.
+
+  Of the lines p, q, r ... no two can meet, for otherwise the lines a,
+  b, c which cut them would also lie in their plane. There is a single
+  infinite number of them, for one passes through each point of a. These
+  lines are said to form a set of lines on the surface.
+
+  If now three of the lines p, q, r be taken, then every line d cutting
+  them will have three points in common with the surface, and will
+  therefore lie altogether on it. This gives rise to a second set of
+  lines on the surface. From what has been said the theorem follows:
+
+  _A ruled quadric surface contains two sets of straight lines. Every
+  line of one set cuts every line of the other, but no two lines of the
+  same set meet._
+
+  _Any two lines of the same set may be taken as bases of two projective
+  rows, or of two projective pencils which generate the surface. They
+  are cut by the lines of the other set in two projective rows._
+
+  The plane at infinity like every other plane cuts the surface either
+  in a conic proper or in a line-pair. In the first case the surface is
+  called an _Hyperboloid of one sheet_, in the second an _Hyperbolic
+  Paraboloid_.
+
+  The latter may be generated by a line cutting three lines of which one
+  lies at infinity, that is, cutting two lines and remaining parallel to
+  a given plane.
+
+
+  QUADRIC SURFACES
+
+  S 91. The conics, the cones of the second order, and the ruled quadric
+  surfaces complete the figures which can be generated by projective
+  rows or flat and axial pencils, that is, by those aggregates of
+  elements which are of one dimension (SS 5, 6). We shall now consider
+  the simpler figures which are generated by aggregates of two
+  dimensions. The space at our disposal will not, however, allow us to
+  do more than indicate a few of the results.
+
+  S 92. We establish a correspondence between the lines and planes in
+  pencils in space, or reciprocally between the points and lines in two
+  or more planes, but consider principally pencils.
+
+  In two pencils we may either make planes correspond to planes and
+  lines to lines, or else planes to lines and lines to planes. If hereby
+  the condition be satisfied that to a flat, or axial, pencil
+  corresponds in the first case a projective flat, or axial, pencil, and
+  in the second a projective axial, or flat, pencil, the pencils are
+  said to be _projective_ in the first case and _reciprocal_ in the
+  second.
+
+  For instance, two pencils which join two points S1 and S2 to the
+  different points and lines in a given plane [pi] are projective (and
+  in perspective position), if those lines and planes be taken as
+  corresponding which meet the plane [pi] in the same point or in the
+  same line. In this case every plane through both centres S1 and S2 of
+  the two pencils will correspond to itself. If these pencils are
+  brought into any other position they will be projective (but not
+  perspective).
+
+  _The correspondence between two projective pencils is uniquely
+  determined, if to four rays (or planes) in the one the corresponding
+  rays (or planes) in the other are given, provided that no three rays
+  of either set lie in a plane._
+
+  Let a, b, c, d be four rays in the one, a', b', c', d' the
+  corresponding rays in the other pencil. We shall show that we can find
+  for every ray e in the first a single corresponding ray e' in the
+  second. To the axial pencil a (b, c, d ...) formed by the planes which
+  join a to b, c, d ..., respectively corresponds the axial pencil a'
+  (b', c', d' ... ), and this correspondence is determined. Hence, the
+  plane a'e' which corresponds to the plane ae is determined. Similarly
+  the plane b'e' may be found and both together determine the ray e'.
+
+  Similarly the correspondence between two reciprocal pencils is
+  determined if for four rays in the one the corresponding planes in the
+  other are given.
+
+  S 93. We may now combine--
+
+  1. Two reciprocal pencils.
+
+    Each ray cuts its corresponding plane in a point, the locus of these
+    points is a quadric surface.
+
+  2. Two projective pencils.
+
+    Each plane cuts its corresponding plane in a line, but a ray as a
+    rule does not cut its corresponding ray. The locus of points where a
+    ray cuts its corresponding ray is a twisted cubic. The lines where a
+    plane cuts its corresponding plane are secants.
+
+  3. Three projective pencils.
+
+    The locus of intersection of corresponding planes is a cubic
+    surface.
+
+  Of these we consider only the first two cases.
+
+  S 94. If two pencils are reciprocal, then to a plane in either
+  corresponds a line in the other, to a flat pencil an axial pencil, and
+  so on. Every line cuts its corresponding plane in a point. If S1 and
+  S2 be the centres of the two pencils, and P be a point where a line a1
+  in the first cuts its corresponding plane [alpha]2, _then the line b2
+  in the pencil S2 which passes through P will meet its corresponding
+  plane [beta]1 in P_. For b2 is a line in the plane [alpha]2. The
+  corresponding plane [beta]1 must therefore pass through the line a1,
+  hence through P.
+
+  The points in which the lines in S1 cut the planes corresponding to
+  them in S2 are therefore the same as the points in which the lines in
+  S2 cut the planes corresponding to them in S1.
+
+  _The locus of these points is a surface which is cut by a plane in a
+  conic or in a line-pair and by a line in not more than two points
+  unless it lies altogether on the surface. The surface itself is
+  therefore called a quadric surface, or a surface of the second order._
+
+  To prove this we consider any line p in space.
+
+  The flat pencil in S1 which lies in the plane drawn through p and the
+  corresponding axial pencil in S2 determine on p two projective rows,
+  and those points in these which coincide with their corresponding
+  points lie on the surface. But there exist only two, or one, or no
+  such points, unless every point coincides with its corresponding
+  point. In the latter case the line lies altogether on the surface.
+
+  This proves also that a plane cuts the surface in a curve of the
+  second order, as no line can have more than two points in common with
+  it. To show that this is a curve of the same kind as those considered
+  before, we have to show that it can be generated by projective flat
+  pencils. We prove first that this is true for any plane through the
+  centre of one of the pencils, and afterwards that every point on the
+  surface may be taken as the centre of such pencil. Let then [alpha]1
+  be a plane through S1. To the flat pencil in S1 which it contains
+  corresponds in S2 a projective axial pencil with axis a2 and this cuts
+  [alpha]1 in a second flat pencil. These two flat pencils in [alpha]1
+  are projective, and, in general, neither concentric nor perspective.
+  They generate therefore a conic. But if the line a2 passes through S1
+  the pencils will have S1 as common centre, and may therefore have two,
+  or one, or no lines united with their corresponding lines. The section
+  of the surface by the plane [alpha]1 will be accordingly a line-pair
+  or a single line, or else the plane [alpha]1 will have only the point
+  S1 in common with the surface.
+
+  Every line l1 through S1 cuts the surface in two points, viz. first in
+  S1 and then at the point where it cuts its corresponding plane. If now
+  the corresponding plane passes through S1, as in the case just
+  considered, then the two points where l1 cuts the surface coincide at
+  S1, and the line is called a tangent to the surface with S1 as point
+  of contact. Hence if l1 be a tangent, it lies in that plane [tau]1
+  which corresponds to the line S2S1 as a line in the pencil S2. The
+  section of this plane has just been considered. It follows that--
+
+  _All tangents to quadric surface at the centre of one of the
+  reciprocal pencils lie in a plane which is called the tangent plane to
+  the surface at that point as point of contact._
+
+  _To the line joining the centres of the two pencils as a line in one
+  corresponds in the other the tangent plane at its centre._
+
+  _The tangent plane to a quadric surface either cuts the surface in two
+  lines, or it has only a single line, or else only a single point in
+  common with the surface._
+
+  _In the first case the point of contact is said to be hyperbolic, in
+  the second parabolic, in the third elliptic._
+
+  S 95. It remains to be proved that every point S on the surface may be
+  taken as centre of one of the pencils which generate the surface. Let
+  S be any point on the surface [Phi]' generated by the reciprocal
+  pencils S1 and S2. We have to establish a reciprocal correspondence
+  between the pencils S and S1, so that the surface generated by them is
+  identical with [Phi]. To do this we draw two planes [alpha]1 and
+  [beta]1 through S1, cutting the surface [Phi] in two conics which we
+  also denote by [alpha]1 and [beta]1. These conics meet at S1, and at
+  some other point T where the line of intersection of [alpha]1 and
+  [beta]1 cuts the surface.
+
+  In the pencil S we draw some plane [sigma] which passes through T, but
+  not through S1 or S2. It will cut the two conics first at T, and
+  therefore each at some other point which we call A and B respectively.
+  These we join to S by lines a and b, and now establish the required
+  correspondence between the pencils S1 and S as follows:--To S1T shall
+  correspond the plane [sigma], to the plane [alpha]1 the line a, and to
+  [beta]1 the line b, hence to the flat pencil in [alpha]1 the axial
+  pencil a. These pencils are made projective by aid of the conic in
+  [alpha]1.
+
+  In the same manner the flat pencil in [beta]1 is made projective to
+  the axial pencil b by aid of the conic in [beta]1, corresponding
+  elements being those which meet on the conic. This determines the
+  correspondence, for we know for more than four rays in S1 the
+  corresponding planes in S. The two pencils S and S1 thus made
+  reciprocal generate a quadric surface [Phi]', which passes through the
+  point S and through the two conics [alpha]1 and [beta]1.
+
+  The two surfaces [Phi] and [Phi]' have therefore the points S and S1
+  and the conics [alpha]1 and [beta]1 in common. To show that they are
+  identical, we draw a plane through S and S2, cutting each of the
+  conics [alpha]1 and [beta]1 in two points, which will always be
+  possible. This plane cuts [Phi] and [Phi]' in two conics which have
+  the point S and the points where it cuts [alpha]1 and [beta]1 in
+  common, that is five points in all. The conics therefore coincide.
+
+  This proves that all those points P on [Phi]' lie on [Phi] which have
+  the property that the plane SS2P cuts the conics [alpha]1, [beta]1 in
+  two points each. If the plane SS2P has not this property, then we draw
+  a plane SS1P. This cuts each surface in a conic, and these conics have
+  in common the points S, S1, one point on each of the conics [alpha]1,
+  [beta]1, and one point on one of the conics through S and S2 which lie
+  on both surfaces, hence five points. They are therefore coincident,
+  and our theorem is proved.
+
+  S 96. The following propositions follow:--
+
+  _A quadric surface has at every point a tangent plane._
+
+  _Every plane section of a quadric surface is a conic or a line-pair._
+
+  _Every line which has three points in common with a quadric surface
+  lies on the surface._
+
+  _Every conic which has five points in common with a quadric surface
+  lies on the surface._
+
+  _Through two conics which lie in different planes, but have two points
+  in common, and through one external point always one quadric surface
+  may be drawn._
+
+  S 97. _Every plane which cuts a quadric surface in a line-pair is a
+  tangent plane._ For every line in this plane through the centre of the
+  line-pair (the point of intersection of the two lines) cuts the
+  surface in two coincident points and is therefore a tangent to the
+  surface, _the centre of the line-pair being the point of contact_.
+
+  _If a quadric surface contains a line, then every plane through this
+  line cuts the surface in a line-pair (or in two coincident lines)._
+  For this plane cannot cut the surface in a conic. Hence:--
+
+  _If a quadric surface contains one line p then it contains an infinite
+  number of lines, and through every point Q on the surface, one line q
+  can be drawn which cuts p._ For the plane through the point Q and the
+  line p cuts the surface in a line-pair which must pass through Q and
+  of which p is one line.
+
+  _No two such lines q on the surface can meet_. For as both meet p
+  their plane would contain p and therefore cut the surface in a
+  triangle.
+
+  _Every line which cuts three lines q will be on the surface_; for it
+  has three points in common with it.
+
+  _Hence the quadric surfaces which contain lines are the same as the
+  ruled quadric surfaces considered in_ SS 89-93, but with one important
+  exception. In the last investigation we have left out of consideration
+  the possibility of a plane having only one line (two coincident lines)
+  in common with a quadric surface.
+
+  S 98. To investigate this case we suppose first that there is one
+  point A on the surface through which two different lines a, b can be
+  drawn, which lie altogether on the surface.
+
+  If P is any other point on the surface which lies neither on a nor b,
+  then the plane through P and a will cut the surface in a second line
+  a' which passes through P and which cuts a. Similarly there is a line
+  b' through P which cuts b. These two lines a' and b' _may_ coincide,
+  but then they must coincide with PA.
+
+  If this happens for one point P, it happens for every other point Q.
+  For if two different lines could be drawn through Q, then by the same
+  reasoning the line PQ would be altogether on the surface, hence two
+  lines would be drawn through P against the assumption. From this
+  follows:--
+
+  _If there is one point on a quadric surface through which one, but
+  only one, line can be drawn on the surface, then through every point
+  one line can be drawn, and all these lines meet in a point. The
+  surface is a cone of the second order_.
+
+  _If through one point on a quadric surface, two, and only two, lines
+  can be drawn on the surface, then through every point two lines may be
+  drawn, and the surface is ruled quadric surface._
+
+  _If through one point on a quadric surface no line on the surface can
+  be drawn, then the surface contains no lines._
+
+  Using the definitions at the end of S 95, we may also say:--
+
+  _On a quadric surface the points are all hyperbolic, or all parabolic,
+  or all elliptic._
+
+  As an example of a quadric surface with elliptical points, we mention
+  the sphere which may be generated by two reciprocal pencils, where to
+  each line in one corresponds the plane perpendicular to it in the
+  other.
+
+  S 99. _Poles and Polar Planes._--The theory of poles and polars with
+  regard to a conic is easily extended to quadric surfaces.
+
+  Let P be a point in space not on the surface, which we suppose not to
+  be a cone. On every line through P which cuts the surface in two
+  points we determine the harmonic conjugate Q of P with regard to the
+  points of intersection. Through one of these lines we draw two planes
+  [alpha] and [beta]. The locus of the points Q in [alpha] is a line a,
+  the polar of P with regard to the conic in which [alpha] cuts the
+  surface. Similarly the locus of points Q in [beta] is a line b. This
+  cuts a, because the line of intersection of [alpha] and [beta]
+  contains but one point Q. The locus of all points Q therefore is a
+  plane. _This plane is called the polar plane of the point P, with
+  regard to the quadric surface. If P lies on the surface we take the
+  tangent plane of P as its polar._
+
+  The following propositions hold:--
+
+  1. _Every point has a polar plane_, which is constructed by drawing
+  the polars of the point with regard to the conics in which two planes
+  through the point cut the surface.
+
+  2. _If Q is a point in the polar of P, then P is a point in the polar
+  of Q_, because this is true with regard to the conic in which a plane
+  through PQ cuts the surface.
+
+  3. _Every plane is the polar plane of one point, which is called the
+  Pole of the plane._
+
+  The pole to a plane is found by constructing the polar planes of three
+  points in the plane. Their intersection will be the pole.
+
+  4. _The points in which the polar plane of P cuts the surface are
+  points of contact of tangents drawn from P to the surface_, as is
+  easily seen. Hence:--
+
+  5. _The tangents drawn from a point P to a quadric surface form a cone
+  of the second order_, for the polar plane of P cuts it in a conic.
+
+  6. _If the pole describes a line a, its polar plane will turn about
+  another line a'_, as follows from 2. _These lines a and a' are said to
+  be conjugate with regard to the surface._
+
+  S 100. The pole of the line at infinity is called the _centre_ of the
+  surface. If it lies at the infinity, the plane at infinity is a
+  tangent plane, and the surface is called a _paraboloid_.
+
+  _The polar plane to any point at infinity passes through the centre,
+  and is called a diametrical plane._
+
+  _A line through the centre is called a diameter. It is bisected at the
+  centre. The line conjugate to it lies at infinity._
+
+  _If a point moves along a diameter its polar plane turns about the
+  conjugate line at infinity_; that is, _it moves parallel to itself,
+  its centre moving on the first line._
+
+  _The middle points of parallel chords lie in a plane_, viz. in the
+  polar plane of the point at infinity through which the chords are
+  drawn.
+
+  _The centres of parallel sections lie in a diameter which is a line
+  conjugate to the line at infinity in which the planes meet._
+
+
+  TWISTED CUBICS
+
+  S 101. If two pencils with centres S1 and S2 are made projective, then
+  to a ray in one corresponds a ray in the other, to a plane a plane, to
+  a flat or axial pencil a projective flat or axial pencil, and so on.
+
+  There is a double infinite number of lines in a pencil. We shall see
+  that a single infinite number of lines in one pencil meets its
+  corresponding ray, and that the points of intersection form a curve in
+  space.
+
+  Of the double infinite number of planes in the pencils each will meet
+  its corresponding plane. This gives a system of a double infinite
+  number of lines in space. We know (S 5) that there is a quadruple
+  infinite number of lines in space. From among these we may select
+  those which satisfy one or more given conditions. The systems of lines
+  thus obtained were first systematically investigated and classified by
+  Plucker, in his _Geometrie des Raumes_. He uses the following names:--
+
+  A _treble infinite_ number of lines, that is, all lines which satisfy
+  one condition, are said to form a _complex of lines_; e.g. all lines
+  cutting a given line, or all lines touching a surface.
+
+  A _double infinite_ number of lines, that is, all lines which satisfy
+  two conditions, or which are common to two complexes, are said to form
+  a _congruence of lines_; e.g. all lines in a plane, or all lines
+  cutting two curves, or all lines cutting a given curve twice.
+
+  A _single infinite_ number of lines, that is, all lines which satisfy
+  three conditions, or which belong to three complexes, form a _ruled
+  surface_; e.g. one set of lines on a ruled quadric surface, or
+  developable surfaces which are formed by the tangents to a curve.
+
+  It follows that all lines in which corresponding planes in two
+  projective pencils meet form a congruence. We shall see this
+  congruence consists of all lines which cut a twisted cubic twice, or
+  of all _secants_ to a twisted cubic.
+
+  S 102. Let l1 be the line S1S2 as a line in the pencil S1. To it
+  corresponds a line l2 in S2. _At each of the centres two corresponding
+  lines meet._ The two axial pencils with l1 and l2 as axes are
+  projective, and, as, their axes meet at S2, the intersections of
+  corresponding planes form a cone of the second order (S 58), with S2
+  as centre. If [pi]1 and [pi]2 be corresponding planes, then their
+  intersection will be a line p2 which passes through S2. Corresponding
+  to it in S1 will be a line p1 which lies in the plane [pi]1, and which
+  therefore meets p2 at some point P. Conversely, if p2 be any line in
+  S2 which meets its corresponding line p1 at a point P, then to the
+  plane l2p2 will correspond the plane l1p1, that is, the plane S1S2P.
+  These planes intersect in p2, so that p2 is a line on the quadric cone
+  generated by the axial pencils l1 and l2. Hence:--
+
+  _All lines in one pencil which meet their corresponding lines in the
+  other form a cone of the second order which has its centre at the
+  centre of the first pencil, and passes through the centre of the
+  second._
+
+  From this follows that the points in which corresponding rays meet lie
+  on two cones of the second order which have the ray joining their
+  centres in common, and form therefore, together with the line S1S2 or
+  l1, the intersection of these cones. Any plane cuts each of the cones
+  in a conic. These two conics have necessarily that point in common in
+  which it cuts the line l1, and therefore besides either one or three
+  other points. It follows that the curve is of the third order as a
+  plane may cut it in three, but not in more than three, points.
+  Hence:--
+
+  _The locus of points in which corresponding lines on two projective
+  pencils meet is a curve of the third order or a "twisted cubic" k,
+  which passes through the centres of the pencils, and which appears as
+  the intersection of two cones of the second order, which have one line
+  in common._
+
+  _A line belonging to the congruence determined by the pencils is a
+  secant of the cubic; it has two, or one, or no points in common with
+  this cubic, and is called accordingly a secant proper, a tangent, or a
+  secant improper of the cubic._ A secant improper may be considered, to
+  use the language of coordinate geometry, as a secant with imaginary
+  points of intersection.
+
+  S 103. If a1 and a2 be any two corresponding lines in the two pencils,
+  then corresponding planes in the axial pencils having a1 and a2 as
+  axes generate a ruled quadric surface. If P be any point on the cubic
+  k, and if p1, p2 be the corresponding rays in S1 and S2 which meet at
+  P, then to the plane a1p1 in S1 corresponds a2p2 in S2. These
+  therefore meet in a line through P.
+
+  This may be stated thus:--
+
+  _Those secants of the cubic which cut a ray a1, drawn through the
+  centre S1 of one pencil, form a ruled quadric surface which passes
+  through both centres, and which contains the twisted cubic k. Of such
+  surfaces an infinite number exists. Every ray through S1 or S2 which
+  is not a secant determines one of them._
+
+  If, however, the rays a1 and a2 are secants meeting at A, then the
+  ruled quadric surface becomes a cone of the second order, having A as
+  centre. Or _all lines of the congruence which pass through a point on
+  the twisted cubic k form a cone of the second order_. In other words,
+  the projection of a twisted cubic from any point in the curve on to
+  any plane is a conic.
+
+  If a1 is not a secant, but made to pass through any point Q in space,
+  the ruled quadric surface determined by a1 will pass through Q. _There
+  will therefore be one line of the congruence passing through Q, and
+  only one._ For if two such lines pass through Q, then the lines S1Q
+  and S2Q will be corresponding lines; hence Q will be a point on the
+  cubic k, and an infinite number of secants will pass through it.
+  Hence:--
+
+  _Through every point in space not on the twisted cubic one and only
+  one secant to the cubic can be drawn._
+
+  S 104. The fact that all the secants through a point on the cubic form
+  a quadric cone shows that the centres of the projective pencils
+  generating the cubic are not distinguished from any other points on
+  the cubic. If we take any two points S, S' on the cubic, and draw the
+  secants through each of them, we obtain two quadric cones, which have
+  the line SS' in common, and which intersect besides along the cubic.
+  If we make these two pencils having S and S' as centres projective by
+  taking four rays on the one cone as corresponding to the four rays on
+  the other which meet the first on the cubic, the correspondence is
+  determined. These two pencils will generate a cubic, and the two cones
+  of secants having S and S' as centres will be identical with the above
+  cones, for each has five rays in common with one of the first, viz.
+  the line SS' and the four lines determined for the correspondence;
+  therefore these two cones intersect in the original cubic. This gives
+  the theorem:--
+
+  _On a twisted cubic any two points may be taken as centres of
+  projective pencils which generate the cubic, corresponding planes
+  being those which meet on the same secant._
+
+  Of the two projective pencils at S and S' we may keep the first fixed,
+  and move the centre of the other along the curve. The pencils will
+  hereby remain projective, and a plane [alpha] in S will be cut by its
+  corresponding plane [alpha]' always in the same secant a. Whilst S'
+  moves along the curve the plane [alpha]' will turn about a, describing
+  an axial pencil.
+
+  AUTHORITIES.--In this article we have given a purely geometrical
+  theory of conics, cones of the second order, quadric surfaces, &c. In
+  doing so we have followed, to a great extent, Reye's _Geometrie der
+  Lage_, and to this excellent work those readers are referred who wish
+  for a more exhaustive treatment of the subject. Other works especially
+  valuable as showing the development of the subject are: Monge,
+  _Geometrie descriptive_: Carnot, _Geometrie de position_ (1803),
+  containing a theory of transversals; Poncelet's great work _Traite des
+  proprietes projectives des figures_ (1822); Mobins, _Barycentrischer
+  Calcul_ (1826); Steiner, _Abhangigkeit geometrischer Gestalten_
+  (1832), containing the first full discussion of the projective
+  relations between rows, pencils, &c.; Von Staudt, _Geometrie der Lage_
+  (1847) and _Beitrage zur Geometrie der Lage_ (1856-1860), in which a
+  system of geometry is built up from the beginning without any
+  reference to number, so that ultimately a number itself gets a
+  geometrical definition, and in which imaginary elements are
+  systematically introduced into pure geometry; Chasles, _Apercu
+  historique_ (1837), in which the author gives a brilliant account of
+  the progress of modern geometrical methods, pointing out the
+  advantages of the different purely geometrical methods as compared
+  with the analytical ones, but without taking as much account of the
+  German as of the French authors; Id., _Rapport sur les progres de la
+  geometrie_ (1870), a continuation of the _Apercu_; Id., _Traite de
+  geometrie superieure_ (1852); Cremona, _Introduzione ad una teoria
+  geometrica delle curve piane_ (1862) and its continuation _Preliminari
+  di una teoria geometrica delle superficie_ (German translations by
+  Curtze). As more elementary books, we mention: Cremona, _Elements of
+  Projective Geometry_, translated from the Italian by C. Leudesdorf
+  (2nd ed., 1894); J.W. Russell, _Pure Geometry_ (2nd ed., 1905).
+       (O. H.)
+
+
+III. DESCRIPTIVE GEOMETRY
+
+This branch of geometry is concerned with the methods for representing
+solids and other figures in three dimensions by drawings in one plane.
+The most important method is that which was invented by Monge towards
+the end of the 18th century. It is based on parallel projections to a
+plane by rays perpendicular to the plane. Such a projection is called
+orthographic (see PROJECTION, S 18). If the plane is horizontal the
+projection is called the plan of the figure, and if the plane is
+vertical the elevation. In Monge's method a figure is represented by its
+plan and elevation. It is therefore often called drawing in plan and
+elevation, and sometimes simply orthographic projection.
+
+  S 1. We suppose then that we have two planes, one horizontal, the
+  other vertical, and these we call the planes of plan and of elevation
+  respectively, or the horizontal and the vertical plane, and denote
+  them by the letters [pi]1 and [pi]2. Their line of intersection is
+  called the axis, and will be denoted by xy.
+
+  If the surface of the drawing paper is taken as the plane of the plan,
+  then the vertical plane will be the plane perpendicular to it through
+  the axis xy. To bring this also into the plane of the drawing paper we
+  turn it about the axis till it coincides with the horizontal plane.
+  This process of turning one plane down till it coincides with another
+  is called _rabatting_ one to the other. Of course there is no
+  necessity to have one of the two planes horizontal, but even when this
+  is not the case it is convenient to retain the above names.
+
+  [Illustration: FIG. 37.]
+
+  [Illustration: FIG. 38.]
+
+  The whole arrangement will be better understood by referring to fig.
+  37. A point A in space is there projected by the perpendicular AA1 and
+  AA2 to the planes [pi]1 and [pi]2 so that A1 and A2 are the horizontal
+  and vertical projections of A.
+
+  If we remember that a line is perpendicular to a plane that is
+  perpendicular to every line in the plane if only it is perpendicular
+  to any two intersecting lines in the plane, we see that the axis which
+  is perpendicular both to AA1 and to AA2 is also perpendicular to A1A0
+  and to A2A0 because these four lines are all in the same plane. Hence,
+  if the plane [pi]2 be turned about the axis till it coincides with the
+  plane [pi]1, then A2A0 will be the continuation of A1A0. This position
+  of the planes is represented in fig. 38, in which the line A1A2 is
+  perpendicular to the axis x.
+
+  Conversely any two points A1, A2 in a line perpendicular to the axis
+  will be the projections of some point in space when the plane [pi]2 is
+  turned about the axis till it is perpendicular to the plane [pi]1,
+  because in this position the two perpendiculars to the planes [pi]1
+  and [pi]2 through the points A1 and A2 will be in a plane and
+  therefore meet at some point A.
+
+  _Representation of Points._--We have thus the following method of
+  representing in a single plane the position of points in space:--_we
+  take in the plane a line xy as the axis, and then any pair of points
+  A1, A2 in the plane on a line perpendicular to the axis represent a
+  point A in space_. If the line A1A2 cuts the axis at A0, and if at A1
+  a perpendicular be erected to the plane, then the point A will be in
+  it at a height A1A = A0A2 above the plane. This gives the position of
+  the point A relative to the plane [pi]1. In the same way, if in a
+  perpendicular to [pi]2 through A2 a point A be taken such that A2A =
+  A0A1, then this will give the point A relative to the plane [pi]2.
+
+  [Illustration: FIG. 39.]
+
+  S 2. The two planes [pi]1, [pi]2 in their original position divide
+  space into four parts. These are called the four quadrants. We suppose
+  that the plane [pi]2 is turned as indicated in fig. 37, so that the
+  point P comes to Q and R to S, then the quadrant in which the point A
+  lies is called the first, and we say that in the first quadrant a
+  point lies above the horizontal and in front of the vertical plane.
+  Now we go round the axis in the sense in which the plane [pi]2 is
+  turned and come in succession to the second, third and fourth
+  quadrant. In the second a point lies above the plane of the plan and
+  behind the plane of elevation, and so on. In fig. 39, which represents
+  a side view of the planes in fig. 37 the quadrants are marked, and in
+  each a point with its projection is taken. Fig. 38 shows how these are
+  represented when the plane [pi]2 is turned down. We see that
+
+  _A point lies in the first quadrant if the plan lies below, the
+  elevation above the axis; in the second if plan and elevation both lie
+  above; in the third if the plan lies above, the elevation below; in
+  the fourth if plan and elevation both lie below the axis._
+
+  _If a point lies in the horizontal plane_, its elevation lies in the
+  axis and the plan coincides with the point itself. _If a point lies in
+  the vertical plane_, its plan lies in the axis and the elevation
+  coincides with the point itself. _If a point lies in the axis_, both
+  its plan and elevation lie in the axis and coincide with it.
+
+  Of each of these propositions, which will easily be seen to be true,
+  the converse holds also.
+
+  S 3. _Representation of a Plane._--As we are thus enabled to represent
+  points in a plane, we can represent any finite figure by representing
+  its separate points. It is, however, not possible to represent a plane
+  in this way, for the projections of its points completely cover the
+  planes [pi]1 and [pi]2, and no plane would appear different from any
+  other. But any plane [alpha] cuts each of the planes [pi]1, [pi]2 in a
+  line. These are called the traces of the plane. They cut each other in
+  the axis at the point where the latter cuts the plane [alpha].
+
+  _A plane is determined by its two traces, which are two lines that
+  meet on the axis_, and, conversely, _any two lines which meet on the
+  axis determine a plane_.
+
+  _If the plane is parallel to the axis its traces are parallel to the
+  axis._ Of these one may be at infinity; then the plane will cut one of
+  the planes of projection at infinity and will be parallel to it. Thus
+  a plane parallel to the horizontal plane of the plan has only one
+  finite trace, viz. that with the plane of elevation.
+
+  [Illustration: FIG. 40.]
+
+  _If the plane passes through the axis both its traces coincide with
+  the axis._ This is the only case in which the representation of the
+  plane by its two traces fails. A third plane of projection is
+  therefore introduced, which is best taken perpendicular to the other
+  two. We call it simply the third plane and denote it by [pi]3. As it
+  is perpendicular to [pi]1, it may be taken as the plane of elevation,
+  its line of intersection [gamma] with [pi]1 being the axis, and be
+  turned down to coincide with [pi]1. This is represented in fig. 40. OC
+  is the axis xy whilst OA and OB are the traces of the third plane.
+  They lie in one line [gamma]. The plane is rabatted about [gamma] to
+  the horizontal plane. A plane [alpha] through the axis xy will then
+  show in it a trace [alpha]3. In fig. 40 the lines OC and OP will thus
+  be the traces of a plane through the axis xy, which makes an angle POQ
+  with the horizontal plane.
+
+  We can also find the trace which any other plane makes with [pi]3. In
+  rabatting the plane [pi]3 its trace OB with the plane [pi]2 will come
+  to the position OD. Hence a plane [beta] having the traces CA and CB
+  will have with the third plane the trace [beta]3, or AD if OD = OB.
+
+  It also follows immediately that--
+
+  _If a plane [alpha] is perpendicular to the horizontal plane, then
+  every point in it has its horizontal projection in the horizontal
+  trace of the plane_, as all the rays projecting these points lie in
+  the plane itself.
+
+  _Any plane which is perpendicular to the horizontal plane has its
+  vertical trace perpendicular to the axis._
+
+  _Any plane which is perpendicular to the vertical plane has its
+  horizontal trace perpendicular to the axis and the vertical
+  projections of all points in the plane lie in this trace._
+
+  S 4. _Representation of a Line._--A line is determined either by two
+  points in it or by two planes through it. We get accordingly two
+  representations of it either by projections or by traces.
+
+  First.--_A line a is represented by its projections a1 and a2 on the
+  two planes [pi]1 and [pi]2._ These may be any two lines, for, bringing
+  the planes [pi]1, [pi]2 into their original position, the planes
+  through these lines perpendicular to [pi]1 and [pi]2 respectively will
+  intersect in some line a which has a1, a2 as its projections.
+
+  Secondly.--_A line a is represented by its traces--that is, by the
+  points in which it cuts the two planes [pi]1, [pi]2._ Any two points
+  may be taken as the traces of a line in space, for it is determined
+  when the planes are in their original position as the line joining the
+  two traces. This representation becomes undetermined if the two traces
+  coincide in the axis. In this case we again use a third plane, or else
+  the projections of the line.
+
+  The fact that there are different methods of representing points and
+  planes, and hence two methods of representing lines, suggests the
+  principle of duality (section ii., _Projective Geometry_, S 41). It is
+  worth while to keep this in mind. It is also worth remembering that
+  traces of planes or lines always lie in the planes or lines which they
+  represent. Projections do not as a rule do this excepting when the
+  point or line projected lies in one of the planes of projection.
+
+  Having now shown how to represent points, planes and lines, we have to
+  state the conditions which must hold in order that these elements may
+  lie one in the other, or else that the figure formed by them may
+  possess certain metrical properties. It will be found that the former
+  are very much simpler than the latter.
+
+  Before we do this, however, we shall explain the notation used; for it
+  is of great importance to have a systematic notation. We shall denote
+  points in space by capitals A, B, C; planes in space by Greek letters
+  [alpha], [beta], [gamma]; lines in space by small letters a, b, c;
+  horizontal projections by suffixes 1, like A1, a1; vertical
+  projections by suffixes 2, like A2, a2; traces by single and double
+  dashes [alpha]' [alpha]", a', a". Hence P1 will be the horizontal
+  projection of a point P in space; a line a will have the projections
+  a1, a2 and the traces a' and a"; a plane [alpha] has the traces
+  [alpha]' and [alpha]".
+
+  S 5. _If a point lies in a line, the projections of the point lie in
+  the projections of the line._
+
+  _If a line lies in a plane, the traces of the line lie in the traces
+  of the plane._
+
+  These propositions follow at once from the definitions of the
+  projections and of the traces.
+
+  If a point lies in two lines its projections must lie in the
+  projections of both. Hence
+
+  _If two lines, given by their projections, intersect, the intersection
+  of their planes and the intersection of their elevations must lie in a
+  line perpendicular to the axis_, because they must be the projections
+  of the point common to the two lines.
+
+  Similarly--_If two lines given by their traces lie in the same plane
+  or intersect, then the lines joining their horizontal and vertical
+  traces respectively must meet on the axis_, because they must be the
+  traces of the plane through them.
+
+  S 6. _To find the projections of a line which joins two points A, B
+  given by their projections A1, A2 and B1, B2_, we join A1, B1 and A2,
+  B2; these will be the projections required. For example, the traces of
+  a line are two points in the line whose projections are known or at
+  all events easily found. They are the traces themselves and the feet
+  of the perpendiculars from them to the axis.
+
+  Hence _if a' a" (fig. 41) are the traces of a line a, and if the
+  perpendiculars from them cut the axis in P and Q respectively, then
+  the line a'Q will be the horizontal and a"P the vertical projection of
+  the line_.
+
+  [Illustration: FIG. 41.]
+
+  Conversely, if the projections a1, a2 of a line are given, and if
+  these cut the axis in Q and P respectively, then _the perpendiculars
+  Pa' and Qa" to the axis drawn through these points cut the projections
+  a1 and a2 in the traces a' and a"_.
+
+  _To find the line of intersection of two planes_, we observe that this
+  line lies in both planes; its traces must therefore lie in the traces
+  of both. Hence the points where the horizontal traces of the given
+  planes meet will be the horizontal, and the point where the vertical
+  traces meet the vertical trace of the line required.
+
+  S 7. _To decide whether a point A, given by its projections, lies in a
+  plane [alpha], given by its traces_, we draw a line p by joining A to
+  some point in the plane [alpha] and determine its traces. If these lie
+  in the traces of the plane, then the line, and therefore the point A,
+  lies in the plane; otherwise not. This is conveniently done by joining
+  A1 to some point p' in the trace [alpha]'; this gives p1; and the
+  point where the perpendicular from p' to the axis cuts the latter we
+  join to A2; this gives p2. If the vertical trace of this line lies in
+  the vertical trace of the plane, then, and then only, does the line p,
+  and with it the point A, lie in the plane [alpha].
+
+  S 8. _Parallel planes have parallel traces_, because parallel planes
+  are cut by any plane, hence also by [pi]1 and by [pi]2, in parallel
+  lines.
+
+  _Parallel lines have parallel projections_, because points at infinity
+  are projected to infinity.
+
+  _If a line is parallel to a plane, then lines through the traces of
+  the line and parallel to the traces of the plane must meet on the
+  axis_, because these lines are the traces of a plane parallel to the
+  given plane.
+
+  S 9. _To draw a plane through two intersecting lines or through two
+  parallel lines_, we determine the traces of the lines; the lines
+  joining their horizontal and vertical traces respectively will be the
+  horizontal and vertical traces of the plane. They will meet, at a
+  finite point or at infinity, on the axis if the lines do intersect.
+
+  _To draw a plane through a line and a point without the line_, we join
+  the given point to any point in the line and determine the plane
+  through this and the given line.
+
+  _To draw a plane through three points which are not in a line_, we
+  draw two of the lines which each join two of the given points and draw
+  the plane through them. If the traces of all three lines AB, BC, CA be
+  found, these must lie in two lines which meet on the axis.
+
+  S 10. We have in the last example got more points, or can easily get
+  more points, than are necessary for the determination of the figure
+  required--in this case the traces of the plane. This will happen in a
+  great many constructions and is of considerable importance. It may
+  happen that some of the points or lines obtained are not convenient in
+  the actual construction. The horizontal traces of the lines AB and AC
+  may, for instance, fall very near together, in which case the line
+  joining them is not well defined. Or, one or both of them may fall
+  beyond the drawing paper, so that they are practically non-existent
+  for the construction. In this case the traces of the line BC may be
+  used. Or, if the vertical traces of AB and AC are both in convenient
+  position, so that the vertical trace of the required plane is found
+  and one of the horizontal traces is got, then we may join the latter
+  to the point where the vertical trace cuts the axis.
+
+  The draughtsman must remember that the lines which he draws are not
+  mathematical lines without thickness, and therefore every drawing is
+  affected by some errors. It is therefore very desirable to be able
+  constantly to check the latter. Such checks always present themselves
+  when the same result can be obtained by different constructions, or
+  when, as in the above case, some lines must meet on the axis, or if
+  three points must lie in a line. A careful draughtsman will always
+  avail himself of these checks.
+
+  S 11. _To draw a plane through a given point parallel to a given plane
+  [alpha]_, we draw through the point two lines which are parallel to
+  the plane [alpha], and determine the plane through them; or, as we
+  know that the traces of the required plane are parallel to those of
+  the given one (S 8), we need only draw one line l through the point
+  parallel to the plane and find one of its traces, say the vertical
+  trace l"; a line through this parallel to the vertical trace of
+  [alpha] will be the vertical trace [beta]" of the required plane
+  [beta], and a line parallel to the horizontal trace of [alpha] meeting
+  [beta]" on the axis will be the horizontal trace [beta]'.
+
+  [Illustration: FIG. 42.]
+
+  Let A1 A2 (fig. 42) be the given point, [alpha]' [alpha]" the given
+  plane, a line l1 through A1, parallel to [alpha]' and a horizontal
+  line l2 through A2 will be the projections of a line l through A
+  parallel to the plane, because the horizontal plane through this line
+  will cut the plane [alpha] in a line c which has its horizontal
+  projection c1 parallel to [alpha]'.
+
+  S 12. We now come to the metrical properties of figures.
+
+  _A line is perpendicular to a plane if the projections of the line are
+  perpendicular to the traces of the plane._ We prove it for the
+  horizontal projection. If a line p is perpendicular to a plane
+  [alpha], every plane through p is perpendicular to [alpha]; hence also
+  the vertical plane which projects the line p to p1. As this plane is
+  perpendicular both to the horizontal plane and to the plane [alpha],
+  it is also perpendicular to their intersection--that is, to the
+  horizontal trace of [alpha]. It follows that every line in this
+  projecting plane, therefore also p1, the plan of p, is perpendicular
+  to the horizontal trace of [alpha].
+
+  _To draw a plane through a given point A perpendicular to a given line
+  p_, we first draw through some point O in the axis lines [gamma]',
+  [gamma]" perpendicular respectively to the projections p1 and p2 of
+  the given line. These will be the traces of a plane [gamma] which is
+  perpendicular to the given line. We next draw through the given point
+  A a plane parallel to the plane [gamma]; this will be the plane
+  required.
+
+  Other metrical properties depend on the determination of the real size
+  or shape of a figure.
+
+  In general the projection of a figure differs both in size and shape
+  from the figure itself. But figures in a plane parallel to a plane of
+  projection will be identical with their projections, and will thus be
+  given in their true dimensions. In other cases there is the problem,
+  constantly recurring, either to find the true shape and size of a
+  plane figure when plan and elevation are given, or, conversely, to
+  find the latter from the known true shape of the figure itself. To do
+  this, the plane is turned about one of its traces till it is laid down
+  into that plane of projection to which the trace belongs. This is
+  technically called rabatting the plane respectively into the plane of
+  the plan or the elevation. As there is no difference in the treatment
+  of the two cases, we shall consider only the case of rabatting a plane
+  [alpha] into the plane of the plan. The plan of the figure is a
+  parallel (orthographic) projection of the figure itself. The results
+  of parallel projection (see PROJECTION, SS 17 and 18) may therefore
+  now be used. The trace [alpha]' will hereby take the place of what
+  formerly was called the axis of projection. Hence we see that
+  corresponding points in the plan and in the rabatted plane are joined
+  by lines which are perpendicular to the trace [alpha]' and that
+  corresponding lines meet on this trace. We also see that the
+  correspondence is completely determined if we know for one point or
+  one line in the plan the corresponding point or line in the rabatted
+  plane.
+
+  Before, however, we treat of this we consider some special cases.
+
+  S 13. _To determine the distance between two points A, B given by
+  their projections A1, B1 and A2, B2, or, in other words, to determine
+  the true length of a line the plan and elevation of which are given._
+
+  [Illustration: FIG. 43.]
+
+  _Solution._--The two points A, B in space lie vertically above their
+  plans A1, B1 (fig. 43) and A1A = A0A2, B1B = B0B2. The four points A,
+  B, A1, B1 therefore form a plane quadrilateral on the base A1B1 and
+  having right angles at the base. This plane we rabatt about A1B1 by
+  drawing A1A and B1B perpendicular to A1B1 and making A1A = A0A2, B1B =
+  B0B2. Then AB will give the length required.
+
+  The construction might have been performed in the elevation by making
+  A2A = A0A1 and B2B = B0B1 on lines perpendicular to A2B2. Of course AB
+  must have the same length in both cases.
+
+  This figure may be turned into a model. Cut the paper along A1A, AB
+  and BB1, and fold the piece A1ABB1 over along A1B1 till it stands
+  upright at right angles to the horizontal plane. The points A, B will
+  then be in their true position in space relative to [pi]1. Similarly
+  if B2BAA2 be cut out and turned along A2B2 through a right angle we
+  shall get AB in its true position relative to the plane [pi]2. Lastly
+  we fold the whole plane of the paper along the axis x till the plane
+  [pi]2 is at right angles to [pi]1. In this position the two sets of
+  points AB will coincide if the drawing has been accurate.
+
+  Models of this kind can be made in many cases and their construction
+  cannot be too highly recommended in order to realize orthographic
+  projection.
+
+  S 14. _To find the angle between two given lines a, b of which the
+  projections a1, b1 and a2, b2 are given._
+
+  [Illustration: FIG. 44.]
+
+  _Solution._--Let a1, b1 (fig. 44) meet in P1, a2, b2 in T, then if the
+  line P1T is not perpendicular to the axis the two lines will not meet.
+  In this case we draw a line parallel to b to meet the line a. This is
+  easiest done by drawing first the line P1P2 perpendicular to the axis
+  to meet a2 in P2, and then drawing through P2 a line c2 parallel to
+  b2; then b1, c2 will be the projections of a line c which is parallel
+  to b and meets a in P. The plane [alpha] which these two lines
+  determine we rabatt to the plan. We determine the traces a' and c' of
+  the lines a and c; then a'c' is the trace [alpha]' of their plane. On
+  rabatting the point P comes to a point S on the line P1Q perpendicular
+  to a'c', so that QS = QP. But QP is the hypotenuse of a triangle PP1Q
+  with a right angle P1. This we construct by making QR = P0P2; then P1R
+  = PQ. The lines a'S and c'S will therefore include angles equal to
+  those made by the given lines. It is to be remembered that two lines
+  include two angles which are supplementary. Which of these is to be
+  taken in any special case depends upon the circumstances.
+
+  _To determine the angle between a line and a plane_, we draw through
+  any point in the line a perpendicular to the plane (S 12) and
+  determine the angle between it and the given line. The complement of
+  this angle is the required one.
+
+  _To determine the angle between two planes_, we draw through any point
+  two lines perpendicular to the two planes and determine the angle
+  between the latter as above.
+
+  In special cases it is simpler to determine at once the angle between
+  the two planes by taking a plane section perpendicular to the
+  intersection of the two planes and rabatt this. This is especially the
+  case if one of the planes is the horizontal or vertical plane of
+  projection.
+
+  Thus in fig. 45 the angle P1QR is the angle which the plane [alpha]
+  makes with the horizontal plane.
+
+  S 15. We return to the general case of rabatting a plane [alpha] of
+  which the traces [alpha]' [alpha]" are given.
+
+  [Illustration: FIG. 45.]
+
+  Here it will be convenient to determine first the position which the
+  trace [alpha]"--which is a line in [alpha]--assumes when rabatted.
+  Points in this line coincide with their elevations. Hence it is given
+  in its true dimension, and we can measure off along it the true
+  distance between two points in it. If therefore (fig. 45) P is any
+  point in [alpha]" originally coincident with its elevation P2, and if
+  O is the point where [alpha]" cuts the axis xy, so that O is also in
+  [alpha]', then the point P will after rabatting the plane assume such
+  a position that OP = OP2. At the same time the plan is an orthographic
+  projection of the plane [alpha]. Hence the line joining P to the plan
+  P1 will after rabatting be perpendicular to [alpha]'. But P1 is known;
+  it is the foot of the perpendicular from P2 to the axis xy. We draw
+  therefore, to find P, from P1 a perpendicular P1Q to [alpha]' and find
+  on it a point P such that OP = OP2. Then the line OP will be the
+  position of [alpha]" when rabatted. This line corresponds therefore to
+  the plan of [alpha]"--that is, to the axis xy, corresponding points on
+  these lines being those which lie on a perpendicular to [alpha]'.
+
+  We have thus one pair of corresponding lines and can now find for any
+  point B1 in the plan the corresponding point B in the rabatted plane.
+  We draw a line through B1, say B1P1, cutting [alpha]' in C. To it
+  corresponds the line CP, and the point where this is cut by the
+  projecting ray through B1, perpendicular to [alpha]', is the required
+  point B.
+
+  Similarly any figure in the rabatted plane can be found when the plan
+  is known; but this is usually found in a different manner without any
+  reference to the general theory of parallel projection. As this method
+  and the reasoning employed for it have their peculiar advantages, we
+  give it also.
+
+  Supposing the planes [pi]1 and [pi]2 to be in their positions in space
+  perpendicular to each other, we take a section of the whole figure by
+  a plane perpendicular to the trace [alpha]' about which we are going
+  to rabatt the plane [alpha]. Let this section pass through the point Q
+  in [alpha]'. Its traces will then be the lines QP1 and P1P2 (fig. 9).
+  These will be at right angles, and will therefore, together with the
+  section QP2 of the plane [alpha], form a right-angled triangle QP1P2
+  with the right angle at P1, and having the sides P1Q and P1P2 which
+  both are given in their true lengths. This triangle we rabatt about
+  its base P1Q, making P1R = P1P2. The line QR will then give the true
+  length of the line QP in space. If now the plane [alpha] be turned
+  about [alpha]' the point P will describe a circle about Q as centre
+  with radius QP = QR, in a plane perpendicular to the trace [alpha]'.
+  Hence when the plane [alpha] has been rabatted into the horizontal
+  plane the point P will lie in the perpendicular P1Q to [alpha]', so
+  that QP = QR.
+
+  If A1 is the plan of a point A in the plane [alpha], and if A1 lies in
+  QP1, then the point A will lie vertically above A1 in the line QP. On
+  turning down the triangle QP1P2, the point A will come to A0, the line
+  A1A0 being perpendicular to QP1. Hence A will be a point in QP such
+  that QA = QA0.
+
+  If B1 is the plan of another point, but such that A1B1 is parallel to
+  [alpha]', then the corresponding line AB will also be parallel to
+  [alpha]'. Hence, if through A a line AB be drawn parallel to [alpha]',
+  and B1B perpendicular to [alpha]', then their intersection gives the
+  point B. Thus of any point given in plan the real position in the
+  plane [alpha], when rabatted, can be found by this second method. This
+  is the one most generally given in books on geometrical drawing. The
+  first method explained is, however, in most cases preferable as it
+  gives the draughtsman a greater variety of constructions. It requires
+  a somewhat greater amount of theoretical knowledge.
+
+  If instead of our knowing the plan of a figure the latter is itself
+  given, then the process of finding the plan is the reverse of the
+  above and needs little explanation. We give an example.
+
+  S 16. _It is required to draw the plan and elevation of a polygon of
+  which the real shape and position in a given plane [alpha] are known._
+
+  We first rabatt the plane [alpha] (fig. 46) as before so that P1 comes
+  to P, hence OP1 to OP. Let the given polygon in [alpha] be the figure
+  ABCDE. We project, not the vertices, but the sides. To project the
+  line AB, we produce it to cut [alpha]' in F and OP in G, and draw GG1
+  perpendicular to [alpha]'; then G1 corresponds to G, therefore FG1 to
+  FG. In the same manner we might project all the other sides, at least
+  those which cut OF and OP in convenient points. It will be best,
+  however, first to produce all the sides to cut OP and [alpha]' and
+  then to draw all the projecting rays through A, B, C ... perpendicular
+  to [alpha]', and in the same direction the lines G, G1, &c. By drawing
+  FG we get the points A1, B1 on the projecting ray through A and B. We
+  then join B to the point M where BC produced meets the trace [alpha]'.
+  This gives C1. So we go on till we have found E1. The line A1 E1 must
+  then meet AE in [alpha]', and this gives a check. If one of the sides
+  cuts [alpha]' or OP beyond the drawing paper this method fails, but
+  then we may easily find the projection of some other line, say of a
+  diagonal, or directly the projection of a point, by the former
+  methods. The diagonals may also serve to check the drawing, for two
+  corresponding diagonals must meet in the trace [alpha]'.
+
+  [Illustration: FIG. 46.]
+
+  Having got the plan we easily find the elevation. The elevation of G
+  is above G1 in [alpha]", and that of F is at F2 in the axis. This
+  gives the elevation F2G2 of FG and in it we get A2B2 in the verticals
+  through A1 and B1. As a check we have OG = OG2. Similarly the
+  elevation of the other sides and vertices are found.
+
+  S 17. We proceed to give some applications of the above principles to
+  the representation of solids and of the solution of problems connected
+  with them.
+
+  _Of a pyramid are given its base, the length of the perpendicular from
+  the vertex to the base, and the point where this perpendicular cuts
+  the base; it is required first to develop the whole surface of the
+  pyramid into one plane, and second to determine its section by a plane
+  which cuts the plane of the base in a given line and makes a given
+  angle with it._
+
+  1. As the planes of projection are not given we can take them as we
+  like, and we select them in such a manner that the solution becomes as
+  simple as possible. We take the plane of the base as the horizontal
+  plane and the vertical plane perpendicular to the plane of the
+  section. Let then (fig. 47) ABCD be the base of the pyramid, V1 the
+  plan of the vertex, then the elevations of A, B, C, D will be in the
+  axis at A2, B2, C2, D2, and the vertex at some point V2 above V1 at a
+  known distance from the axis. The lines V1A, V1B, &c., will be the
+  plans and the lines V2A2, V2B2, &c., the elevations of the edges of
+  the pyramid, of which thus plan and elevation are known.
+
+  We develop the surface into the plane of the base by turning each
+  lateral face about its lower edge into the horizontal plane by the
+  method used in S 14. If one face has been turned down, say ABV to ABP,
+  then the point Q to which the vertex of the next face BCV comes can be
+  got more simply by finding on the line V1Q perpendicular to BC the
+  point Q such that BQ = BP, for these lines represent the same edge BV
+  of the pyramid. Next R is found by making CR = CQ, and so on till we
+  have got the last vertex--in this case S. The fact that AS must equal
+  AP gives a convenient check.
+
+  2. The plane [alpha] whose section we have to determine has its
+  horizontal trace given perpendicular to the axis, and its vertical
+  trace makes the given angle with the axis. This determines it. To find
+  the section of the pyramid by this plane there are two methods
+  applicable: we find the sections of the plane either with the faces or
+  with the edges of the pyramid. We use the latter.
+
+  As the plane [alpha] is perpendicular to the vertical plane, the trace
+  [alpha]" contains the projection of every figure in it; the points
+  E2, F2, G2, H2 where this trace cuts the elevations of the edges will
+  therefore be the elevations of the points where the edges cut [alpha].
+  From these we find the plans E1, F1, G1, H1, and by joining them the
+  plan of the section. If from E1, F1 lines be drawn perpendicular to
+  AB, these will determine the points E, F on the developed face in
+  which the plane [alpha] cuts it; hence also the line EF. Similarly on
+  the other faces. Of course BF must be the same length on BP and on BQ.
+  If the plane [alpha] be rabatted to the plan, we get the real shape of
+  the section as shown in the figure in EFGH. This is done easily by
+  making F0F = OF2, &c. If the figure representing the development of
+  the pyramid, or better a copy of it, is cut out, and if the lateral
+  faces be bent along the lines AB, BC, &c., we get a model of the
+  pyramid with the section marked on its faces. This may be placed on
+  its plan ABCD and the plane of elevation bent about the axis x. The
+  pyramid stands then in front of its elevations. If next the plane
+  [alpha] with a hole cut out representing the true section be bent
+  along the trace [alpha]' till its edge coincides with [alpha]", the
+  edges of the hole ought to coincide with the lines EF, FG, &c., on the
+  faces.
+
+  S 18. Polyhedra like the pyramid in S 17 are represented by the
+  projections of their edges and vertices. But solids bounded by curved
+  surfaces, or surfaces themselves, cannot be thus represented.
+
+  For a surface we may use, as in case of the plane, its traces--that
+  is, the curves in which it cuts the planes of projection. We may also
+  project points and curves on the surface. A ray cuts the surface
+  generally in more than one point; hence it will happen that some of
+  the rays touch the surface, if two of these points coincide. The
+  points of contact of these rays will form some curve on the surface,
+  and this will appear from the centre of projection as the boundary of
+  the surface or of part of the surface. The outlines of all surfaces of
+  solids which we see about us are formed by the points at which rays
+  through our eye touch the surface. The projections of these contours
+  are therefore best adapted to give an idea of the shape of a surface.
+
+  [Illustration: FIG. 47.]
+
+  Thus the tangents drawn from any finite centre to a sphere form a
+  right circular cone, and this will be cut by any plane in a conic. It
+  is often called the projection of a sphere, but it is better called
+  the contour-line of the sphere, as it is the boundary of the
+  projections of all points on the sphere.
+
+  If the centre is at infinity the tangent cone becomes a right circular
+  cylinder touching the sphere along a great circle, and if the
+  projection is, as in our case, orthographic, then the section of this
+  cone by a plane of projection will be a circle equal to the great
+  circle of the sphere. We get such a circle in the plan and another in
+  the elevation, their centres being plan and elevation of the centre of
+  the sphere.
+
+  Similarly the rays touching a cone of the second order will lie in two
+  planes which pass through the vertex of the cone, the contour-line of
+  the projection of the cone consists therefore of two lines meeting in
+  the projection of the vertex. These may, however, be invisible if no
+  real tangent rays can be drawn from the centre of projection; and this
+  happens when the ray projecting the centre of the vertex lies within
+  the cone. In this case the traces of the cone are of importance. Thus
+  in representing a cone of revolution with a vertical axis we get in
+  the plan a circular trace of the surface whose centre is the plan of
+  the vertex of the cone, and in the elevation the contour, consisting
+  of a pair of lines intersecting in the elevation of the vertex of the
+  cone. The circle in the plan and the pair of lines in the elevation do
+  not determine the surface, for an infinite number of surfaces might be
+  conceived which pass through the circular trace and touch two planes
+  through the contour lines in the vertical plane. The surface becomes
+  only completely defined if we write down to the figure that it shall
+  represent a cone. The same holds for all surfaces. Even a plane is
+  fully represented by its traces only under the silent understanding
+  that the traces are those of a plane.
+
+  S 19. Some of the simpler problems connected with the representation
+  of surfaces are the determination of plane sections and of the curves
+  of intersection of two such surfaces. The former is constantly used in
+  nearly all problems concerning surfaces. Its solution depends of
+  course on the nature of the surface.
+
+  To determine the curve of intersection of two surfaces, we take a
+  plane and determine its section with each of the two surfaces,
+  rabatting this plane if necessary. This gives two curves which lie in
+  the same plane and whose intersections will give us points on both
+  surfaces. It must here be remembered that two curves in space do not
+  necessarily intersect, hence that the points in which their
+  projections intersect are not necessarily the projections of points
+  common to the two curves. This will, however, be the case if the two
+  curves lie in a common plane. By taking then a number of plane
+  sections of the surfaces we can get as many points on their curve of
+  intersection as we like. These planes have, of course, to be selected
+  in such a way that the sections are curves as simple as the case
+  permits of, and such that they can be easily and accurately drawn.
+  Thus when possible the sections should be straight lines or circles.
+  This not only saves time in drawing but determines all points on the
+  sections, and therefore also the points where the two curves meet,
+  with equal accuracy.
+
+  S 20. We give a few examples how these sections have to be selected. A
+  cone is cut by every plane through the vertex in lines, and if it is a
+  cone of revolution by planes perpendicular to the axis in circles.
+
+  A cylinder is cut by every plane parallel to the axis in lines, and if
+  it is a cylinder of revolution by planes perpendicular to the axis in
+  circles.
+
+  A sphere is cut by every plane in a circle.
+
+  Hence in case of two cones situated anywhere in space we take sections
+  through both vertices. These will cut both cones in lines. Similarly
+  in case of two cylinders we may take sections parallel to the axis of
+  both. In case of a sphere and a cone of revolution with vertical axis,
+  horizontal sections will cut both surfaces in circles whose plans are
+  circles and whose elevations are lines, whilst vertical sections
+  through the vertex of the cone cut the latter in lines and the sphere
+  in circles. To avoid drawing the projections of these circles, which
+  would in general be ellipses, we rabatt the plane and then draw the
+  circles in their real shape. And so on in other cases.
+
+  Special attention should in all cases be paid to those points in which
+  the tangents to the projection of the curve of intersection are
+  parallel or perpendicular to the axis x, or where these projections
+  touch the contour of one of the surfaces.     (O. H.)
+
+
+IV. ANALYTICAL GEOMETRY
+
+1. In the name _geometry_ there is a lasting record that the science had
+its origin in the knowledge that two distances may be compared by
+measurement, and in the idea that measurement must be effectual in the
+dissociation of different directions as well as in the comparison of
+distances in the same direction. The distance from an observer's eye of
+an object seen would be specified as soon as it was ascertained that a
+rod, straight to the eye and of length taken as known, could be given
+the direction of the line of vision, and had to be moved along it a
+certain number of times through lengths equal to its own in order to
+reach the object from the eye. Moreover, if a field had for two of its
+boundaries lines straight to the eye, one running from south to north
+and the other from west to east, the position of a point in the field
+would be specified if the rod, when directed west, had to be shifted
+from the point one observed number of times westward to meet the former
+boundary, and also, when directed south, had to be shifted another
+observed number of times southward to meet the latter. Comparison by
+measurement, the beginning of geometry, involved counting, the basis of
+arithmetic; and the science of number was marked out from the first as
+of geometrical importance.
+
+But the arithmetic of the ancients was inadequate as a science of
+number. Though a length might be recognized as known when measurement
+certified that it was so many times a standard length, it was not every
+length which could be thus specified in terms of the same standard
+length, even by an arithmetic enriched with the notion of fractional
+number. The idea of possible incommensurability of lengths was
+introduced into Europe by Pythagoras; and the corresponding idea of
+irrationality of number was absent from a crude arithmetic, while there
+were great practical difficulties in the way of its introduction. Hence
+perhaps it arose that, till comparatively modern times, appeal to
+arithmetical aid in geometrical reasoning was in all possible ways
+restrained. Geometry figured rather as the helper of the more difficult
+science of arithmetic.
+
+2. It was reserved for algebra to remove the disabilities of arithmetic,
+and to restore the earliest ideas of the land-measurer to the position
+of controlling ideas in geometrical investigation. This unified science
+of pure number made comparatively little headway in the hands of the
+ancients, but began to receive due attention shortly after the revival
+of learning. It expresses whole classes of arithmetical facts in single
+statements, gives to arithmetical laws the form of equations involving
+symbols which may mean any known or sought numbers, and provides
+processes which enable us to analyse the information given by an
+equation and derive from that equation other equations, which express
+laws that are in effect consequences or causes of a law started from,
+but differ greatly from it in form. Above all, for present purposes, it
+deals not only with integral and fractional number, but with number
+regarded as capable of continuous growth, just as distance is capable of
+continuous growth. The difficulty of the arithmetical expression of
+irrational number, a difficulty considered by the modern school of
+analysts to have been at length surmounted (see FUNCTION), is not vital
+to it. It can call the ratio of the diagonal of a square to a side, for
+instance, or that of the circumference of a circle to a diameter, a
+number, and let a or x denote that number, just as properly as it may
+allow either letter to denote any rational number which may be greater
+or less than the ratio in question by a difference less than any minute
+one we choose to assign.
+
+Counting only, and not the counting of objects, is of the essence of
+arithmetic, and of algebra. But it is lawful to count objects, and in
+particular to count equal lengths by measure. The widened idea is that
+even when a or x is an irrational number we may speak of a or x unit
+lengths by measure. We may give concrete interpretation to an
+algebraical equation by allowing its terms all to mean numbers of times
+the same unit length, or the same unit area, or &c. and in any equation
+lawfully derived from the first by algebraical processes we may do the
+same. Descartes in his _Geometrie_ (1637) was the first to systematize
+the application of this principle to the inherent first notions of
+geometry; and the methods which he instituted have become the most
+potent methods of all in geometrical research. It is hardly too much to
+say that, when known facts as to a geometrical figure have once been
+expressed in algebraical terms, all strictly consequential facts as to
+the figure can be deduced by almost mechanical processes. Some may well
+be unexpected consequences; and in obtaining those of which there has
+been suggestion beforehand the often bewildering labour of constant
+attention to the figure is obviated. These are the methods of what is
+now called _analytical_, or sometimes _algebraical_, _geometry_.
+
+3. The modern use of the term "analytical" in geometry has obscured, but
+not made obsolete, an earlier use, one as old as Plato. There is nothing
+algebraical in this analysis, as distinguished from synthesis, of the
+Greeks, and of the expositors of pure geometry. It has reference to an
+order of ideas in demonstration, or, more frequently, in discovering
+means to effect the geometrical construction of a figure with an
+assigned special property. We have to suppose hypothetically that the
+construction has been performed, drawing a rough figure which exhibits
+it as nearly as is practicable. We then analyse or critically examine
+the figure, treated as correct, and ascertain other properties which it
+can only possess in association with the one in question. Presently one
+of these properties will often be found which is of such a character
+that the construction of a figure possessing it is simple. The means of
+effecting synthetically a construction such as was desired is thus
+brought to light by what Plato called _analysis_. Or again, being asked
+to prove a theorem A, we ascertain that it must be true if another
+theorem B is, that B must be if C is, and so on, thus eventually finding
+that the theorem A is the consequence, through a chain of
+intermediaries, of a theorem Z of which the establishment is easy. This
+geometrical analysis is not the subject of the present article; but in
+the reasoning from form to form of an equation or system of equations,
+with the object of basing the algebraical proof of a geometrical fact on
+other facts of a more obvious character, the same logic is utilized, and
+the name "analytical geometry" is thus in part explained.
+
+4. In algebra real positive number was alone at first dealt with, and in
+geometry actual signless distance. But in algebra it became of
+importance to say that every equation of the first degree has a root,
+and the notion of negative number was introduced. The negative unit had
+to be defined as what can be added to the positive unit and produce the
+sum zero. The corresponding notion was readily at hand in geometry,
+where it was clear that a unit distance can be measured to the left or
+down from the farther end of a unit distance already measured to the
+right or up from a point O, with the result of reaching O again. Thus,
+to give full interpretation in geometry to the algebraically negative,
+it was only necessary to associate distinctness of sign with
+oppositeness of direction. Later it was discovered that algebraical
+reasoning would be much facilitated, and that conclusions as to the real
+would retain all their soundness, if a pair of imaginary units
+[+-][root]-1 of what might be called number were allowed to be
+contemplated, the pair being defined, though not separately, by the two
+properties of having the real sum 0 and the real product 1. Only in
+these two real combinations do they enter in conclusions as to the real.
+An advantage gained was that every quadratic equation, and not some
+quadratics only, could be spoken of as having two roots. These
+admissions of new units into algebra were final, as it admitted of proof
+that all equations of degrees higher than two have the full numbers of
+roots possible for their respective degrees in any case, and that every
+root has a value included in the form a + b [root]-1, with a, b, real.
+The corresponding enrichment could be given to geometry, with
+corresponding advantages and the same absence of danger, and this was
+done. On a line of measurement of distance we contemplate as existing,
+not only an infinite continuum of points at real distances from an
+origin of measurement O, but a doubly infinite continuum of points, all
+but the singly infinite continuum of real ones imaginary, and imaginary
+in conjugate pairs, a conjugate pair being at imaginary distances from
+O, which have a real arithmetic and a real geometric mean. To geometry
+enriched with this conception all algebra has its application.
+
+5. Actual geometry is one, two or three-dimensional, i.e. lineal, plane
+or solid. In one-dimensional geometry positions and measurements in a
+single line only are admitted. Now descriptive constructions for points
+in a line are impossible without going out of the line. It has therefore
+been held that there is a sense in which no science of geometry strictly
+confined to one dimension exists. But an algebra of one variable can be
+applied to the study of distances along a line measured from a chosen
+point on it, so that the idea of construction as distinct from
+measurement is not essential to a one-dimensional geometry aided by
+algebra. In geometry of two dimensions, the flat of the land-measurer,
+the passage from one point O to any other point, can be effected by two
+successive marches, one east or west and one north or south, and, as
+will be seen, an algebra of two variables suffices for geometrical
+exploitation. In geometry of three dimensions, that of space, any point
+can be reached from a chosen one by three marches, one east or west, one
+north or south, and one up or down; and we shall see that an algebra of
+three variables is all that is necessary. With three dimensions actual
+geometry stops; but algebra can supply any number of variables. Four or
+more variables have been used in ways analogous to those in which one,
+two and three variables are used for the purposes of one, two and
+three-dimensional geometry, and the results have been expressed in
+quasi-geometrical language on the supposition that a higher space can be
+conceived of, though not realized, in which four independent directions
+exist, such that no succession of marches along three of them can effect
+the same displacement of a point as a march along the fourth; and
+similarly for higher numbers than four. Thus analytical, though not
+actual, geometries exist for four and more dimensions. They are in fact
+algebras furnished with nomenclature of a geometrical cast, suggested by
+convenient forms of expression which actual geometry has, in return for
+benefits received, conferred on algebras of one, two and three
+variables.
+
+We will confine ourselves to the dimensions of actual geometry, and will
+devote no space to the one-dimensional, except incidentally as existing
+within the two-dimensional. The analytical method will now be explained
+for the cases of two and three dimensions in succession. The form of it
+originated by Descartes, and thence known as Cartesian, will alone be
+considered in much detail.
+
+
+  I. _Plane Analytical Geometry._
+
+  [Illustration: FIG. 48.]
+
+  [Illustration: FIG. 49.]
+
+  6. _Coordinates._--It is assumed that the points, lines and figures
+  considered lie in one and the same plane, which plane therefore need
+  not be in any way referred to. In the plane a point O, and two lines
+  x'Ox, y'Oy, intersecting in O, are taken once for all, and regarded as
+  fixed. O is called the origin, and x'Ox, y'Oy the axes of x and y
+  respectively. Other positions in the plane are specified in relation
+  to this fixed origin and these fixed axes. From any point P we suppose
+  PM drawn parallel to the axis of y to meet the axis of x in M, and may
+  also suppose PN drawn parallel to the axis of x to meet the axis of y
+  in N, so that OMPN is a parallelogram. The position of P is determined
+  when we know OM ( = NP) and MP ( = ON). If OM is x times the unit of a
+  scale of measurement chosen at pleasure, and MP is y times the unit,
+  so that x and y have numerical values, we call x and y the (Cartesian)
+  coordinates of P. To distinguish them we often speak of y as the
+  ordinate, and of x as the abscissa.
+
+  It is necessary to attend to signs; x has one sign or the other
+  according as the point P is on one side or the other of the axis of y,
+  and y one sign or the other according as P is on one side or the other
+  of the axis of x. Using the letters N, E, S, W, as in a map, and
+  considering the plane as divided into four quadrants by the axes, the
+  signs are usually taken to be:
+
+    x   y   For quadrant
+
+    +   +      N  E
+    +   -      S  E
+    -   +      N  W
+    -   -      S  W
+
+  A point is referred to as the point (a, b), when its coordinates are x
+  = a, y = b. A point may be fixed, or it may be variable, i.e. be
+  regarded for the time being as free to move in the plane. The
+  coordinates (x, y) of a variable point are algebraic variables, and
+  are said to be "current coordinates."
+
+  The axes of x and y are usually (as in fig. 48) taken at right angles
+  to one another, and we then speak of them as rectangular axes, and of
+  x and y as "rectangular coordinates" of a point P; OMPN is then a
+  rectangle. Sometimes, however, it is convenient to use axes which are
+  oblique to one another, so that (as in fig. 49) the angle xOy between
+  their positive directions is some known angle [omega] distinct from a
+  right angle, and OMPN is always an oblique parallelogram with given
+  angles; and we then speak of x and y as "oblique coordinates." The
+  coordinates are as a rule taken to be rectangular in what follows.
+
+  7. _Equations and loci._ If (x, y) is the point P, and if we are given
+  that x = 0, we are told that, in fig. 48 or fig. 49, the point M lies
+  at O, whatever value y may have, i.e. we are told the one fact that P
+  lies on the axis of y. Conversely, if P lies anywhere on the axis of
+  y, we have always OM = 0, i.e. x = 0. Thus the equation x = 0 is one
+  satisfied by the coordinates (x, y) of every point in the axis of y,
+  and not by those of any other point. We say that x = 0 is the equation
+  of the axis of y, and that the axis of y is the locus represented by
+  the equation x = 0. Similarly y = 0 is the equation of the axis of x.
+  An equation x = a, where a is a constant, expresses that P lies on a
+  parallel to the axis of y through a point M on the axis of x such that
+  OM = a. Every line parallel to the axis of y has an equation of this
+  form. Similarly, every line parallel to the axis of x has an equation
+  of the form y = b, where b is some definite constant.
+
+  These are simple cases of the fact that a single equation in the
+  current coordinates of a variable point (x, y) imposes one limitation
+  on the freedom of that point to vary. The coordinates of a point taken
+  at random in the plane will, as a rule, not satisfy the equation, but
+  infinitely many points, and in most cases infinitely many real ones,
+  have coordinates which do satisfy it, and these points are exactly
+  those which lie upon some locus of one dimension, a straight line or
+  more frequently a curve, which is said to be represented by the
+  equation. Take, for instance, the equation y = mx, where m is a given
+  constant. It is satisfied by the coordinates of every point P, which
+  is such that, in fig. 48, the distance MP, with its proper sign, is m
+  times the distance OM, with its proper sign, i.e. by the coordinates
+  of every point in the straight line through O which we arrive at by
+  making a line, originally coincident with x'Ox, revolve about O in the
+  direction opposite to that of the hands of a watch through an angle of
+  which m is the tangent, and by those of no other points. That line is
+  the locus which it represents. Take, more generally, the equation y =
+  [phi](x), where [phi](x) is any given non-ambiguous function of x.
+  Choosing any point M on x'Ox in fig. 1, and giving to x the value of
+  the numerical measure of OM, the equation determines a single
+  corresponding y, and so determines a single point P on the line
+  through M parallel to y'Oy. This is one point whose coordinates
+  satisfy the equation. Now let M move from the extreme left to the
+  extreme right of the line x'Ox, regarded as extended both ways as far
+  as we like, i.e. let x take all real values from -[oo] to [oo]. With
+  every value goes a point P, as above, on the parallel to y'Oy through
+  the corresponding M; and we thus find that there is a path from the
+  extreme left to the extreme right of the figure, all points P along
+  which are distinguished from other points by the exceptional property
+  of satisfying the equation by their coordinates. This path is a locus;
+  and the equation y = [phi](x) represents it. More generally still,
+  take an equation f(x, y) = 0 which involves both x and y under a
+  functional form. Any particular value given to x in it produces from
+  it an equation for the determination of a value or values of y, which
+  go with that value of x in specifying a point or points (x, y), of
+  which the coordinates satisfy the equation f(x, y) = 0. Here again, as
+  x takes all values, the point or points describe a path or paths,
+  which constitute a locus represented by the equation. Except when y
+  enters to the first degree only in f(x, y), it is not to be expected
+  that all the values of y, determined as going with a chosen value of
+  x, will be necessarily real; indeed it is not uncommon for all to be
+  imaginary for some ranges of values of x. The locus may largely
+  consist of continua of imaginary points; but the real parts of it
+  constitute a real curve or real curves. Note that we have to allow x
+  to admit of all imaginary, as well as of all real, values, in order to
+  obtain all imaginary parts of the locus.
+
+  A locus or curve may be algebraically specified in another way; viz.
+  we may be given two equations x = f([theta]), y = F([theta]), which
+  express the coordinates of any point of it as two functions of the
+  same variable parameter [theta] to which all values are open. As
+  [theta] takes all values in turn, the point (x, y) traverses the
+  curve.
+
+  It is a good exercise to trace a number of curves, taken as defined by
+  the equations which represent them. This, in simple cases, can be done
+  approximately by plotting the values of y given by the equation of a
+  curve as going with a considerable number of values of x, and
+  connecting the various points (x, y) thus obtained. But methods exist
+  for diminishing the labour of this tentative process.
+
+  Another problem, which will be more attended to here, is that of
+  determining the equations of curves of known interest, taken as
+  defined by geometrical properties. It is not a matter for surprise
+  that the curves which have been most and longest studied geometrically
+  are among those represented by equations of the simplest character.
+
+  8. _The Straight Line._--This is the simplest type of locus. Also the
+  simplest type of equation in x and y is Ax + By + C = 0, one of the
+  first degree. Here the coefficients A, B, C are constants. They are,
+  like the current coordinates, x, y, numerical. But, in giving
+  interpretation to such an equation, we must of course refer to numbers
+  Ax, By, C of unit magnitudes of the same kind, of units of counting
+  for instance, or unit lengths or unit squares. It will now be seen
+  that every straight line has an equation of the first degree, and that
+  every equation of the first degree represents a straight line.
+
+  [Illustration: FIG. 50.]
+
+  It has been seen (S 7) that lines parallel to the axes have equations
+  of the first degree, free from one of the variables. Take now a
+  straight line ABC inclined to both axes. Let it make a given angle
+  [alpha] with the positive direction of the axis of x, i.e. in fig. 50
+  let this be the angle through which Ax must be revolved
+  counter-clockwise about A in order to be made coincident with the
+  line. Let C, of coordinates (h, k), be a fixed point on the line, and
+  P(x, y) any other point upon it. Draw the ordinates CD, PM of C and P,
+  and let the parallel to the axis of x through C meet PM, produced if
+  necessary, in R. The right-angled triangle CRP tells us that, with the
+  signs appropriate to their directions attached to CR and RP,
+
+    RP = CR tan [alpha], i.e. MP - DC = (OM - OD) tan [alpha],
+
+  and this gives that
+
+    y - k = tan [alpha] (x - h),
+
+  an equation of the first degree satisfied by x and y. No point not on
+  the line satisfies the same equation; for the line from C to any point
+  off the line would make with CR some angle [beta] different from
+  [alpha], and the point in question would satisfy an equation y - k =
+  tan [beta](x - h), which is inconsistent with the above equation.
+
+  The equation of the line may also be written y = mx + b, where m = tan
+  [alpha], and b = k - h tan [alpha]. Here b is the value obtained for y
+  from the equation when 0 is put for x, i.e. it is the numerical
+  measure, with proper sign, of OB, the intercept made by the line on
+  the axis of y, measured from the origin. For different straight lines,
+  m and b may have any constant values we like.
+
+  Now the general equation of the first degree Ax + By + C = 0 may be
+  written y = -(A/B)x - C/B, unless B = 0, in which case it represents a
+  line parallel to the axis of y; and -A/B, -C/B are values which can be
+  given to m and b, so that every equation of the first degree
+  represents a straight line. It is important to notice that the general
+  equation, which in appearance contains three constants A, B, C, in
+  effect depends on two only, the ratios of two of them to the third. In
+  virtue of this last remark, we see that two distinct conditions
+  suffice to determine a straight line. For instance, it is easy from
+  the above to see that
+
+    x    y
+    -- + -- = 1
+    a    b
+
+  is the equation of a straight line determined by the two conditions
+  that it makes intercepts OA, OB on the two axes, of which a and b are
+  the numerical measures with proper signs: note that in fig. 50 a is
+  negative. Again,
+
+             y2 - y1
+    y - y1 = ------- (x - x1),
+             x2 - x1
+
+  i.e.
+
+    (y1 - y2)x - (x1 - x2)y + x1y2 - x2y1 = 0,
+
+  represents the line determined by the data that it passes through two
+  given points (x1, y1) and (x2, y2). To prove this find m in the
+  equation y - y1 = m(x - x1) of a line through (x1, y1), from the
+  condition that (x2, y2) lies on the line.
+
+  In this paragraph the coordinates have been assumed rectangular. Had
+  they been oblique, the doctrine of similar triangles would have given
+  the same results, except that in the forms of equation y - k = m(x -
+  h), y = mx + b, we should not have had m = tan [alpha].
+
+  9. _The Circle._--It is easy to write down the equation of a given
+  circle. Let (h, k) be its given centre C, and [rho] the numerical
+  measure of its given radius. Take P (x, y) any point on its
+  circumference, and construct the triangle CRP, in fig. 50 as above.
+  The fact that this is right-angled tells us that
+
+    CR^2 + RP^2 = CP^2,
+
+  and this at once gives the equation
+
+    (x - h)^2 + (y - k)^2 = [rho]^2.
+
+  A point not upon the circumference of the particular circle is at some
+  distance from (h, k) different from [rho], and satisfies an equation
+  inconsistent with this one; which accordingly represents the
+  circumference, or, as we say, the circle.
+
+  The equation is of the form
+
+    x^2 + y^2 + 2Ax + 2By + C = 0.
+
+  Conversely every equation of this form represents a circle: we have
+  only to take -A, -B, A^2 + B^2 - C for h, k, [rho]^2 respectively, to
+  obtain its centre and radius. But this statement must appear too
+  unrestricted. Ought we not to require A^2 + B^2 - C to be positive?
+  Certainly, if by circle we are only to mean the visible round
+  circumference of the geometrical definition. Yet, analytically, we
+  contemplate altogether imaginary circles, for which [rho]^2 is
+  negative, and circles, for which [rho] = 0, with all their reality
+  condensed into their centres. Even when [rho]^2 is positive, so that a
+  visible round circumference exists, we do not regard this as
+  constituting the whole of the circle. Giving to x any value whatever
+  in (x - h)^2 + (y - k)^2 = [rho]^2, we obtain two values of y, real,
+  coincident or imaginary, each of which goes with the abscissa x as the
+  ordinate of a point, real or imaginary, on what is represented by the
+  equation of the circle.
+
+  The doctrine of the imaginary on a circle, and in geometry generally,
+  is of purely algebraical inception; but it has been in its entirety
+  accepted by modern pure geometers, and signal success has attended the
+  efforts of those who, like K.G.C. von Staudt, have striven to base its
+  conclusions on principles not at all algebraical in form, though of
+  course cognate to those adopted in introducing the imaginary into
+  algebra.
+
+  A circle with its centre at the origin has an equation x^2 + y^2 =
+  [rho]^2.
+
+  In oblique coordinates the general equation of a circle is x^2 + 2xy
+  cos [omega] + y^2 + 2Ax + 2By + C = 0.
+
+  10. The conic sections are the next simplest loci; and it will be seen
+  later that they are the loci represented by equations of the second
+  degree. Circles are particular cases of conic sections; and they have
+  just been seen to have for their equations a particular class of
+  equations of the second degree. Another particular class of such
+  equations is that included in the form (Ax + By + C)(A'x + B'y + C') =
+  0, which represents two straight lines, because the product on the
+  left vanishes if, and only if, one of the two factors does, i.e. if,
+  and only if, (x, y) lies on one or other of two straight lines. The
+  condition that ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0, which is often
+  written (a, b, c, f, g, h)(x, y, I)^2 = 0, takes this form is abc +
+  2fgh-af^2-bg^2 - ch^2 = 0. Note that the two lines may, in particular
+  cases, be parallel or coincident.
+
+  Any equation like F1(x, y) F2(x, y) ... F_n(x, y) = 0, of which the
+  left-hand side breaks up into factors, represents all the loci
+  separately represented by F1(x, y) = 0, F2(x, y) = 0, ... F_n(x, y) =
+  0. In particular an equation of degree n which is free from x
+  represents n straight lines parallel to the axis of x, and one of
+  degree n which is homogeneous in x and y, i.e. one which upon division
+  by x^n, becomes an equation in the ratio y/x, represents n straight
+  lines through the origin.
+
+  Curves represented by equations of the third degree are called cubic
+  curves. The general equation of this degree will be written (*)(x, y,
+  I)^3 = 0.
+
+  11. _Descriptive Geometry._--A geometrical proposition is either
+  descriptive or metrical: in the former case the statement of it is
+  independent of the idea of magnitude (length, inclination, &c.), and
+  in the latter it has reference to this idea. The method of coordinates
+  seems to be by its inception essentially metrical. Yet in dealing by
+  this method with descriptive propositions we are eminently free from
+  metrical considerations, because of our power to use general
+  equations, and to avoid all assumption that measurements implied are
+  any particular measurements.
+
+  [Illustration: FIG. 51.]
+
+  12. It is worth while to illustrate this by the instance of the
+  well-known theorem of the radical centre of three circles. The theorem
+  is that, given any three circles A, B, C (fig. 51), the common chords
+  [alpha][alpha]', [beta][beta]', [gamma][gamma]' of the three pairs of
+  circles meet in a point.
+
+  The geometrical proof is metrical throughout:--
+
+  Take O the point of intersection of [alpha][alpha]', [beta][beta]',
+  and joining this with [gamma]', suppose that [gamma]'O does not pass
+  through [gamma], but that it meets the circles A, B in two distinct
+  points [gamma]2, [gamma]1 respectively. We have then the known
+  metrical property of intersecting chords of a circle; viz. in circle
+  C, where [alpha][alpha]', [beta][beta]', are chords meeting at a point
+  O,
+
+    O[alpha].O[alpha]' = O[beta].O[beta]',
+
+  where, as well as in what immediately follows, O[alpha], &c. denote,
+  of course, _lengths_ or _distances_.
+
+  Similarly in circle A,
+
+    O[beta].O[beta]' = O[gamma]2.O[gamma]',
+
+  and in circle B,
+
+    O[alpha].O[alpha]' = O[gamma]1.O[gamma]'.
+
+  Consequently O[gamma]1.O[gamma]' = O[gamma]2.O[gamma]', that is,
+  O[gamma]1 = O[gamma]2, or the points [gamma]1 and [gamma]2 coincide;
+  that is, they each coincide with [gamma].
+
+  We contrast this with the analytical method:--
+
+  Here it only requires to be known that an equation Ax + By + C = 0
+  represents a line, and an equation x^2 + y^2 + Ax + By + C = 0
+  represents a circle. A, B, C have, in the two cases respectively,
+  metrical significations; but these we are not concerned with. Using S
+  to denote the function x^2 + y^2 + Ax + By + C, the equation of a
+  circle is S = o. Let the equation of any other circle be S', = x^2 +
+  y^2 + A'x + B'y + C' = 0; the equation S - S' = 0 is a linear equation
+  (S - S' is in fact = (A - A')x + (B - B')y + C - C), and it thus
+  represents a line; this equation is satisfied by the coordinates of
+  each of the points of intersection of the two circles (for at each of
+  these points S = 0 and S' = 0, therefore also S - S' = 0); hence the
+  equation S - S' = 0 is that of the line joining the two points of
+  intersection of the two circles, or say it is the equation of the
+  common chord of the two circles. Considering then a third circle S", =
+  x^2 + y^2 + A"x + B"y + C" = 0, the equations of the common chords are
+  S-S' = 0, S - S" = 0, S' - S" = 0 (each of these a linear equation);
+  at the intersection of the first and second of these lines S = S' and
+  S = S", therefore also S' = S", or the equation of the third line is
+  satisfied by the coordinates of the point in question; that is, the
+  three chords intersect in a point O, the coordinates of which are
+  determined by the equations S = S' = S".
+
+  It further appears that if the two circles S = 0, S' = 0 do not
+  intersect in any real points, they must be regarded as intersecting in
+  two imaginary points, such that the line joining them is the real line
+  represented by the equation S - S' = 0; or that two circles, whether
+  their intersections be real or imaginary, have always a real common
+  chord (or radical axis), and that for _any_ three circles the common
+  chords intersect in a point (of course real) which is the radical
+  centre. And by this very theorem, given two circles with imaginary
+  intersections, we can, by drawing circles which meet each of them in
+  real points, construct the radical axis of the first-mentioned two
+  circles.
+
+  13. The principle employed in showing that the equation of the common
+  chord of two circles is S - S' = 0 is one of very extensive
+  application, and some more illustrations of it may be given.
+
+  Suppose S = 0, S' = 0 are lines (that is, let S, S' now denote linear
+  functions Ax + By + C, A'x + B'y + C'), then S - kS' = 0 (k an
+  arbitrary constant) is the equation of any line passing through the
+  point of intersection of the two given lines. Such a line may be made
+  to pass through any given point, say the point (x0, y0); if S0, S'0
+  are what S, S' respectively become on writing for (x, y) the values
+  (x0, y0), then the value of k is k = S0 : S'0. The equation in fact is
+  SS'0 - S0S' = 0; and starting from this equation we at once verify it
+  _a posteriori_; the equation is a linear equation satisfied by the
+  values of (x, y) which make S = 0, S' = 0; and satisfied also by the
+  values (x0, y0); and it is thus the equation of the line in question.
+
+  If, as before, S = 0, S' = 0 represent circles, then (k being
+  arbitrary) S - kS' = 0 is the equation of any circle passing through
+  the two points of intersection of the two circles; and to make this
+  pass through a given point (x0, y0) we have again k = S0 : S'0. In the
+  particular case k = 1, the circle becomes the common chord (more
+  accurately it becomes the common chord together with the line
+  infinity; see S 23 below).
+
+  If S denote the general quadric function,
+
+    S = ax^2 +2hxy + by^2 + 2fy + 2gx + c,
+
+  then the equation S = 0 represents a conic; assuming this, then, if S'
+  = 0 represents another conic, the equation S - kS' = 0 represents
+  _any_ conic through the four points of intersection of the two conics.
+
+  [Illustration: FIG. 52.]
+
+  14. The object still being to illustrate the mode of working with
+  coordinates for descriptive purposes, we consider the theorem of the
+  polar of a point in regard to a circle. Given a circle and a point O
+  (fig. 52), we draw through O any two lines meeting the circle in the
+  points A, A' and B, B' respectively, and then taking Q as the
+  intersection of the lines AB' and A'B, the theorem is that the locus
+  of the point Q is a right line depending only upon O and the circle,
+  but independent of the particular lines OAA' and OBB'.
+
+  Taking O as the origin, and for the axes any two lines through O at
+  right angles to each other, the equation of the circle will be
+
+    x^2 + y^2 + 2Ax + 2By + C = 0;
+
+  and if the equation of the line OAA' is taken to be y = mx, then the
+  points A, A' are found as the intersections of the straight line with
+  the circle; or to determine x we have
+
+    x^2(1 + m^2) + 2x(A + Bm) + C = 0.
+
+  If(x1, y1) are the coordinates of A, and (x2, y2) of A', then the
+  roots of this equation are x1, x2, whence easily
+
+    1    1        A + Bm
+    -- + --  = -2 ------.
+    x1   x2         C
+
+  And similarly, if the equation of the line OBB' is taken to be y =
+  m'x1 and the coordinates of B, B' to be (x3, y3) and (x4, y4)
+  respectively, then
+
+    1    1         A + Bm'
+    -- + --  =  -2 -------.
+    x3   x4           C'
+
+  We have then by S 8
+
+    x(y1 - y4) - y(x1 - x4) + x1y4 - x4y1 = 0,
+
+    x(y2 - y3) - y(x2 - x3) + x2y3 - x3y2 = 0,
+
+  as the equations of the lines AB' and A'B respectively. Reducing by
+  means of the relations y1 - mx1 = 0, y2 - mx2 = 0, y3 - m'x3 = 0, y4 -
+  m'x4 = 0, the two equations become
+
+    x(mx1 - m'x4) - y(x1 - x4) + (m'- m)x1x4 = 0,
+
+    x(mx2 - m'x3) - y(x2 - x3) + (m'- m)x2x3 = 0,
+
+  and if we divide the first of these equations by x1x4, and the second
+  by x2x3 and then add, we obtain
+      _                               _      _                       _
+     |   / 1    1  \       / 1    1 \  |    |  1    1     / 1    1 \  |
+    x| m(  -- + --  ) - m'(  -- + -- ) | - y|  -- + -- - (  -- + -- ) |
+     |_  \ x3   x4 /       \ x1   x2/ _|    |_ x3   x4    \ x1   x2/ _|
+
+      + 2m' - 2m = 0,
+
+  or, what is the same thing,
+
+     / 1    1  \              / 1    1  \
+    (  -- + --  )(y - m'x) - (  -- + --  )(y - mx) + 2m' - 2m = 0,
+     \ x1   x2 /              \ x3   x4 /
+
+  which by what precedes is the equation of a line through the point Q.
+  Substituting herein for 1/x1 + 1/x2, 1/x3 + 1/x4 their foregoing
+  values, the equation becomes
+
+    -(A + Bm)(y - m'x) + (A + Bm')(y - mx) + C(m' - m) = 0;
+
+  that is,
+
+    (m - m')(Ax + By + C) = 0;
+
+  or finally it is Ax + By + C = 0, showing that the point Q lies in a
+  line the position of which is independent of the particular lines
+  OAA', OBB' used in the construction. It is proper to notice that there
+  is no correspondence to each other of the points A, A' and B, B'; the
+  grouping might as well have been A, A' and B', B; and it thence
+  appears that the line Ax + By + C = 0 just obtained is in fact the
+  line joining the point Q with the point R which is the intersection of
+  AB and A'B'.
+
+  15. In S 8 it has been seen that two conditions determine the equation
+  of a straight line, because in Ax + By + C = 0 one of the coefficients
+  may be divided out, leaving only two parameters to be determined.
+  Similarly five conditions instead of six determine an equation of the
+  second degree (a, b, c, f, g, h)(x, y, 1)^2 = 0, and nine instead of
+  ten determine a cubic (*)(x, y, 1)^3 = 0. It thus appears that a cubic
+  can be made to pass through 9 given points, and that the cubic so
+  passing through 9 given points is completely determined. There is,
+  however, a remarkable exception. Considering two given cubic curves S
+  = 0, S' = 0, these intersect in 9 points, and through these 9 points
+  we have the whole series of cubics S - kS' = 0, where k is an
+  arbitrary constant: k may be determined so that the cubic shall pass
+  through a given tenth point (k = S0 : S'0, if the coordinates are (x0,
+  y0), and S0, S'0 denote the corresponding values of S, S'). The
+  resulting curve SS'0 - S'S0 = 0 may be regarded as the cubic
+  determined by the conditions of passing through 8 of the 9 points and
+  through the given point (x0, y0); and from the equation it thence
+  appears that the curve passes through the remaining one of the 9
+  points. In other words, we thus have the theorem, any cubic curve
+  which passes through 8 of the 9 intersections of two given cubic
+  curves passes through the 9th intersection.
+
+  The applications of this theorem are very numerous; for instance, we
+  derive from it Pascal's theorem of the inscribed hexagon. Consider a
+  hexagon inscribed in a conic. The three alternate sides constitute a
+  cubic, and the other three alternate sides another cubic. The cubics
+  intersect in 9 points, being the 6 vertices of the hexagon, and the 3
+  Pascalian points, or intersections of the pairs of opposite sides of
+  the hexagon. Drawing a line through two of the Pascalian points, the
+  conic and this line constitute a cubic passing through 8 of the 9
+  points of intersection, and it therefore passes through the remaining
+  point of intersection--that is, the third Pascalian point; and since
+  obviously this does not lie on the conic, it must lie on the
+  line--that is, we have the theorem that the three Pascalian points (or
+  points of intersection of the pairs of opposite sides) lie on a line.
+
+  16. _Metrical Theory resumed. Projections and Perpendiculars._--It
+  is a metrical fact of fundamental importance, already used in S 8,
+  that, if a finite line PQ be projected on any other line OO' by
+  perpendiculars PP', QQ' to OO', the length of the projection P'Q' is
+  equal to that of PQ multiplied by the cosine of the acute angle
+  between the two lines. Also the algebraical sum of the projections of
+  the sides of any closed polygon upon any line is zero, because as a
+  point goes round the polygon, from any vertex A to A again, the point
+  which is its projection on the line passes from A' the projection of A
+  to A' again, i.e. traverses equal distances along the line in positive
+  and negative senses. If we consider the polygon as consisting of two
+  broken lines, each extending from the same initial to the same
+  terminal point, the sum of the projections of the lines which compose
+  the one is equal, in sign and magnitude, to the sum of the projections
+  of the lines composing the other. Observe that the projection on a
+  line of a length perpendicular to the line is zero.
+
+  Let us hence find the equation of a straight line such that the
+  perpendicular OD on it from the origin is of length [rho] taken as
+  positive, and is inclined to the axis of x at an angle xOD = [alpha],
+  measured counter-clockwise from Ox. Take any point P (x, y) on the
+  line, and construct OM and MP as in fig. 48. The sum of the
+  projections of OM and MP on OD is OD itself; and this gives the
+  equation of the line
+
+    x cos [alpha] + y sin [alpha] = [rho].
+
+  Observe that cos [alpha] and sin [alpha] here are the sin [alpha] and
+  -cos [alpha], or the -sin [alpha] and cos [alpha] of S 8 according to
+  circumstances.
+
+  We can write down an expression for the perpendicular distance from
+  this line of any point (x', y') which does not lie upon it. If the
+  parallel through (x', y') to the line meet OD in E, we have x' cos
+  [alpha] + y' sin [alpha] = OE, and the perpendicular distance required
+  is OD - OE, i.e. [rho] - x' cos [alpha] - y' sin [alpha]; it is the
+  perpendicular distance taken positively or negatively according as
+  (x', y') lies on the same side of the line as the origin or not.
+
+  The general equation Ax + By + C = 0 may be given the form x cos
+  [alpha] + y sin [alpha] - [rho] = 0 by dividing it by [root](A^2 +
+  B^3). Thus (Ax' + By' + C) / [root](A^2 + B^2) is in absolute value
+  the perpendicular distance of (x', y') from the line Ax + By + C = 0.
+  Remember, however, that there is an essential ambiguity of sign
+  attached to a square root. The expression found gives the distance
+  taken positively when (x', y') is on the origin side of the line, if
+  the sign of C is given to [root](A^2 + B^2).
+
+  17. _Transformation of Coordinates._--We often need to adopt new axes
+  of reference in place of old ones; and the above principle of
+  projections readily expresses the old coordinates of any point in
+  terms of the new.
+
+  [Illustration: FIG. 53.]
+
+  Suppose, for instance, that we want to take for new origin the point
+  O' of old coordinates OA = h, AO' = k, and for new axes of X and Y
+  lines through O' obtained by rotating parallels to the old axes of x
+  and y through an angle [theta] counter-clockwise. Construct (fig. 53)
+  the old and new coordinates of any point P. Expressing that the
+  projections, first on the old axis of x and secondly on the old axis
+  of y, of OP are equal to the sums of the projections, on those axes
+  respectively, of the parts of the broken line OO'M'P, we obtain:
+
+    x = h + X cos [theta] + Y cos ([theta] + 1/2[pi]) = h + X cos [theta] -
+      Y sin [theta],
+
+  and
+
+    y = k + X cos (1/2[pi] - [theta]) + Y cos [theta] = k + X sin [theta] +
+      Y cos [theta].
+
+  Be careful to observe that these formulae do not apply to every
+  conceivable change of reference from one set of rectangular axes to
+  another. It might have been required to take O'X, O'Y' for the
+  positive directions of the new axes, so that the change of directions
+  of the axes could not be effected by rotation. We must then write -Y
+  for Y in the above.
+
+  Were the new axes oblique, making angles [alpha], [beta] respectively
+  with the old axis of x, and so inclined at the angle [beta] - [alpha],
+  the same method would give the formulae
+
+    x = h + X cos [alpha] + Y cos [beta], y = k + X sin [alpha] + Y sin [beta].
+
+  18. _The Conic Sections._--The conics, as they are now called, were at
+  first defined as curves of intersection of planes and a cone; but
+  Apollonius substituted a definition free from reference to space of
+  three dimensions. This, in effect, is that a conic is the locus of a
+  point the distance of which from a given point, called the focus, has
+  a given ratio to its distance from a given line, called the directrix
+  (see CONIC SECTION). If e : 1 is the ratio, e is called the
+  eccentricity. The distances are considered signless.
+
+  Take (h, k) for the focus, and x cos [alpha] + y sin [alpha] - p = 0
+  for the directrix. The absolute values of [root] {(x - h)^2 + (y -
+  k)^2} and p - x cos [alpha] -y sin [alpha] are to have the ratio e :
+  1; and this gives
+
+    (x - h)^2 + (y - k)^2 = e^2(p - x cos [alpha] - y sin [alpha])^2
+
+  as the general equation, in rectangular coordinates, of a conic.
+
+  It is of the second degree, and is the general equation of that
+  degree. If, in fact, we multiply it by an unknown [lambda], we can, by
+  solving six simultaneous equations in the six unknowns [lambda], h, k,
+  e, p, [alpha], so choose values for these as to make the coefficients
+  in the equation equal to those in any equation of the second degree
+  which may be given. There is no failure of this statement in the
+  special case when the given equation represents two straight lines, as
+  in S 10, but there is speciality: if the two lines intersect, the
+  intersection and either bisector of the angle between them are a focus
+  and directrix; if they are united in one line, any point on the line
+  and a perpendicular to it through the point are: if they are parallel,
+  the case is a limiting one in which e and h^2 + k^2 have become
+  infinite while e^(-2)(h^2 + k^2) remains finite. In the case (S 9) of
+  an equation such as represents a circle there is another instance of
+  proceeding to a limit: e has to become 0, while ep remains finite:
+  moreover [alpha] is indeterminate. The centre of a circle is its
+  focus, and its directrix has gone to infinity, having no special
+  direction. This last fact illustrates the necessity, which is also
+  forced on plane geometry by three-dimensional considerations, of
+  treating all points at infinity in a plane as lying on a single
+  straight line.
+
+  Sometimes, in reducing an equation to the above focus and directrix
+  form, we find for h, k, e, p, tan [alpha], or some of them, only
+  imaginary values, as quadratic equations have to be solved; and we
+  have in fact to contemplate the existence of entirely imaginary
+  conics. For instance, no real values of x and y satisfy x^2 + 2y^2 + 3
+  = 0. Even when the locus represented is real, we obtain, as a rule,
+  four sets of values of h, k, e, p, of which two sets are imaginary; a
+  real conic has, besides two real foci and corresponding directrices,
+  two others that are imaginary.
+
+  In oblique as well as rectangular coordinates equations of the second
+  degree represent conics.
+
+  19. _The three Species of Conics._--A real conic, which does not
+  degenerate into straight lines, is called an ellipse, parabola or
+  hyperbola according as e <, = , or > 1. To trace the three forms it is best so to
+  choose the axes of reference as to simplify their equations.
+
+  In the case of a parabola, let 2c be the distance between the given
+  focus and directrix, and take axes referred to which these are the
+  point (c, 0) and the line x = - c. The equation becomes (x - c)^2 +
+  y^2 = (x + c)^2, i.e. y^2 = 4cx.
+
+  In the other cases, take a such that a(e ~ e^(-1)) is the distance of
+  focus from directrix, and so choose axes that these are (ae, 0) and x
+  = ae^(-1), thus getting the equation(x - ae)^2 + y^2 = e^2(x -
+  ae^(-1))^2, i.e. (1 - e^2)x^2 + y^2 = a^2(1 - e^2). When e < 1, i.e.
+  in the case of an ellipse, this may be written x^2/a^2 + y^2/b^2 = 1,
+  where b^2 = a^2(1 - e^2); and when e > 1, i.e. in the case of an
+  hyperbola, x^2/a^2 - y^2/b^2 = 1, where b^2 = a^2(e^2 - 1). The axes
+  thus chosen for the ellipse and hyperbola are called the principal
+  axes.
+
+  In figs. 54, 55, 56 in order, conics of the three species, thus
+  referred, are depicted.
+
+  [Illustration: FIG. 54]
+
+  [Illustration: FIG. 55]
+
+  [Illustration: FIG. 56.]
+
+  The oblique straight lines in fig. 56 are the _asymptotes_ x/a =
+  [+-]y/b of the hyperbola, lines to which the curve tends with
+  unlimited closeness as it goes to infinity. The hyperbola would have
+  an equation of the form xy = c if referred to its asymptotes as axes,
+  the coordinates being then oblique, unless a = b, in which case the
+  hyperbola is called rectangular. An ellipse has two imaginary
+  asymptotes. In particular a circle x^2 + y^2 = a^2, a particular
+  ellipse, has for asymptotes the imaginary lines x = [+-]y [root]-1.
+  These run from the centre to the so-called circular points at
+  infinity.
+
+  20. _Tangents and Curvature._--Let (x', y') and (x' + h, y' + k) be
+  two neighbouring points P, P' on a curve. The equation of the line on
+  which both lie is h(y - y') = k(x - x'). Now keep P fixed, and let P'
+  move towards coincidence with it along the curve. The connecting line
+  will tend towards a limiting position, to which it can never attain as
+  long as P and P' are distinct. The line which occupies this limiting
+  position is the tangent at P. Now if we subtract the equation of the
+  curve, with (x', y') for the coordinates in it, from the like equation
+  in (x' + h, y' + k), we obtain a relation in h and k, which will, as a
+  rule, be of the form 0 = Ah + Bk + terms of higher degrees in h and k,
+  where A, B and the other coefficients involve x' and y'. This gives
+  k/h = -A/B + terms which tend to vanish as h and k do, so that -A : B
+  is the limiting value tended to by k : h. Hence the equation of the
+  tangent is B(y - y') + A(x - x') = 0.
+
+  The _normal_ at (x', y') is the line through it at right angles to the
+  tangent, and its equation is A(y - y') - B(x - x') = 0.
+
+  In the case of the conic (a, b, c, f, g, h) (x, y, 1)^2 = 0 we find
+  that A/B = (ax' + hy' + g)/(hx' + by' + f).
+
+  We can obtain the coordinates of Q, the intersection of the normals
+  QP, QP' at (x', y') and (x' + h, y' + k), and then, using the limiting
+  value of k : h, deduce those of its limiting position as P' moves up
+  to P. This is the _centre of curvature_ of the curve at P (x', y'),
+  and is so called because it is the centre of the circle of closest
+  contact with the curve at that point. That it is so follows from the
+  facts that the closest circle is the limit tended to by the circle
+  which touches the curve at P and passes through P', and that the arc
+  from P to P' of this circle lies between the circles of centre Q and
+  radii QP, QP', which circles tend, not to different limits as P' moves
+  up to P, but to one. The distance from P to the centre of curvature is
+  the _radius of curvature_.
+
+  21. _Differential Plane Geometry._--The language and notation of the
+  differential calculus are very useful in the study of tangents and
+  curvature. Denoting by ([xi], [eta]) the current coordinates, we find,
+  as above, that the tangent at a point (x, y) of a curve is [eta] - y =
+  ([xi] - x)dy/dx, where dy/dx is found from the equation of the curve.
+  If this be f(x, y) = 0 the tangent is ([xi] - x) (dPf/dPx) + ([eta] -
+  y) (dPf/dPy) = 0. If [rho] and ([alpha], [beta]) are the radius and
+  centre of curvature at (x, y), we find that q([alpha] - x) = -p(1 +
+  p^2), q([beta] - y) = 1 + p^2, q^2[rho]^2 = (1 + p^2)^3, where p, q
+  denote dy/dx, d^2y/dx^2 respectively. (See INFINITESIMAL CALCULUS.)
+
+  In any given case we can, at all events in theory, eliminate x, y
+  between the above equations for [alpha] - x and [beta] - y, and the
+  equation of the curve. The resulting equation in ([alpha], [beta])
+  represents the locus of the centre of curvature. This is the _evolute_
+  of the curve.
+
+  22. _Polar Coordinates._--In plane geometry the distance of any point
+  P from a fixed origin (or pole) O, and the inclination xOP of OP to a
+  fixed line Ox, determine the point: r, the numerical measure of OP,
+  the _radius vector_, and [theta], the circular measure of xOP, the
+  _inclination_, are called polar coordinates of P. The formulae x = r
+  cos [theta], y = r sin [theta] connect Cartesian and polar
+  coordinates, and make transition from either system to the other easy.
+  In polar coordinates the equations of a circle through O, and of a
+  conic with O as focus, take the simple forms r = 2a cos
+  ([theta]-[alpha]), r {1 - e cos ([theta]-[alpha])} = l. The use of
+  polar coordinates is very convenient in discussing curves which have
+  properties of symmetry akin to that of a regular polygon, such curves
+  for instance as r = a cos m [theta], with m integral, and also the
+  curves called spirals, which have equations giving r as functions of
+  [theta] itself, and not merely of sin [theta] and cos [theta]. In the
+  geometry of motion under central forces the advantage of working with
+  polar coordinates is great.
+
+  23. _Trilinear and Areal Coordinates._--Consider a fixed triangle ABC,
+  and regard its sides as produced without limit. Denote, as in
+  trigonometry, by a, b, c the positive numbers of units of a chosen
+  scale contained in the lengths BC, CA, AB, by A, B, C the angles, and
+  by [Delta] the area, of the triangle. We might, as in S 6, take CA, CB
+  as axes of x and y, inclined at an angle C. Any point P (x, y) in the
+  plane is at perpendicular distances y sin C and x sin C from CA and
+  CB. Call these [beta] and [alpha] respectively. The signs of [beta]
+  and [alpha] are those of y and x, i.e. [beta] is positive or negative
+  according as P lies on the same side of CA as B does or the opposite,
+  and similarly for [alpha]. An equation in (x, y) of any degree may,
+  upon replacing in it x and y by [alpha] cosec C and [beta] cosec C, be
+  written as one of the same degree in ([alpha], [beta]). Now let
+  [gamma] be the perpendicular distance of P from the third side AB,
+  taken as positive or negative as P is on the C side of AB or not. The
+  geometry of the figure tells us that a[alpha] + b[beta] + c[gamma] =
+  2[Delta]. By means of this relation in [alpha], [beta], [gamma] we can
+  give an equation considered countless other forms, involving two or
+  all of [alpha], [beta], [gamma]. In particular we may make it
+  _homogeneous_ in [alpha], [beta], [gamma]: to do this we have only to
+  multiply the terms of every degree less than the highest present in
+  the equation by a power of (a[alpha] + b[beta] + c[gamma])/2[Delta]
+  just sufficient to raise them, in each case, to the highest degree.
+
+  We call ([alpha], [beta], [gamma]) _trilinear coordinates_, and an
+  equation in them the trilinear equation of the locus represented.
+  Trilinear equations are, as a rule, dealt with in their homogeneous
+  forms. An advantage thus gained is that we need not mean by ([alpha],
+  [beta], [gamma]) the actual measures of the perpendicular distances,
+  but any properly signed numbers which have the same ratio two and two
+  as these distances.
+
+  In place of [alpha], [beta], [gamma] it is lawful to use, as
+  coordinates specifying the position of a point in the plane of a
+  triangle of reference ABC, any given multiples of these. For instance,
+  we may use x = a[alpha]/2[Delta], y = b[beta]/2[Delta], z =
+  c[gamma]/2[Delta], the properly signed ratios of the triangular areas
+  PBC, PCA, PAB to the triangular area ABC. These are called the _areal_
+  coordinates of P. In areal coordinates the relation which enables us
+  to make any equation homogeneous takes the simple form x + y + z = 1;
+  and, as before, we need mean by x, y, z, in a homogeneous equation,
+  only signed numbers in the right ratios.
+
+  Straight lines and conics are represented in trilinear and in areal,
+  because in Cartesian, coordinates by equations of the first and second
+  degrees respectively, and these degrees are preserved when the
+  equations are made homogeneous. What must be said about points
+  infinitely far off in order to make universal the statement, to which
+  there is no exception as long as finite distances alone are
+  considered, that _every_ homogeneous equation of the first degree
+  represents a straight line? Let the point of areal coordinates (x',
+  y', z') move infinitely far off, and mean by x, y, z finite quantities
+  in the ratios which x', y', z' tend to assume as they become infinite.
+  The relation x' + y' + z' = 1 gives that the limiting state of things
+  tended to is expressed by x + y + z = 0. This particular equation of
+  the first degree is satisfied by no point at a finite distance; but we
+  see the propriety of saying that it has to be taken as satisfied by
+  all the points conceived of as actually at infinity. Accordingly the
+  special property of these points is expressed by saying that they lie
+  on a special straight line, of which the areal equation is x + y + z =
+  0. In trilinear coordinates this _line at infinity_ has for equation
+  a[alpha] + b[beta] + c[gamma] = 0.
+
+  On the one special line at infinity parallel lines are treated as
+  meeting. There are on it two special (imaginary) points, the circular
+  points at infinity of S 19, through which all circles pass in the same
+  sense. In fact if S = O be one circle, in areal coordinates, S + (x +
+  y + z)(lx + my + nz) = 0 may, by proper choice of l, m, n, be made any
+  other; since the added terms are once lx + my + nz, and have the
+  generality of any expression like a'x + b'y + c' in Cartesian
+  coordinates. Now these two circles intersect in the two points where
+  either meets x + y + z = 0 as well as in two points on the radical
+  axis lx + my + nz = 0.
+
+  24. Let us consider the perpendicular distance of a point ([alpha]',
+  [beta]', [gamma]') from a line l[alpha] + m[beta] + n[gamma]. We can
+  take rectangular axes of Cartesian coordinates (for clearness as to
+  equalities of angle it is best to choose an origin inside ABC), and
+  refer to them, by putting expressions p - x cos[theta] - y sin[theta],
+  &c., for [alpha] &c.; we can then apply S 16 to get the perpendicular
+  distance; and finally revert to the trilinear notation. The result is
+  to find that the required distance is
+
+    (l[alpha]' + m[beta]' + n[gamma]')/{l, m, n},
+
+  where {l, m, n}^2 = l^2 + m^2 + n^2 - 2mn cos A - 2nl cos B - 2lm cos C.
+
+  In areal coordinates the perpendicular distance from (x', y', z') to
+  lx + my + nz = 0 is 2[Delta](lx' + my' + nz')/{al, bm, cn}. In both
+  cases the coordinates are of course actual values.
+
+  Now let [xi], [eta], [zeta] be the perpendiculars on the line from the
+  vertices A, B, C, i.e. the points (1, 0, 0), (0, 1, 0), (0, 0, 1),
+  with signs in accord with a convention that oppositeness of sign
+  implies distinction between one side of the line and the other. Three
+  applications of the result above give
+
+    [xi]/l = 2[Delta]/{al, bm, cn} = [eta]/m = [zeta]/n;
+
+  and we thus have the important fact that [xi]x' + [eta]y' + [zeta]z'
+  is the perpendicular distance between a point of areal coordinates
+  (x'y'z') and a line on which the perpendiculars from A, B, C are [xi],
+  [eta], [zeta] respectively. We have also that [xi]x + [eta]y + [zeta]z
+  = 0 is the areal equation of the line on which the perpendiculars are
+  [xi], [eta], [zeta]; and, by equating the two expressions for the
+  perpendiculars from (x', y', z') on the line, that in all cases
+  {a[xi], b[eta], c[zeta]}^2 = 4[Delta]^2.
+
+  25. _Line-coordinates. Duality._--A quite different order of ideas
+  may be followed in applying analysis to geometry. The notion of a
+  straight line specified may precede that of a point, and points may be
+  dealt with as the intersections of lines. The specification of a line
+  may be by means of coordinates, and that of a point by an equation,
+  satisfied by the coordinates of lines which pass through it. Systems
+  of _line-coordinates_ will here be only briefly considered. Every such
+  system is allied to some system of point-coordinates; and space will
+  be saved by giving prominence to this fact, and not recommencing _ab
+  initio_.
+
+  Suppose that any particular system of point-coordinates, in which lx +
+  my + nz = 0 may represent any straight line, is before us: notice that
+  not only are trilinear and areal coordinates such systems, but
+  Cartesian coordinates also, since we may write x/z, y/z for the
+  Cartesian x, y, and multiply through by z. The line is exactly
+  assigned if l, m, n, or their mutual ratios, are known. Call (l, m, n)
+  the _coordinates_ of the line. Now keep x, y, z constant, and let the
+  coordinates of the line vary, but always so as to satisfy the
+  equation. This equation, which we now write xl + ym + zn = 0, is
+  satisfied by the coordinates of every line through a certain fixed
+  point, and by those of no other line; it is the equation of that point
+  in the line-coordinates l, m, n.
+
+  Line-coordinates are also called _tangential_ coordinates. A curve is
+  the envelope of lines which touch it, as well as the locus of points
+  which lie on it. A homogeneous equation of degree above the first in
+  l, m, n is a relation connecting the coordinates of every line which
+  touches some curve, and represents that curve, regarded as an
+  envelope. For instance, the condition that the line of coordinates (l,
+  m, n), i.e. the line of which the allied point-coordinate equation is
+  lx + my + nz = 0, may touch a conic (a, b, c, f, g, h) (x, y, z)^2 =
+  0, is readily found to be of the form (A, B, C, F, G, H) (l, m, n)^2 =
+  0, i.e. to be of the second degree in the line-coordinates. It is not
+  hard to show that the _general_ equation of the second degree in l, m,
+  n thus represents a conic; but the degenerate conics of
+  line-coordinates are not line-pairs, as in point-coordinates, but
+  point-pairs.
+
+  The degree of the point-coordinate equation of a curve is the _order_
+  of the curve, the number of points in which it cuts a straight line.
+  That of the line-coordinate equation is its _class_, the number of
+  tangents to it from a point. The order and class of a curve are
+  generally different when either exceeds two.
+
+  26. The system of line-coordinates allied to the areal system of
+  point-coordinates has special interest.
+
+  The l, m, n of this system are the perpendiculars [xi], [eta], [zeta]
+  of S 24; and x'[xi] + y'[eta] + z'[zeta] = 0 is the equation of the
+  point of areal coordinates (x', y', z'), i.e. is a relation which the
+  perpendiculars from the vertices of the triangle of reference on every
+  line through the point, but no other line, satisfy. Notice that a
+  non-homogeneous equation of the first degree in [xi], [eta], [zeta]
+  does not, as a homogeneous one does, represent a point, but a circle.
+  In fact x'[xi] + y'[eta] + z'[zeta] = R expresses the constancy of the
+  perpendicular distance of the fixed point x'[xi] + y'[eta] + z'[zeta]
+  = 0 from the variable line ([xi], [eta], [zeta]), i.e. the fact that
+  ([xi], [eta], [zeta]) touches a circle with the fixed point for
+  centre. The relation in any [xi], [eta], [zeta] which enables us to
+  make an equation homogeneous is not linear, as in point-coordinates,
+  but quadratic, viz. it is the relation {a[xi], b[eta], c[zeta]}^2 =
+  4[Delta]^2 of S 24. Accordingly the homogeneous equation of the above
+  circle is
+
+    4[Delta]^2(x'[xi] + y'[eta] + z'[zeta])^2 = R^2{a[xi], b[eta], c[zeta]}^2.
+
+  Every circle has an equation of this form in the present system of
+  line-coordinates. Notice that the equation of any circle is satisfied
+  by those coordinates of lines which satisfy both x'[xi] + y'[eta] +
+  z'[zeta] = 0, the equation of its centre, and {a[xi], b[eta],
+  c[zeta]}^2 = 0. This last equation, of which the left-hand side
+  satisfies the condition for breaking up into two factors, represents
+  the two imaginary circular points at infinity, through which all
+  circles and their asymptotes pass.
+
+  There is strict duality in descriptive geometry between
+  point-line-locus and line-point-envelope theorems. But in metrical
+  geometry duality is encumbered by the fact that there is in a plane
+  one special line only, associated with distance, while of special
+  points, associated with direction, there are two: moreover the line is
+  real, and the points both imaginary.
+
+
+  II. _Solid Analytical Geometry._
+
+  27. Any point in space may be specified by three coordinates. We
+  consider three fixed planes of reference, and generally, as in all
+  that follows, three which are at right angles two and two. They
+  intersect, two and two, in lines x'Ox, y'Oy, z'Oz, called the axes of
+  x, y, z respectively, and divide all space into eight parts called
+  octants. If from any point P in space we draw PN parallel to zOz' to
+  meet the plane xOy in N, and then from N draw NM parallel to yOy' to
+  meet x'Ox in M, the coordinates (x, y, z) of P are the numerical
+  measures of OM, MN, NP; in the case of rectangular coordinates these
+  are the perpendicular distances of P from the three planes of
+  reference. The sign of each coordinate is positive or negative as P
+  lies on one side or the other of the corresponding plane. In the
+  octant delineated the signs are taken all positive.
+
+  [Illustration: FIG. 57.]
+
+  [Illustration: FIG. 58.]
+
+  In fig. 57 the delineation is on a plane of the paper taken parallel
+  to the plane zOx, the points of a solid figure being projected on that
+  plane by parallels to some chosen line through O in the positive
+  octant. Sometimes it is clearer to delineate, as in fig. 58, by
+  projection parallel to that line in the octant which is equally
+  inclined to Ox, Oy, Oz upon a plane of the paper perpendicular to it.
+  It is possible by parallel projection to delineate equal scales along
+  Ox, Oy, Oz by scales having any ratios we like along lines in a plane
+  having any mutual inclinations we like.
+
+  [Illustration: FIG. 59.]
+
+  For the delineation of a surface of simple form it frequently suffices
+  to delineate the sections by the coordinate planes; and, in
+  particular, when the surface has symmetry about each coordinate plane,
+  to delineate the quarter-sections belonging to a single octant. Thus
+  fig. 59 conveniently represents an octant of the wave surface, which
+  cuts each coordinate plane in a circle and an ellipse. Or we may
+  delineate a series of contour lines, i.e. sections by planes parallel
+  to xOy, or some other chosen plane; of course other sections may be
+  indicated too for greater clearness. For the delineation of a curve a
+  good method is to represent, as above, a series of points P thereof,
+  each accompanied by its ordinate PN, which serves to refer it to the
+  plane of xy. The employment of stereographic projection is also
+  interesting.
+
+  28. In plane geometry, reckoning the line as a curve of the first
+  order, we have only the point and the curve. In solid geometry,
+  reckoning a line as a curve of the first order, and the plane as a
+  surface of the first order, we have the point, the curve and the
+  surface; but the increase of complexity is far greater than would
+  hence at first sight appear. In plane geometry a curve is considered
+  in connexion with lines (its tangents); but in solid geometry the
+  curve is considered in connexion with lines and planes (its tangents
+  and osculating planes), and the surface also in connexion with lines
+  and planes (its tangent lines and tangent planes); there are surfaces
+  arising out of the line--cones, skew surfaces, developables, doubly
+  and triply infinite systems of lines, and whole classes of theories
+  which have nothing analogous to them in plane geometry: it is thus a
+  very small part indeed of the subject which can be even referred to in
+  the present article.
+
+  In the case of a surface we have between the coordinates (x, y, z) a
+  single, or say a onefold relation, which can be represented by a
+  single relation [f](x, y, z) = 0; or we may consider the coordinates
+  expressed each of them as a given function of two variable parameters
+  p, q; the form z = [f](x, y) is a particular case of each of these
+  modes of representation; in other words, we have in the first mode
+  [f](x, y, z) = z - [f](x, y), and in the second mode x = p, y = q for
+  the expression of two of the coordinates in terms of the parameters.
+
+  In the case of a curve we have between the coordinates (x, y, z) a
+  twofold relation: two equations [f](x, y, z) = 0, [phi](x, y, z) = 0
+  give such a relation; i.e. the curve is here considered as the
+  intersection of two surfaces (but the curve is not always the complete
+  intersection of two surfaces, and there are hence difficulties); or,
+  again, the coordinates may be given each of them as a function of a
+  single variable parameter. The form y = [phi](x), z = [psi](x), where
+  two of the coordinates are given in terms of the third, is a
+  particular case of each of these modes of representation.
+
+  29. The remarks under plane geometry as to descriptive and metrical
+  propositions, and as to the non-metrical character of the method of
+  coordinates when used for the proof of a descriptive proposition,
+  apply also to solid geometry; and they might be illustrated in like
+  manner by the instance of the theorem of the radical centre of four
+  spheres. The proof is obtained from the consideration that S and S'
+  being each of them a function of the form x^2 + y^2 + z^2 + ax + by +
+  cz + d, the difference S-S' is a mere linear function of the
+  coordinates, and consequently that S-S' = 0 is the equation of the
+  plane containing the circle of intersection of the two spheres S = 0
+  and S' = 0.
+
+  [Illustration: FIG. 60.]
+
+  30. _Metrical Theory._--The foundation in solid geometry of the
+  metrical theory is in fact the before-mentioned theorem that if a
+  finite right line PQ be projected upon any other line OO' by lines
+  perpendicular to OO', then the length of the projection P'Q' is equal
+  to the length of PQ into the cosine of its inclination to P'Q'--or (in
+  the form in which it is now convenient to state the theorem) the
+  perpendicular distance P'Q' of two parallel planes is equal to the
+  inclined distance PQ into the cosine of the inclination. The principle
+  of S 16, that the algebraical sum of the projections of the sides of
+  any closed polygon on any line is zero, or that the two sets of sides
+  of the polygon which connect a vertex A and a vertex B have the same
+  sum of projections on the line, in sign and magnitude, as we pass from
+  A to B, is applicable when the sides do not all lie in one plane.
+
+  31. Consider the skew quadrilateral QMNP, the sides QM, MN, NP being
+  respectively parallel to the three rectangular axes Ox, Oy, Oz; let
+  the lengths of these sides be [xi], [eta], [zeta], and that of the
+  side QP be = [rho]; and let the cosines of the inclinations (or say
+  the cosine-inclinations) of [rho] to the three axes be [alpha],
+  [beta], [gamma]; then projecting successively on the three sides and
+  on QP we have
+
+    [xi], [eta], [zeta] = [rho][alpha], [rho][beta], [rho][gamma],
+
+  and
+
+    [rho] = [alpha][xi] + [beta][eta] + [gamma][zeta],
+
+  whence [rho]^2 = [xi]^2 + [eta]^2 + [zeta]^2, which is the relation
+  between a distance [rho] and its projections [xi], [eta], [zeta] upon
+  three rectangular axes. And from the same equations we obtain
+  [alpha]^2 + [beta]^2 + [gamma]^2 = 1, which is a relation connecting
+  the cosine-inclinations of a line to three rectangular axes.
+
+  Suppose we have through Q any other line QT, and let the
+  cosine-inclinations of this to the axes be [alpha]', [beta]',
+  [gamma]', and [delta] be its cosine-inclination to QP; also let [rho]
+  be the length of the projection of QP upon QT; then projecting on QT
+  we have
+
+    [rho] = [alpha]'[xi] + [beta]'[eta] + [gamma]'[zeta] = [rho][delta].
+
+  And in the last equation substituting for [xi], [eta], [zeta] their
+  values [rho][alpha], [rho][beta], [rho][gamma] we find
+
+    [delta] = [alpha][alpha]' + [beta][beta]' + [gamma][gamma]',
+
+  which is an expression for the mutual cosine-inclination of two lines,
+  the cosine-inclinations of which to the axes are [alpha], [beta],
+  [gamma] and [alpha]', [beta]', [gamma]' respectively. We have of
+  course [alpha]^2 + [beta]^2 + [gamma]^2 = 1 and [alpha]'^2 + [beta]'^2
+  + [gamma]'^2 = 1; and hence also
+
+    1 - [delta]^2 = ([alpha]^2 + [beta]^2 + [gamma]^2)([alpha]'^2 + [beta]'^2 + [gamma]'^2)
+      - ([alpha][alpha]' + [beta][beta]' + [gamma][gamma]')^2,
+
+    = ([beta][gamma]' - [beta]'[gamma])^2 + ([gamma][alpha]' - [gamma]'[alpha])^2 +
+      ([alpha][beta]' - [alpha]'[beta])^2;
+
+  so that the sine of the inclination can only be expressed as a square
+  root. These formulae are the foundation of spherical trigonometry.
+
+  32. _Straight Lines, Planes and Spheres._--The foregoing formulae give
+  at once the equations of these loci.
+
+  For first, taking Q to be a fixed point, coordinates (a, b, c), and
+  the cosine-inclinations ([alpha], [beta], [gamma]) to be constant,
+  then P will be a point in the line through Q in the direction thus
+  determined; or, taking (x, y, z) for its coordinates, these will be
+  the current coordinates of a point in the line. The values of [xi],
+  [eta], [zeta] then are x - a, y - b, z - c, and we thus have
+
+     x - a    y - b    z - c
+    ------- = ----- = ------- (= [rho]),
+    [alpha]  [beta]   [gamma]
+
+  which (omitting the last equation, = [rho]) are the equations of the
+  line through the point (a, b, c), the cosine-inclinations to the axes
+  being [alpha], [beta], [gamma], and these quantities being connected
+  by the relation [alpha]^2 + [beta]^2 + [gamma]^2 = 1. This equation
+  may be omitted, and then [alpha], [beta], [gamma], instead of being
+  equal, will only be proportional, to the cosine-inclinations.
+
+  Using the last equation, and writing
+
+    x, y, z = a + [alpha][rho], b + [beta][rho], c + [gamma][rho],
+
+  these are expressions for the current coordinates in terms of a
+  parameter [rho], which is in fact the distance from the fixed point
+  (a, b, c).
+
+  It is easy to see that, if the coordinates (x, y, z) are connected by
+  any two linear equations, these equations can always be brought into
+  the foregoing form, and hence that the two linear equations represent
+  a line.
+
+  Secondly, taking for greater simplicity the point Q to be coincident
+  with the origin, and [alpha]', [beta]', [gamma]', p to be constant,
+  then p is the perpendicular distance of a plane from the origin, and
+  [alpha]', [beta]', [gamma]' are the cosine-inclinations of this
+  distance to the axes ([alpha]'^2 + [beta]'^2 + [gamma]'^2 = 1). P is
+  any point in this plane, and taking its coordinates to be (x, y, z)
+  then ([xi], [eta], [zeta]) are = (x, y, z), and the foregoing equation
+  p = [alpha]'[xi] + [beta]'[eta] + [gamma]'[zeta] becomes
+
+    [alpha]'x + [beta]'y + [gamma]'z = p,
+
+  which is the equation of the plane in question.
+
+  If, more generally, Q is not coincident with the origin, then, taking
+  its coordinates to be (a, b, c), and writing p1 instead of p, the
+  equation is
+
+    [alpha]'(x - a) + [beta]'(y - b) + [gamma]'(z - c) = p1;
+
+  and we thence have p1 = p - (a[alpha]' + b[beta]' + c[gamma]'), which
+  is an expression for the perpendicular distance of the point (a, b, c)
+  from the plane in question.
+
+  It is obvious that any linear equation Ax + By + Cz + D = O between
+  the coordinates can always be brought into the foregoing form, and
+  hence that such an equation represents a plane.
+
+  Thirdly, supposing Q to be a fixed point, coordinates (a, b, c), and
+  the distance QP = [rho], to be constant, say this is = d, then, as
+  before, the values of [xi], [eta], [zeta] are x - a, y - b, z - c, and
+  the equation [xi]^2 + [eta]^2 + [zeta]^2 = [rho]^2 becomes
+
+    (x - a)^2 + (y - b)^2 + (z - c)^2 = d^2,
+
+  which is the equation of the sphere, coordinates of the centre = (a,
+  b, c), and radius = d.
+
+  A quadric equation wherein the terms of the second order are x^2 + y^2
+  + z^2, viz. an equation
+
+    x^2 + y^2 + z^2 + Ax + By + Cz + D = 0,
+
+  can always, it is clear, be brought into the foregoing form; and it
+  thus appears that this is the equation of a sphere, coordinates of the
+  centre -1/2A, -1/2B, -1/2C, and squared radius = 1/4(A^2 + B^2 + C^2)
+  - D.
+
+  33. _Cylinders, Cones, ruled Surfaces._--If the two equations of a
+  straight line involve a parameter to which any value may be given, we
+  have a singly infinite system of lines. They cover a surface, and the
+  equation of the surface is obtained by eliminating the parameter
+  between the two equations.
+
+  If the lines all pass through a given point, then the surface is a
+  cone; and, in particular, if the lines are all parallel to a given
+  line, then the surface is a cylinder.
+
+  Beginning with this last case, suppose the lines are parallel to the
+  line x = mz, y = nz, the equations of a line of the system are x = mz
+  + a, y = nz + b,--where a, b are supposed to be functions of the
+  variable parameter, or, what is the same thing, there is between them
+  a relation f(a, b) = 0: we have a = x - mz, b = y - nz, and the result
+  of the elimination of the parameter therefore is [f](x - mz, y - nz) =
+  0, which is thus the general equation of the cylinder the generating
+  lines whereof are parallel to the line x = mz, y = nz. The equation of
+  the section by the plane z = 0 is [f](x, y) = 0, and conversely if the
+  cylinder be determined by means of its curve of intersection with the
+  plane z = 0, then, taking the equation of this curve to be f(x, y) =
+  0, the equation of the cylinder is [f](x - mz, y - nz) = 0. Thus, if
+  the curve of intersection be the circle (x - [alpha])^2 + (y -
+  [beta])^2 = [gamma]^2, we have (x - mz - [alpha])^2 + (y - nz -
+  [beta])^2 = [gamma]^2 as the equation of an oblique cylinder on this
+  base, and thus also (x - [alpha])^2 + (y - [beta])^2 = [gamma]^2 as
+  the equation of the right cylinder.
+
+  If the lines all pass through a given point (a, b, c), then the
+  equations of a line are x - a = [alpha](z - c), y - b = [beta](z - c),
+  where [alpha], [beta] are functions of the variable parameter, or,
+  what is the same thing, there exists between them an equation
+  f([alpha], [beta]) = 0; the elimination of the parameter gives,
+  therefore, f[(x - a)/(x - c'), (y - b)/(z - c)] = 0; and this
+  equation, or, what is the same thing, any homogeneous equation f(x -
+  a, y - b, z - c) = 0, or, taking f to be a rational and integral
+  function of the order n, say (*)(x - a, y - b, z - c)^n = 0, is the
+  general equation of the cone having the point (a, b, c) for its
+  vertex. Taking the vertex to be at the origin, the equation is (*)(x,
+  y, z)^n = 0; and, in particular, (*)(x, y, z)^2 = 0 is the equation of
+  a cone of the second order, or quadricone, having the origin for its
+  vertex.
+
+  34. In the general case of a singly infinite system of lines, the
+  locus is a ruled surface (or _regulus_). Now, when a line is changing
+  its position in space, it may be looked upon as in a state of turning
+  about some point in itself, while that point is, as a rule, in a state
+  of moving out of the plane in which the turning takes place. If
+  instantaneously it is only in a state of turning, it is usual, though
+  not strictly accurate, to say that it intersects its consecutive
+  position. A regulus such that consecutive lines on it do not
+  intersect, in this sense, is called a skew surface, or _scroll_; one
+  on which they do is called a developable surface or _torse_.
+
+  Suppose, for instance, that the equations of a line (depending on the
+  variable parameter [theta]) are x/a + y/c = [theta] (1 + y/b), x/a -
+  z/c = 1/[theta] (1 - y/b); then, eliminating [theta] we have x^2/a^2 -
+  z^2/c^2 = 1 - y^2/b^2, or say, x^2/a^2 + z^2/b^2 - z^2/c^2 = 1, the
+  equation of a quadric surface, afterwards called the hyperboloid of
+  one sheet; this surface is consequently a scroll. It is to be remarked
+  that we have upon the surface a second singly infinite series of
+  lines; the equations of a line of this second system (depending on the
+  variable parameter [phi]) are
+
+    x    z          /    y \   x    z      1    /    y \
+    -- + -- = [phi]( 1 - -- ), -- - -- = ----- ( 1 + -- ).
+    a    c          \    b /   a    c    [phi]  \    b /
+
+  It is easily shown that any line of the one system intersects every
+  line of the other system.
+
+  Considering any curve (of double curvature) whatever, the tangent
+  lines of the curve form a singly infinite system of lines, each line
+  intersecting the consecutive line of the system,--that is, they form a
+  developable, or torse; the curve and torse are thus inseparably
+  connected together, forming a single geometrical figure. An osculating
+  plane of the curve (see S 38 below) is a tangent plane of the torse
+  all along a generating line.
+
+  35. _Transformation of Coordinates._--There is no difficulty in
+  changing the origin, and it is for brevity assumed that the origin
+  remains unaltered. We have, then, two sets of rectangular axes, Ox,
+  Oy, Oz, and Ox1, Oy1, Ozx1, the mutual cosine-inclinations being shown
+  by the diagram--
+
+        |    x    |    y   |    z    |
+    ----+---------+--------+---------+
+    x1  | [alpha] | [beta] | [gamma] |
+    ----+---------+--------+---------+
+    y1  | [alpha] | [beta]'| [gamma]'|
+    ----+---------+--------+---------+
+    z1  | [alpha]"| [beta]"| [gamma]"|
+    ----+---------+--------+---------+
+
+  that is, [alpha], [beta], [gamma] are the cosine-inclinations of Ox1
+  to Ox, Oy, Oz; [alpha]', [beta]', [gamma]' those of Oy1, &c.
+
+  And this diagram gives also the linear expressions of the coordinates
+  (x1, y1, z1) or (x, y, z) of either set in terms of those of the other
+  set; we thus have
+
+    x1 = [alpha] x + [beta] y + [gamma] z,
+     x = [alpha]x1 + [alpha]'y1 + [alpha]"z1,
+
+    y1 = [alpha]'x + [beta]'y + [gamma]'z,
+     y = [beta]x1 + [beta]'y1 + [beta]"z1,
+
+    z1 = [alpha]"x + [beta]"y + [gamma]"z,
+     z = [gamma]x1 + [gamma]'y1 + [gamma]"z1,
+
+  which are obtained by projection, as above explained. Each of these
+  equations is, in fact, nothing else than the before-mentioned equation
+  p = [alpha]'[xi] + [beta]'[eta] + [gamma]'[zeta], adapted to the
+  problem in hand.
+
+  But we have to consider the relations between the nine coefficients.
+  By what precedes, or by the consideration that we must have
+  identically x^2 + y^2 + z^2 = x1^2 + y1^2 + z1^2, it appears that
+  these satisfy the relations--
+
+    a^2 + [beta]^2 + [gamma]^2 = 1,
+    [alpha]^2 + [alpha]'^2 + [alpha]"^2 = 1,
+
+    [alpha]'^2 + [beta]'^2 + [gamma]'^2 = 1,
+    [beta]^2 + [beta]'^2 + [beta]"^2 = 1,
+
+    [alpha]"^2 + [beta]"^2 + [gamma]"^2 = 1,
+    [gamma]^2 + [gamma]'^2 + [gamma]"^2 = 1,
+
+    a'a" + [beta]'[beta]" + [gamma]'[gamma]" = 0,
+    [beta][gamma] +[beta]'[gamma]' + [beta]"[gamma]" = 0,
+
+    [alpha]"[alpha] + [beta]"[beta] + [gamma]"[gamma] = 0,
+    [gamma][alpha] + [gamma]'[alpha]' + [gamma]"[alpha]" = 0,
+
+    [alpha][alpha]' + [beta][beta]' + [gamma][gamma]' = 0,
+    [alpha][beta] +[alpha]'[beta]' + [alpha]"[beta]" = 0,
+
+  either set of six equations being implied in the other set.
+
+  It follows that the square of the determinant
+
+    |[alpha],  [beta],  [gamma] |
+    |                           |
+    |[alpha]', [beta]', [gamma]'|
+    |                           |
+    |[alpha]", [beta]", [gamma]"|
+
+  is = 1; and hence that the determinant itself is = [+-] 1. The
+  distinction of the two cases is an important one: if the determinant
+  is = + 1, then the axes Ox1, Oy1, Oz1 are such that they can by a
+  rotation about O be brought to coincide with Ox, Oy, Oz respectively;
+  if it is = -1, then they cannot. But in the latter case, by measuring
+  x1, y1, z1 in the opposite directions we change the signs of all the
+  coefficients and so make the determinant to be = + 1; hence the former
+  case need alone be considered, and it is accordingly assumed that the
+  determinant is = + 1. This being so, it is found that we have the
+  equality [alpha] = [beta]'[gamma]" - [beta]"[gamma]', and eight like
+  ones, obtained from this by cyclical interchanges of the letters
+  [alpha], [beta], [gamma], and of unaccented, singly and doubly
+  accented letters.
+
+  36. The nine cosine-inclinations above are, as has been seen,
+  connected by six equations. It ought then to be possible to express
+  them all in terms of three parameters. An elegant means of doing this
+  has been given by Rodrigues, who has shown that the tabular expression
+  of the formulae of transformation may be written
+
+        |               x                |               y                |               z                |
+    ----+--------------------------------+--------------------------------+--------------------------------+
+    x1  |1 + [lambda]^2 - [mu]^2 - [nu]^2|      2([lambda][mu] - [nu])    |     2([nu][lambda] + [mu])     |
+    ----+--------------------------------+--------------------------+-----+--------------------------------+
+    y1  |     2([lambda][mu] + [nu])     |1 - [lambda]^2 + [mu]^2 - [nu]^2|     2([mu][nu] + [lambda])     |
+    ----+--------------------------------+--------------------------+-----+--------------------------------+
+    z1  |     2([nu][lambda] - [mu])     |      2([mu][nu] + [lambda])    |1 - [lambda]^2 - [mu]^2 + [nu]^2|
+    ----+--------------------------------+--------------------------------+--------------------------------+
+       /(1 + [lambda]^2 + [mu]^2 + [nu]^2),
+
+  the meaning being that the coefficients in the transformation are
+  fractions, with numerators expressed as in the table, and the common
+  denominator.
+
+  37. _The Species of Quadric Surfaces_.--Surfaces represented by
+  equations of the second degree are called _quadric_ surfaces. Quadric
+  surfaces are either _proper_ or _special_. The special ones arise when
+  the coefficients in the general equation are limited to satisfy
+  certain special equations; they comprise (1) plane-pairs, including in
+  particular one plane twice repeated, and (2) cones, including in
+  particular cylinders; there is but one form of cone, but cylinders may
+  be elliptic, parabolic or hyperbolic.
+
+  A discussion of the general equation of the second degree shows that
+  the _proper_ quadric surfaces are of five kinds, represented
+  respectively, when referred to the most convenient axes of reference,
+  by equations of the five types (a and b positive):
+
+             x^2   y^2
+    (1)  z = --- + ---, elliptic paraboloid.
+             2a    2b
+
+             x^2   y^2
+    (2)  z = --- - ---, hyperbolic paraboloid.
+             2a    2b
+
+         x^2   y^2   z^2
+    (3)  --- + --- + --- = 1, ellipsoid.
+         a^2   b^2   c^2
+
+         x^2   y^2   z^2
+    (4)  --- + --- - --- = 1, hyperboloid of one sheet.
+         a^2   b^2   c^2
+
+         x^2   y^2   z^2
+    (5)  --- + --- - --- = -1, hyperboloid of two sheets.
+         a^2   b^2   c^2
+
+  It is at once seen that these are distinct surfaces; and the equations
+  also show very readily the general form and mode of generation of the
+  several surfaces.
+
+  [Illustration: FIG. 61.]
+
+  In the elliptic paraboloid (fig. 61) the sections by the planes of zx
+  and zy are the parabolas
+
+        x^2      y^2
+    z = ---, z = ---
+        2a       2b
+
+  having the common axes Oz; and the section by any plane z = [gamma]
+  parallel to that of xy is the ellipse
+
+              x^2   y^2
+    [gamma] = --- + ---;
+              2a    2b
+
+  so that the surface is generated by a variable ellipse moving parallel
+  to itself along the parabolas as directrices.
+
+  [Illustration: FIG. 62.]
+
+  [Illustration: FIG. 63.]
+
+  In the hyperbolic paraboloid (figs. 62 and 63) the sections by the
+  planes of zx, zy are the parabolas z = x^2/2a, z = - y^2/2b, having
+  the opposite axes Oz, Oz', and the section by a plane z = [gamma]
+  parallel to that of xy is the hyperbola [gamma] = x^2/2a - y^2/2b,
+  which has its transverse axis parallel to Ox or Oy according as
+  [gamma] is positive or negative. The surface is thus generated by a
+  variable hyperbola moving parallel to itself along the parabolas as
+  directrices. The form is best seen from fig. 63, which represents the
+  sections by planes parallel to the plane of xy, or say the contour
+  lines; the continuous lines are the sections above the plane of xy,
+  and the dotted lines the sections below this plane. The form is, in
+  fact, that of a saddle.
+
+  [Illustration: FIG. 64.]
+
+  In the ellipsoid (fig. 64) the sections by the planes of zx, zy, and
+  xy are each of them an ellipse, and the section by any parallel plane
+  is also an ellipse. The surface may be considered as generated by an
+  ellipse moving parallel to itself along two ellipses as directrices.
+
+  In the hyperboloid of one sheet (fig. 65), the sections by the planes
+  of zx, zy are the hyperbolas
+
+    x^2   z^2      y^2   z^2
+    --- - --- = 1, --- - --- = 1,
+    c^2   c^2      b^2   c^2
+
+  having a common conjugate axis zOz'; the section by the plane of x, y,
+  and that by any parallel plane, is an ellipse; and the surface may be
+  considered as generated by a variable ellipse moving parallel to
+  itself along the two hyperbolas as directrices. If we imagine two
+  equal and parallel circular disks, their points connected by strings
+  of equal lengths, so that these are the generators of a right circular
+  cylinder, and if we turn one of the disks about its centre through an
+  angle in its plane, the strings in their new positions will be one
+  system of generators of a hyperboloid of one sheet, for which a = b;
+  and if we turn it through the same angle in the opposite direction, we
+  get in like manner the generators of the other system; there will be
+  the same general configuration when a = | b. The hyperbolic paraboloid
+  is also covered by two systems of rectilinear generators as a method
+  like that used in S 34 establishes without difficulty. The figures
+  should be studied to see how they can lie.
+
+  [Illustration: FIG. 65.]
+
+  [Illustration: FIG. 66.]
+
+  In the hyperboloid of two sheets (fig. 66) the sections by the planes
+  of zx and zy are the hyperbolas
+
+    z^2   x^2      z^2   y^2
+    --- - --- = 1, --- - --- = 1,
+    c^2   a^2      c^2   b^2
+
+  having a common transverse axis along z'Oz; the section by any plane z
+  = [+-][gamma] parallel to that of xy is the ellipse
+
+    x^2   y^2   [gamma]^2
+    --- + --- = --------- - 1,
+    a^2   b^2      c^2
+
+  provided [gamma]^2 > c^2, and the surface, consisting of two distinct
+  portions or sheets, may be considered as generated by a variable
+  ellipse moving parallel to itself along the hyperbolas as directrices.
+
+  38. _Differential Geometry of Curves._--For convenience consider the
+  coordinates (x, y, z) of a point on a curve in space to be given as
+  functions of a variable parameter [theta], which may in particular be
+  one of themselves. Use the notation x', x" for dx/d[theta],
+  d^2x/d[theta]^2, and similarly as to y and z. Only a few formulae will
+  be given. Call the current coordinates ([xi], [eta], [zeta]).
+
+  The _tangent_ at (x, y, z) is the line tended to as a limit by the
+  connector of (x, y, z) and a neighbouring point of the curve when the
+  latter moves up to the former: its equations are
+
+    ([xi] - x)/x' = ([eta] - y)/y' = ([zeta] - z)/z'.
+
+  The _osculating plane_ at (x, y, z) is the plane tended to as a limit
+  by that through (x, y, z) and two neighbouring points of the curve as
+  these, remaining distinct, both move up to (x, y, z): its one equation
+  is
+
+    ([xi] - x)(y'z" - y"z') + ([eta] - y)(z'x" - z"x') + ([zeta] - z)
+      (x'y" - x"y') = 0.
+
+  The _normal plane_ is the plane through (x, y, z) at right angles to
+  the tangent line, i.e. the plane
+
+    x'([xi] - x) + y'([eta] - y) + z'([zeta] - z) = 0.
+
+  It cuts the osculating plane in a line called the _principal normal_.
+  Every line through (x, y, z) in the normal plane is a normal. The
+  normal perpendicular to the osculating plane is called the _binormal_.
+  A tangent, principal normal, and binormal are a convenient set of
+  rectangular axes to use as those of reference, when the nature of a
+  curve near a point on it is to be discussed.
+
+  Through (x, y, z) and three neighbouring points, all on the curve,
+  passes a single sphere; and as the three points all move up to (x, y,
+  z) continuing distinct, the sphere tends to a limiting size and
+  position. The limit tended to is the sphere of closest contact with
+  the curve at (x, y, z); its centre and radius are called the centre
+  and radius of _spherical curvature_. It cuts the osculating plane in a
+  circle, called the _circle of absolute curvature_; and the centre and
+  radius of this circle are the centre and radius of absolute curvature.
+  The centre of absolute curvature is the limiting position of the point
+  where the principal normal at (x, y, z) is cut by the normal plane at
+  a neighbouring point, as that point moves up to (x, y, z).
+
+  39. _Differential Geometry of Surfaces._--Let (x, y, z) be any chosen
+  point on a surface [f](x, y, z) = 0. As a second point of the surface
+  moves up to (x, y, z), its connector with (x, y, z) tends to a
+  limiting position, a tangent line to the surface at (x, y, z). All
+  these tangent lines at (x, y, z), obtained by approaching (x, y, z)
+  from different directions on a surface, lie in one plane
+
+    dP[f]              dP[f]               dP[f]
+    ----- ([xi] - x) + ----- ([eta] - y) + ----- ([zeta] - z) = 0.
+     dPx                dPy                 dPz
+
+  This plane is called the _tangent plane_ at (x, y, z). One line
+  through (x, y, z) is at right angles to the tangent plane. This is the
+  normal
+
+                /dP[f]                /dP[f]                   /dP[f]
+    ([xi] - x) / ----- = ([eta] - y) / ----- = ([zeta] - z) = / -----.
+              /   dPx               /   dPy                  /   dPz
+
+  The tangent plane is cut by the surface in a curve, real or imaginary,
+  with a node or double point at (x, y, z). Two of the tangent lines
+  touch this curve at the node. They are called the "chief tangents"
+  (_Haupt-tangenten_) at (x, y, z); they have closer contact with the
+  surface than any other tangents.
+
+  In the case of a quadric surface the curve of intersection of a
+  tangent and the surface is of the second order and has a node, it must
+  therefore consist of two straight lines. Consequently a quadric
+  surface is covered by two sets of straight lines, a pair through every
+  point on it; these are imaginary for the ellipsoid, hyperboloid of two
+  sheets, and elliptic paraboloid.
+
+  A surface of any order is covered by two singly infinite systems of
+  curves, a pair through every point, the tangents to which are all
+  chief tangents at their respective points of contact. These are called
+  _chief-tangent curves_; on a quadric surface they are the above
+  straight lines.
+
+  40. The tangents at a point of a surface which bisect the angles
+  between the chief tangents are called the _principal tangents_ at the
+  point. They are at right angles, and together with the normal
+  constitute a convenient set of rectangular axes to which to refer the
+  surface when its properties near the point are under discussion. At a
+  special point which is such that the chief tangents there run to the
+  circular points at infinity in the tangent plane, the principal
+  tangents are indeterminate; such a special point is called an umbilic
+  of the surface.
+
+  There are two singly infinite systems of curves on a surface, a pair
+  cutting one another at right angles through every point upon it, all
+  tangents to which are principal tangents of the surface at their
+  respective points of contact. These are called _lines of curvature_,
+  because of a property next to be mentioned.
+
+  As a point Q moves in an arbitrary direction on a surface from
+  coincidence with a chosen point P, the normal at it, as a rule, at
+  once fails to meet the normal at P; but, if it takes the direction of
+  a line of curvature through P, this is instantaneously not the case.
+  We have thus on the normal two centres of curvature, and the distances
+  of these from the point on the surface are the two _principal radii of
+  curvature_ of the surface at that point; these are also the radii of
+  curvature of the sections of the surface by planes through the normal
+  and the two principal tangents respectively; or say they are the radii
+  of curvature of the normal sections through the two principal tangents
+  respectively. Take at the point the axis of z in the direction of the
+  normal, and those of x and y in the directions of the principal
+  tangents respectively, then, if the radii of curvature be a, b (the
+  signs being such that the coordinates of the two centres of curvature
+  are z = a and z = b respectively), the surface has in the
+  neighbourhood of the point the form of the paraboloid
+
+        x^2   y^2
+    z = --- + ---,
+        2a    2b
+
+  and the chief-tangents are determined by the equation 0 = x^2/2a +
+  y^2/2b. The two centres of curvature may be on the same side of the
+  point or on opposite sides; in the former case a and b have the same
+  sign, the paraboloid is elliptic, and the chief-tangents are
+  imaginary; in the latter case a and b have opposite signs, the
+  paraboloid is hyperbolic, and the chief-tangents are real.
+
+  The normal sections of the surface and the paraboloid by the same
+  plane have the same radius of curvature; and it thence readily follows
+  that the radius of curvature of a normal section of the surface by a
+  plane inclined at an angle [theta] to that of zx is given by the
+  equation
+
+      1     cos^2 [theta]   sin^2 [theta]
+    ----- = ------------- + -------------.
+    [rho]        a               b
+
+  The section in question is that by a plane through the normal and a
+  line in the tangent plane inclined at an angle [theta] to the
+  principal tangent along the axis of x. To complete the theory,
+  consider the section by a plane having the same trace upon the tangent
+  plane, but inclined to the normal at an angle [phi]; then it is shown
+  without difficulty (Meunier's theorem) that the radius of curvature of
+  this inclined section of the surface is = [rho] cos [phi].
+
+  AUTHORITIES.--The above article is largely based on that by Arthur
+  Cayley in the 9th edition of this work. Of early and important recent
+  publications on analytical geometry, special mention is to be made of
+  R. Descartes, _Geometrie_ (Leyden, 1637); John Wallis, _Tractatus de
+  sectionibus conicis nova methodo expositis_ (1655, _Opera
+  mathematica_, i., Oxford, 1695); de l'Hospital, _Traite analytique des
+  sections coniques_ (Paris, 1720); Leonhard Euler, _Introductio in
+  analysin infinitorum_, ii. (Lausanne, 1748); Gaspard Monge,
+  "Application d'algebre a la geometrie" (_Journ. Ecole Polytech._,
+  1801); Julius Plucker, _Analytisch-geometrische Entwickelungen_, 3
+  Bde. (Essen, 1828-1831); _System der analytischen Geometrie_ (Berlin,
+  1835); G. Salmon, _A Treatise on Conic Sections_ (Dublin, 1848; 6th
+  ed., London, 1879); Ch. Briot and J. Bouquet, _Lecons de geometrie
+  analytique_ (Paris, 1851; 16th ed., 1897); M. Chasles, _Traite de
+  geometrie superieure_ (Paris, 1852); Wilhelm Fiedler, _Analytische
+  Geometrie der Kegelschnitte_ nach G. Salmon frei bearbeitet (Leipzig,
+  5te Aufl., 1887-1888); N.M. Ferrers, _An Elementary Treatise on
+  Trilinear Coordinates_ (London, 1861); Otto Hesse, _Vorlesungen aus
+  der analytischen Geometrie_ (Leipzig, 1865, 1881); W.A. Whitworth,
+  _Trilinear Coordinates and other Methods of Modern Analytical
+  Geometry_ (Cambridge, 1866); J. Booth, _A Treatise on Some New
+  Geometrical Methods_ (London, i., 1873; ii., 1877); A. Clebsch-F.
+  Lindemann, _Vorlesungen uber Geometrie_, Bd. i. (Leipzig, 1876, 2te
+  Aufl., 1891); R. Baltser, _Analytische Geometrie_ (Leipzig, 1882);
+  Charlotte A. Scott, _Modern Methods of Analytical Geometry_ (London,
+  1894); G. Salmon, _A Treatise on the Analytical Geometry of three
+  Dimensions_ (Dublin, 1862; 4th ed., 1882); Salmon-Fiedler,
+  _Analytische Geometrie des Raumes_ (Leipzig, 1863; 4te Aufl., 1898);
+  P. Frost, _Solid Geometry_ (London, 3rd ed., 1886; 1st ed., Frost and
+  J. Wolstenholme). See also E. Pascal, _Repertorio di matematiche
+  superiori, II. Geometria_ (Milan, 1900), and articles now appearing in
+  the _Encyklopadie der mathematischen Wissenschaften_, Bd. iii. 1, 2.
+       (E. B. El.)
+
+
+V. LINE GEOMETRY
+
+Line geometry is the name applied to those geometrical investigations in
+which the straight line replaces the point as element. Just as ordinary
+geometry deals primarily with points and systems of points, this theory
+deals in the first instance with straight lines and systems of straight
+lines. In two dimensions there is no necessity for a special line
+geometry, inasmuch as the straight line and the point are
+interchangeable by the principle of duality; but in three dimensions the
+straight line is its own reciprocal, and for the better discussion of
+systems of lines we require some new apparatus, e.g., a system of
+coordinates applicable to straight lines rather than to points. The
+essential features of the subject are most easily elucidated by
+analytical methods: we shall therefore begin with the notion of line
+coordinates, and in order to emphasize the merits of the system of
+coordinates ultimately adopted, we first notice a system without these
+advantages, but often useful in special investigations.
+
+  In ordinary Cartesian coordinates the two equations of a straight line
+  may be reduced to the form y = rx + s, z = tx + u, and r, s, t, u may
+  be regarded as the four coordinates of the line. These coordinates
+  lack symmetry: moreover, in changing from one base of reference to
+  another the transformation is not linear, so that the degree of an
+  equation is deprived of real significance. For purposes of the general
+  theory we employ homogeneous coordinates; if x1y1z1w1 and x2y2z2w2 are
+  two points on the line, it is easily verified that the six
+  determinants of the array
+
+    |x1y1z1w1|
+    |x2y2z2w2|
+
+  are in the same ratios for all point-pairs on the line, and further,
+  that when the point coordinates undergo a linear transformation so
+  also do these six determinants. We therefore adopt these six
+  determinants for the coordinates of the line, and express them by the
+  symbols l, [lambda], m, [mu], n, [nu] where l = x1w2 - x2w1, [lambda]
+  = y1z2 - y2z1, &c. There is the further advantage that if a1b1c1d1 and
+  a2b2c2d2 be two planes through the line, the six determinants
+
+    |a1b1c1d1|
+    |a2b2c2d2|
+
+  are in the same ratios as the foregoing, so that except as regards a
+  factor of proportionality we have [lambda] = b1c2 - b2c1, l = c1d2 -
+  c2d1, &c. The identical relation l[lambda] + m[mu] + n[nu] = o reduces
+  the number of independent constants in the six coordinates to four,
+  for we are only concerned with their mutual ratios; and the quadratic
+  character of this relation marks an essential difference between point
+  geometry and line geometry. The condition of intersection of two lines
+  is
+
+    l[lambda]' + l'[lambda] + m[mu]' + m'[mu] + n[nu]' + n'[nu] = 0
+
+  where the accented letters refer to the second line. If the
+  coordinates are Cartesian and l, m, n are direction cosines, the
+  quantity on the left is the mutual moment of the two lines.
+
+  Since a line depends on four constants, there are three distinct types
+  of configurations arising in line geometry--those containing a
+  triply-infinite, a doubly-infinite and a singly-infinite number of
+  lines; they are called Complexes, Congruences, and Ruled Surfaces or
+  Skews respectively. A _Complex_ is thus a system of lines satisfying
+  one condition--that is, the coordinates are connected by a single
+  relation; and the degree of the complex is the degree of this equation
+  supposing it to be algebraic. The lines of a complex of the nth degree
+  which pass through any point lie on a cone of the nth degree, those
+  which lie in any plane envelop a curve of the nth class and there are
+  n lines of the complex in any plane pencil; the last statement
+  combines the former two, for it shows that the cone is of the nth
+  degree and the curve is of the nth class. To find the lines common to
+  four complexes of degrees n1, n2, n3, n4, we have to solve five
+  equations, viz. the four complex equations together with the quadratic
+  equation connecting the line coordinates, therefore the number of
+  common lines is 2n1n2n3n4. As an example of complexes we have the
+  lines meeting a twisted curve of the nth degree, which form a complex
+  of the nth degree.
+
+  A _Congruence_ is the set of lines satisfying two conditions: thus a
+  finite number m of the lines pass through any point, and a finite
+  number n lie in any plane; these numbers are called the degree and
+  class respectively, and the congruence is symbolically written (m, n).
+
+  The simplest example of a congruence is the system of lines
+  constituted by all those that pass through m points and those that lie
+  in n planes; through any other point there pass m of these lines, and
+  in any other plane there lie n, therefore the congruence is of degree
+  m and class n. It has been shown by G.H. Halphen that the number of
+  lines common to two congruences is mm' + nn', which may be verified by
+  taking one of them to be of this simple type. The lines meeting two
+  fixed lines form the general (1, 1) congruence; and the chords of a
+  twisted cubic form the general type of a (1, 3) congruence; Halphen's
+  result shows that two twisted cubics have in general ten common
+  chords. As regards the analytical treatment, the difficulty is of the
+  same nature as that arising in the theory of curves in space, for a
+  congruence is not in general the complete intersection of two
+  complexes.
+
+  A _Ruled Surface_, _Regulus_ or _Skew_ is a configuration of lines
+  which satisfy three conditions, and therefore depend on only one
+  parameter. Such lines all lie on a surface, for we cannot draw one
+  through an arbitrary point; only one line passes through a point of
+  the surface; the simplest example, that of a quadric surface, is
+  really two skews on the same surface.
+
+  The degree of a ruled surface _qua_ line geometry is the number of its
+  generating lines contained in a linear complex. Now the number which
+  meets a given line is the degree of the surface _qua_ point geometry,
+  and as the lines meeting a given line form a particular case of linear
+  complex, it follows that the degree is the same from whichever point
+  of view we regard it. The lines common to three complexes of degrees,
+  n1n2n3, form a ruled surface of degree 2n1n2n3; but not every ruled
+  surface is the complete intersection of three complexes.
+
+
+    Linear complex.
+
+  In the case of a complex of the first degree (or linear complex) the
+  lines through a fixed point lie in a plane called the polar plane or
+  nul-plane of that point, and those lying in a fixed plane pass through
+  a point called the nul-point or pole of the plane. If the nul-plane of
+  A pass through B, then the nul-plane of B will pass through A; the
+  nul-planes of all points on one line l1 pass through another line l2.
+  The relation between l1 and l2 is reciprocal; any line of the complex
+  that meets one will also meet the other, and every line meeting both
+  belongs to the complex. They are called conjugate or polar lines with
+  respect to the complex. On these principles can be founded a theory of
+  reciprocation with respect to a linear complex.
+
+  This may be aptly illustrated by an elegant example due to A. Voss.
+  Since a twisted cubic can be made to satisfy twelve conditions, it
+  might be supposed that a finite number could be drawn to touch four
+  given lines, but this is not the case. For, suppose one such can be
+  drawn, then its reciprocal with respect to any linear complex
+  containing the four lines is a curve of the third class, i.e. another
+  twisted cubic, touching the same four lines, which are unaltered in
+  the process of reciprocation; as there is an infinite number of
+  complexes containing the four lines, there is an infinite number of
+  cubics touching the four lines, and the problem is poristic.
+
+  The following are some geometrical constructions relating to the
+  unique linear complex that can be drawn to contain five arbitrary
+  lines:
+
+  To construct the nul-plane of any point O, we observe that the two
+  lines which meet any four of the given five are conjugate lines of the
+  complex, and the line drawn through O to meet them is therefore a ray
+  of the complex; similarly, by choosing another four we can find
+  another ray through O: these rays lie in the nul-plane, and there is
+  clearly a result involved that the five lines so obtained all lie in
+  one plane. A reciprocal construction will enable us to find the
+  nul-point of any plane. Proceeding now to the metrical properties and
+  the statical and dynamical applications, we remark that there is just
+  one line such that the nul-plane of any point on it is perpendicular
+  to it. This is called the central axis; if d be the shortest distance,
+  [theta] the angle between it and a ray of the complex, then d tan
+  [theta] = p, where p is a constant called the pitch or parameter. Any
+  system of forces can be reduced to a force R along a certain line, and
+  a couple G perpendicular to that line; the lines of nul-moment for the
+  system form a linear complex of which the given line is the central
+  axis and the quotient G/R is the pitch. Any motion of a rigid body can
+  be reduced to a screw motion about a certain line, i.e. to an angular
+  velocity [omega] about that line combined with a linear velocity u
+  along the line. The plane drawn through any point perpendicular to the
+  direction of its motion is its nul-plane with respect to a linear
+  complex having this line for central axis, and the quotient u/[omega]
+  for pitch (cf. Sir R.S. Ball, _Theory of Screws_).
+
+  The following are some properties of a configuration of two linear
+  complexes:
+
+  The lines common to the two-complexes also belong to an infinite
+  number of linear complexes, of which two reduce to single straight
+  lines. These two lines are conjugate lines with respect to each of the
+  complexes, but they may coincide, and then some simple modifications
+  are required. The locus of the central axis of this system of
+  complexes is a surface of the third degree called the cylindroid,
+  which plays a leading part in the theory of screws as developed
+  synthetically by Ball. Since a linear complex has an invariant of the
+  second degree in its coefficients, it follows that two linear
+  complexes have a lineo-linear invariant. This invariant is
+  fundamental: if the complexes be both straight lines, its vanishing is
+  the condition of their intersection as given above; if only one of
+  them be a straight line, its vanishing is the condition that this line
+  should belong to the other complex. When it vanishes for any two
+  complexes they are said to be in _involution_ or _apolar_; the
+  nul-points P, Q of any plane then divide harmonically the points in
+  which the plane meets the common conjugate lines, and each complex is
+  its own reciprocal with respect to the other. As regards a
+  configuration of these linear complexes, the common lines from one
+  system of generators of a quadric, and the doubly infinite system of
+  complexes containing the common lines, include an infinite number of
+  straight lines which form the other system of generators of the same
+  quadric.
+
+
+    General line coordinates.
+
+  If the equation of a linear complex is Al + Bm + Cn + D[lambda] +
+  E[mu] + F[nu] = 0, then for a line not belonging to the complex we may
+  regard the expression on the left-hand side as a multiple of the
+  moment of the line with respect to the complex, the word moment being
+  used in the statical sense; and we infer that when the coordinates are
+  replaced by linear functions of themselves the new coordinates are
+  multiples of the moments of the line with respect to six fixed
+  complexes. The essential features of this coordinate system are the
+  same as those of the original one, viz. there are six coordinates
+  connected by a quadratic equation, but this relation has in general a
+  different form. By suitable choice of the six fundamental complexes,
+  as they may be called, this connecting relation may be brought into
+  other simple forms of which we mention two: (i.) When the six are
+  mutually in involution it can be reduced to x1^2 + x2^2 + x3^2 + x4^2
+  + x5^2 + x6^2 = 0; (ii.) When the first four are in involution and the
+  other two are the lines common to the first four it is x1^2 + x2^2 +
+  x3^2 + x4^2 - 2x5x6 = 0. These generalized coordinates might be
+  explained without reference to actual magnitude, just as homogeneous
+  point coordinates can be; the essential remark is that the equation of
+  any coordinate to zero represents a linear complex, a point of view
+  which includes our original system, for the equation of a coordinate
+  to zero represents all the lines meeting an edge of the fundamental
+  tetrahedron.
+
+  The system of coordinates referred to six complexes mutually in
+  involution was introduced by Felix Klein, and in many cases is more
+  useful than that derived directly from point coordinates; e.g. in the
+  discussion of quadratic complexes: by means of it Klein has developed
+  an analogy between line geometry and the geometry of spheres as
+  treated by G. Darboux and others. In fact, in that geometry a point is
+  represented by _five_ coordinates, connected by a relation of the same
+  type as the one just mentioned when the five fundamental spheres are
+  mutually at right angles and the equation of a sphere is of the first
+  degree. Extending this to four dimensions of space, we obtain an exact
+  analogue of line geometry, in which (i.) a point corresponds to a
+  line; (ii.) a linear complex to a hypersphere; (iii.) two linear
+  complexes in involution to two orthogonal hyperspheres; (iv.) a linear
+  complex and two conjugate lines to a hypersphere and two inverse
+  points. Many results may be obtained by this principle, and more still
+  are suggested by trying to extend the properties of circles to spheres
+  in three and four dimensions. Thus the elementary theorem, that, given
+  four lines, the circles circumscribed to the four triangles formed by
+  them are concurrent, may be extended to six hyperplanes in four
+  dimensions; and then we can derive a result in line geometry by
+  translating the inverse of this theorem. Again, just as there is an
+  infinite number of spheres touching a surface at a given point, two of
+  them having contact of a closer nature, so there is an infinite number
+  of linear complexes touching a non-linear complex at a given line, and
+  _three_ of these have contact of a closer nature (cf. Klein, _Math.
+  Ann._ v.).
+
+  Sophus Lie has pointed out a different analogy with sphere geometry.
+  Suppose, in fact, that the equation of a sphere of radius r is
+
+    x^2 + y^2 + z^2 + 2ax + 2by + 2cz + d = 0,
+
+  so that r^2 = a^2 + b^2 + c^2 - d; then introducing the quantity e to
+  make this equation homogeneous, we may regard the sphere as given by
+  the six coordinates a, b, c, d, e, r connected by the equation a^2 +
+  b^2 + c^2 - r^2 - de = 0, and it is easy to see that two spheres
+  touch, if the polar form 2aa1 + 2bb1 + 2cc1 - 2rr1 - de1 - d1e
+  vanishes. Comparing this with the equation x1^2 + x2^2 + x3^2 + x4^2 -
+  2x5x6 = 0 given above, it appears that this sphere geometry and line
+  geometry are identical, for we may write a = x1, b = x2, c = x3, r =
+  x4(/[delta] - 1), d = x5, e = 1/2x6; but it is to be noticed that a
+  sphere is really replaced by two lines whose coordinates only differ
+  in the sign of x4, so that they are polar lines with respect to the
+  complex x4 = 0. Two spheres which touch correspond to two lines which
+  intersect, or more accurately to two pairs of lines (p, p') and (q,
+  q'), of which the pairs (p, q) and (p', q') both intersect. By this
+  means the problem of describing a sphere to touch four given spheres
+  is reduced to that of drawing a pair of lines (t, t') (of which t
+  intersects one line of the four pairs (pp'), (qq'), (rr'), (ss'), and
+  t' intersects the remaining four). We may, however, ignore the
+  accented letters in translating theorems, for a configuration of lines
+  and its polar with respect to a linear complex have the same
+  projective properties. In Lie's transformation a linear complex
+  corresponds to the totality of spheres cutting a given sphere at a
+  given angle. A most remarkable result is that lines of curvature in
+  the sphere geometry become asymptotic lines in the line geometry.
+
+  Some of the principles of line geometry may be brought into clearer
+  light by admitting the ideas of space of four and five dimensions.
+
+  Thus, regarding the coordinates of a line as homogeneous coordinates
+  in five dimensions, we may say that line geometry is equivalent to
+  geometry on a quadric surface in five dimensions. A linear complex is
+  represented by a hyperplane section; and if two such complexes are in
+  involution, the corresponding hyperplanes are conjugate with respect
+  to the fundamental quadric. By projecting this quadric
+  stereographically into space of four dimensions we obtain Klein's
+  analogy. In the same way geometry in a linear complex is equivalent to
+  geometry on a quadric in four dimensions; when two lines intersect the
+  representative points are on the same generator of this quadric.
+  Stereographic projection, therefore, converts a curve in a linear
+  complex, i.e. one whose tangents all belong to the complex, into one
+  whose tangents intersect a fixed conic: when this conic is the
+  imaginary circle at infinity the curve is what Lie calls a minimal
+  curve. Curves in a linear complex have been extensively studied. The
+  osculating plane at any point of such a curve is the nul-plane of the
+  point with respect to the complex, and points of superosculation
+  always coincide in pairs at the points of contact of stationary
+  tangents. When a point of such a curve is given, the osculating plane
+  is determined, hence all the curves through a given point with the
+  same tangent have the same torsion.
+
+
+    Non-linear complexes.
+
+  The lines through a given point that belong to a complex of the nth
+  degree lie on a cone of the nth degree: if this cone has a double line
+  the point is said to be a singular point. Similarly, a plane is said
+  to be singular when the envelope of the lines in it has a double
+  tangent. It is very remarkable that the same surface is the locus of
+  the singular points and the envelope of the singular planes: this
+  surface is called the singular surface, and both its degree and class
+  are in general 2n(n - 1)^2, which is equal to four for the quadratic
+  complex.
+
+  The singular lines of a complex F = 0 are the lines common to F and
+  the complex
+
+    [delta]F    [delta]F       [delta]F    [delta]F    [delta]F    [delta]F
+    -------- --------------- + --------  ----------- + --------  ----------- = 0.
+    [delta]l [delta][lambda]   [delta]m  [delta][mu]   [delta]n  [delta][nu]
+
+  As already mentioned, at each line l of a complex there is an infinite
+  number of tangent linear complexes, and they all contain the lines
+  adjacent to l. If now l be a singular line, these complexes all reduce
+  to straight lines which form a plane pencil containing the line l.
+  Suppose the vertex of the pencil is A, its plane a, and one of its
+  lines [xi], then l' being a complex line near l, meets [xi], or more
+  accurately the mutual moment of l', and is of the second order of
+  small quantities. If P be a point on l, a line through P quite near l
+  in the plane a will meet [xi] and is therefore a line of the complex;
+  hence the complex-cones of all points on l touch a and the
+  complex-curves of all planes through l touch l at A. It follows that l
+  is a double line of the complex-cone of A, and a double tangent of the
+  complex-curve of a. Conversely, a double line of a cone or curve is a
+  singular line, and a singular line clearly touches the curves of all
+  planes through it in the same point. Suppose now that the consecutive
+  line l' is also a singular line, A' being the allied singular point,
+  a' the singular plane and [xi]' any line of the pencil (A', a') so
+  that [xi]' is a tangent line at l' to the complex: the mutual moments
+  of the pairs l', [xi] and l, [xi] are each of the second order; hence
+  the plane a' meets the lines l and [xi]' in two points very near A.
+  This being true for all singular planes, near a the point of contact
+  of a with its envelope is in A, i.e. the locus of singular points is
+  the same as the envelope of singular planes. Further, when a line
+  touches a complex it touches the singular surface, for it belongs to a
+  plane pencil like (Aa), and thus in Klein's analogy the analogue of a
+  focus of a hyper-surface being a bitangent line of the complex is also
+  a bitangent line of the singular surface. The theory of cosingular
+  complexes is thus brought into line with that of confocal surfaces in
+  four dimensions, and guided by these principles the existence of
+  cosingular quadratic complexes can easily be established, the analysis
+  required being almost the same as that invented for confocal cyclides
+  by Darboux and others. Of cosingular complexes of higher degree
+  nothing is known.
+
+  Following J. Plucker, we give an account of the lines of a quadratic
+  complex that meet a given line.
+
+  The cones whose vertices are on the given line all pass through eight
+  fixed points and envelop a surface of the fourth degree; the conics
+  whose planes contain the given line all lie on a surface of the fourth
+  class and touch eight fixed planes. It is easy to see by elementary
+  geometry that these two surfaces are identical. Further, the given
+  line contains four singular points A1, A2, A3, A4, and the planes into
+  which their cones degenerate are the eight common tangent planes
+  mentioned above; similarly, there are four singular planes, a1, a2,
+  a3, a4, through the line, and the eight points into which their conics
+  degenerate are the eight common points above. The locus of the pole of
+  the line with respect to all the conics in planes through it is a
+  straight line called the _polar line_ of the given one; and through
+  this line passes the polar plane of the given line with respect to
+  each of the cones. The name polar is applied in the ordinary
+  analytical sense; any line has an infinite number of polar complexes
+  with respect to the given complex, for the equation of the latter can
+  be written in an infinite number of ways; one of these polars is a
+  straight line, and is the polar line already introduced. The surface
+  on which lie all the conics through a line l is called the Plucker
+  surface of that line: from the known properties of (2, 2)
+  correspondences it can be shown that the Plucker surface of l cuts l1
+  in a range of the same cross ratio as that of the range in which the
+  Plucker surface of l1 cuts l. Applying this to the case in which l1 is
+  the polar of l, we find that the cross ratios of (A1, A2, A3, A4) and
+  (a1, a2, a3, a4) are equal. The identity of the locus of the A's with
+  the envelope of the a's follows at once; moreover, a line meets the
+  singular surface in four points having the same cross ratio as that of
+  the four tangent planes drawn through the line to touch the surface.
+  The Plucker surface has eight nodes, eight singular tangent planes,
+  and is a double line. The relation between a line and its polar line
+  is not a reciprocal one with respect to the complex; but W. Stahl has
+  pointed out that the relation is reciprocal as far as the singular
+  surface is concerned.
+
+
+    Quadratic complexes.
+
+  To facilitate the discussion of the general quadratic complex we
+  introduce Klein's canonical form. We have, in fact, to deal with two
+  quadratic equations in six variables; and by suitable linear
+  transformations these can be reduced to the form
+
+    a1x1^2 + a2x2^2 + a3x3^2 + a4x4^2 + a5x5^2 +  a6x6^2 = 0
+     x1^2  +  x2^2  +  x3^2  +  x4^2  +  x5^2  +   x6^2  = 0
+
+  subject to certain exceptions, which will be mentioned later.
+
+  Taking the first equation to be that of the complex, we remark that
+  both equations are unaltered by changing the sign of any coordinate;
+  the geometrical meaning of this is, that the quadratic complex is its
+  own reciprocal with respect to each of the six fundamental complexes,
+  for changing the sign of a coordinate is equivalent to taking the
+  polar of a line with respect to the corresponding fundamental complex.
+  It is easy to establish the existence of six systems of bitangent
+  linear complexes, for the complex l1x1 + l2x2 + l3x3 + l4x4 + l5x5 +
+  l6x6 = 0 is a bitangent when
+
+                  l2^2      l3^2      l4^2      l5^2      l6^2
+    l1 = 0, and ------- + ------- + ------- + ------- + ------- = 0
+                a2 - a1   a3 - a1   a4 - a1   a5 - a1   a6 - a1
+
+  and its lines of contact are conjugate lines with respect to the first
+  fundamental complex. We therefore infer the existence of six systems
+  of bitangent lines of the complex, of which the first is given by
+
+              x2^2      x3^2      x4^2      x5^2      x6^2
+    x1 = 0, ------- + ------- + ------- + ------- + ------- = 0.
+            a2 - a1   a3 - a1   a4 - a1   a5 - a1   a6 - a1
+
+  Each of these lines is a bitangent of the singular surface, which is
+  therefore completely determined as being the focal surface of the (2,
+  2) congruence above. It is thence easy to verify that the two
+  complexes [Sigma]ax^2 = 0 and [Sigma]bx^2 = 0 are cosingular if b_r =
+  a_r[lambda] + [mu]/a_r[nu] + [rho].
+
+  The singular surface of the general quadratic complex is the famous
+  quartic, with sixteen nodes and sixteen singular tangent planes, first
+  discovered by E.E. Kummer.
+
+  We cannot give a full account of its properties here, but we deduce at
+  once from the above that its bitangents break up into six (2, 2)
+  congruences, and the six linear complexes containing these are
+  mutually in involution. The nodes of the singular surface are points
+  whose complex cones are coincident planes, and the complex conic in a
+  singular tangent plane consists of two coincident points. This
+  configuration of sixteen points and planes has many interesting
+  properties; thus each plane contains six points which lie on a conic,
+  while through each point there pass six planes which touch a quadric
+  cone. In many respects the Kummer quartic plays a part in three
+  dimensions analogous to the general quartic curve in two; it further
+  gives a natural representation of certain relations between
+  hyperelliptic functions (cf. R.W.H.T. Hudson, _Kummer's Quartic_,
+  1905).
+
+
+    Classification of quadratic complexes.
+
+  As might be expected from the magnitude of a form in six variables,
+  the number of projectivally distinct varieties of quadratic complexes
+  is very great; and in fact Adolf Weiler, by whom the question was
+  first systematically studied on lines indicated by Klein, enumerated
+  no fewer than forty-nine different types. But the principle of the
+  classification is so important, and withal so simple, that we give a
+  brief sketch which indicates its essential features.
+
+  We have practically to study the intersection of two quadrics F and F'
+  in six variables, and to classify the different cases arising we make
+  use of the results of Karl Weierstrass on the equivalence conditions
+  of two pairs of quadratics. As far as at present required, they are as
+  follows: Suppose that the factorized form of the determinantal
+  equation Disct (F + [lambda]F') = 0 is
+
+    ([lambda] - [alpha])^(s1 + s2 + s3 ...)
+      ([lambda] - [beta])^(t1 + t2 + t3 + ...) ...
+
+  where the root [alpha] occurs s1 + s2 + s3 ... times in the
+  determinant, s2 + s3 ... times in every first minor, s3 + ... times in
+  every second minor, and so on; the meaning of each exponent is then
+  perfectly definite. Every factor of the type ([lambda] - [alpha])^s is
+  called an _elementartheil_ (elementary divisor) of the determinant,
+  and the condition of equivalence of two pairs of quadratics is simply
+  that their determinants have the same elementary divisors. We write
+  the pair of forms symbolically thus [(s1s2 ...), (t1t2 ...), ...],
+  letters in the inner brackets referring to the same factor. Returning
+  now to the two quadratics representing the complex, the sum of the
+  exponents will be six, and two complexes are put in the same class if
+  they have the same symbolical expression; i.e. the actual values of
+  the roots of the determinantal equation need not be the same for both,
+  but their manner of occurrence, as far as here indicated, must be
+  identical in the two. The enumeration of all possible cases is thus
+  reduced to a simple question in combinatorial analysis, and the actual
+  study of any particular case is much facilitated by a useful rule of
+  Klein's for writing down in a simple form two quadratics belonging to
+  a given class--one of which, of course, represents the equation
+  connecting line coordinates, and the other the equation of the
+  complex. The general complex is naturally [111111]; the complex of
+  tangents to a quadric is [(111), (111)] and that of lines meeting a
+  conic is [(222)]. Full information will be found in Weiler's memoir,
+  _Math. Ann._ vol. vii.
+
+  The detailed study of each variety of complex opens up a vast subject;
+  we only mention two special cases, the harmonic complex and the
+  tetrahedral complex.
+
+  The harmonic complex, first studied by Battaglini, is generated in an
+  infinite number of ways by the lines cutting two quadrics
+  harmonically. Taking the most general case, and referring the quadrics
+  to their common self-conjugate tetrahedron, we can find its equation
+  in a simple form, and verify that this complex really depends only on
+  seventeen constants, so that it is not the most general quadratic
+  complex. It belongs to the general type in so far as it is discussed
+  above, but the roots of the determinant are in involution. The
+  singular surface is the "tetrahedroid" discussed by Cayley. As a
+  particular case, from a metrical point of view, we have L.F. Painvin's
+  complex generated by the lines of intersection of perpendicular
+  tangent planes of a quadric, the singular surface now being Fresnel's
+  wave surface. The tetrahedral or Reye complex is the simplest and best
+  known of proper quadratic complexes. It is generated by the lines
+  which cut the faces of a tetrahedron in a constant cross ratio, and
+  therefore by those subtending the same cross ratio at the four
+  vertices. The singular surface is made up of the faces or the vertices
+  of the fundamental tetrahedron, and each edge of this tetrahedron is a
+  double line of the complex. The complex was first discussed by K.T.
+  Reye as the assemblage of lines joining corresponding points in a
+  homographic transformation of space, and this point of view leads to
+  many important and elegant properties. A (metrically) particular case
+  of great interest is the complex generated by the normals to a family
+  of confocal quadrics, and for many investigations it is convenient to
+  deal with this complex referred to the principal axes. For example,
+  Lie has developed the theory of curves in a Reye complex (i.e. curves
+  whose tangents belong to the complex) as solutions of a differential
+  equation of the form (b - c)xdydz + (c - a)ydzdx + (a - b)zdxdy = 0,
+  and we can simplify this equation by a logarithmic transformation.
+  Many theorems connecting complexes with differential equations have
+  been given by Lie and his school. A line complex, in fact, corresponds
+  to a Mongian equation having [oo]^3 line integrals.
+
+
+    Congruences.
+
+  As the coordinates of a line belonging to a congruence are functions
+  of two independent parameters, the theory of congruences is analogous
+  to that of surfaces, and we may regard it as a fundamental inquiry to
+  find the simplest form of surface into which a given congruence can be
+  transformed. Most of those whose properties have been extensively
+  discussed can be represented on a plane by a birational
+  transformation. But in addition to the difficulties of the theory of
+  algebraic surfaces, a subject still in its infancy, the theory of
+  congruences has other difficulties in that a congruence is seldom
+  completely represented, even by two equations.
+
+  A fundamental theorem is that the lines of a congruence are in general
+  bitangents of a surface; in fact, since the condition of intersection
+  of two consecutive straight lines is ld[lambda] + dmd[mu] + dnd[nu] =
+  0, a line l of the congruence meets two adjacent lines, say l1 and l2.
+  Suppose l, l1 lie in the plane pencil (A1a1) and l, l2 in the plane
+  pencil (A2a2), then the locus of the A's is the same as the envelope
+  of the a's, but a2 is the tangent plane at A1 and a1 at A2. This
+  surface is called the focal surface of the congruence, and to it all
+  the lines l are bitangent. The distinctive property of the points A is
+  that two of the congruence lines through them coincide, and in like
+  manner the planes a each contain two coincident lines. The focal
+  surface consists of two sheets, but one or both may degenerate into
+  curves; thus, for example, the normals to a surface are bitangents of
+  the surface of centres, and in the case of Dupin's cyclide this
+  surface degenerates into two conics.
+
+  In the discussion of congruences it soon becomes necessary to
+  introduce another number r, called the rank, which expresses the
+  number of plane pencils each of which contains an arbitrary line and
+  two lines of the congruence. The order of the focal surface is 2m(n -
+  1) - 2r, and its class is m(m - 1) - 2r. Our knowledge of congruences
+  is almost exclusively confined to those in which either m or n does
+  not exceed two. We give a brief account of those of the second order
+  without singular lines, those of order unity not being especially
+  interesting. A congruence generally has singular points through which
+  an infinite number of lines pass; a singular point is said to be of
+  order r when the lines through it lie on a cone of the rth degree. By
+  means of formulae connecting the number of singular points and their
+  orders with the class m of quadratic congruence Kummer proved that the
+  class cannot exceed seven. The focal surface is of degree four and
+  class 2m; this kind of quartic surface has been extensively studied by
+  Kummer, Cayley, Rohn and others. The varieties (2, 2), (2, 3), (2, 4),
+  (2, 5) all belong to at least one Reye complex; and so also does the
+  most important class of (2, 6) congruences which includes all the
+  above as special cases. The congruence (2, 2) belongs to a linear
+  complex and forty different Reye complexes; as above remarked, the
+  singular surface is Kummer's sixteen-nodal quartic, and the same
+  surface is focal for six different congruences of this variety. The
+  theory of (2, 2) congruences is completely analogous to that of the
+  surfaces called cyclides in three dimensions. Further particulars
+  regarding quadratic congruences will be found in Kummer's memoir of
+  1866, and the second volume of Sturm's treatise. The properties of
+  quadratic congruences having singular lines, i.e. degenerate focal
+  surfaces, are not so interesting as those of the above class; they
+  have been discussed by Kummer, Sturm and others.
+
+
+    Ruled surfaces.
+
+  Since a ruled surface contains only [infinity]^1 elements, this theory
+  is practically the same as that of curves. If a linear complex
+  contains more than n generators of a ruled surface of the nth degree,
+  it contains all the generators, hence for n = 2 there are three
+  linearly independent complexes, containing all the generators, and
+  this is a well-known property of quadric surfaces. In ruled cubics the
+  generators all meet two lines which may or may not coincide; these two
+  cases correspond to the two main classes of cubics discussed by Cayley
+  and Cremona. As regards ruled quartics, the generators must lie in one
+  and may lie in two linear complexes. The first class is equivalent to
+  a quartic in four dimensions and is always rational, but the latter
+  class has to be subdivided into the elliptic and the rational, just
+  like twisted quartic curves. A quintic skew may not lie in a linear
+  complex, and then it is unicursal, while of sextics we have two
+  classes not in a linear complex, viz. the elliptic variety, having
+  thirty-six places where a linear complex contains six consecutive
+  generators, and the rational, having six such places.
+
+  The general theory of skews in two linear complexes is identical with
+  that of curves on a quadric in three dimensions and is known. But for
+  skews lying in only one linear complex there are difficulties; the
+  curve now lies in four dimensions, and we represent it in three by
+  stereographic projection as a curve meeting a given plane in n points
+  on a conic. To find the maximum deficiency for a given degree would
+  probably be difficult, but as far as degree eight the space-curve
+  theory of Halphen and Nother can be translated into line geometry at
+  once. When the skew does not lie in a linear complex at all the theory
+  is more difficult still, and the general theory clearly cannot advance
+  until further progress is made in the study of twisted curves.
+
+  REFERENCES.--The earliest works of a general nature are Plucker, _Neue
+  Geometrie des Raumes_ (Leipzig, 1868); and Kummer, "Uber die
+  algebraischen Strahlensysteme," _Berlin Academy_ (1866). Systematic
+  development on purely synthetic lines will be found in the three
+  volumes of Sturm, _Liniengeometrie_ (Leipzig, 1892, 1893, 1896); vol.
+  i. deals with the linear and Reye complexes, vols. ii. and iii. with
+  quadratic congruences and complexes respectively. For a highly
+  suggestive review by Gino Loria see _Bulletin des sciences
+  mathematiques_ (1893, 1897). A shorter treatise, giving a very
+  interesting account of Klein's coordinates, is the work of Koenigs,
+  _La Geometrie reglee et ses applications_ (Paris, 1898). English
+  treatises are C.M. Jessop, _Treatise on the Line Complex_ (1903);
+  R.W.H.T. Hudson, _Kummer's Quartic_ (1905). Many references to memoirs
+  on line geometry will be found in Hagen, _Synopsis der hoheren
+  Mathematik_, ii. (Berlin, 1894); Loria, _Il passato ed il presente
+  delle principali teorie geometriche_ (Milan, 1897); a clear resume of
+  the principal results is contained in the very elegant volume of
+  Pascal, _Repertorio di mathematiche superiori_, ii. (Milan, 1900).
+  Another treatise dealing extensively with line geometry is Lie,
+  _Geometrie der Beruhrungstransformationen_ (Leipzig, 1896). Many
+  memoirs on the subject have appeared in the _Mathematische Annalen_; a
+  full list of these will be found in the index to the first fifty
+  volumes, p. 115. Perhaps the two memoirs which have left most
+  impression on the subsequent development of the subject are Klein,
+  "Zur Theorie der Liniencomplexe des ersten und zweiten Grades," _Math.
+  Ann._ ii.; and Lie, "Uber Complexe, insbesondere Linien- und
+  Kugelcomplexe," _Math. Ann._ v.     (J. H. Gr.)
+
+
+VI. NON-EUCLIDEAN GEOMETRY
+
+The various metrical geometries are concerned with the properties of the
+various types of congruence-groups, which are defined in the study of
+the _axioms_ of _geometry_ and of their immediate consequences. But this
+point of view of the subject is the outcome of recent research, and
+historically the subject has a different origin. Non-Euclidean geometry
+arose from the discussion, extending from the Greek period to the
+present day, of the various assumptions which are implicit in the
+traditional Euclidean system of geometry. In the course of these
+investigations it became evident that metrical geometries, each
+internally consistent but inconsistent in many respects with each other
+and with the Euclidean system, could be developed. A short historical
+sketch will explain this origin of the subject, and describe the famous
+and interesting progress of thought on the subject. But previously a
+description of the chief characteristic properties of elliptic and of
+hyperbolic geometries will be given, assuming the standpoint arrived at
+below under VII. _Axioms of Geometry_.
+
+First assume the equation to the absolute (cf. _loc. cit._) to be w^2 -
+x^2 - y^2 - z^2 = 0. The absolute is then real, and the geometry is
+hyberbolic.
+
+  The distance (d12) between the two points (x1, y1, z1, w1) and (x2,
+  y2, z2, w2) is given by
+
+    cosh (d12/[gamma]) = (w1w2 - x1x2 - y1y2 - z1z2)/[(w1^2 - x1^2 - y1^2 - Z1^2)
+      (w2^2 - x2^2 - y2^2 - z2^2)]1/2     (1)
+
+  The only points to which the metrical geometry applies are those
+  within the region enclosed by the quadric; the other points are
+  "improper ideal points." The angle ([theta]12) between two planes, l1x
+  + m1y + n1z + r1w = 0 and l2x + m2y + n2z + r2w = 0, is given by
+
+  cos [theta]12 = (l1l2 + m1m2 + n1n2 - r1r2)/{(l1^2 + m1^2 + n1^2 - r1^2)
+    (l2^2 + m2^2 + n2^2 - r2^2)}^1/2       (2)
+
+  These planes only have a real angle of inclination if they possess a
+  line of intersection within the actual space, i.e. if they intersect.
+  Planes which do not intersect possess a shortest distance along a line
+  which is perpendicular to both of them. If this shortest distance is
+  [delta]12, we have
+
+    cosh ([delta]12/[gamma]) = (l1l2 + m1m2 + n1n2 - r1r2)/(l1^2 + m1^2 + n1^2 - r1^2)
+      (l2^2 + m2^2 + n2^2 - r2^2)^1/2     (3)
+
+  [Illustration: FIG. 67.]
+
+  Thus in the case of the two planes one and only one of the two,
+  [theta]12 and [delta]12, is real. The same considerations hold for
+  coplanar straight lines (see VII. _Axioms of Geometry_). Let O (fig.
+  67) be the point (0, 0, 0, 1), OX the line y = 0, z = 0, OY the line z
+  = 0, x = 0, and OZ the line x = 0, y = 0. These are the coordinate
+  axes and are at right angles to each other. Let P be any point, and
+  let [rho] be the distance OP, [theta] the angle POZ, and [phi] the
+  angle between the planes ZOX and ZOP. Then the coordinates of P can be
+  taken to be
+
+    sinh ([rho]/[gamma]) sin [theta] cos [phi], sinh ([rho]/[gamma]) sin [theta]
+      sin [phi], sinh ([rho]/[gamma]) cos[theta], cosh ([rho]/[gamma]).
+
+  [Illustration: FIG. 68.]
+
+  If ABC is a triangle, and the sides and angles are named according to
+  the usual convention, we have
+
+    sinh (a/[gamma])/sin A = sinh (b/[gamma])/sin B = sinh (c/[gamma])/sin C, (4)
+
+  and also
+
+    cosh (a/[gamma]) = cosh (b/[gamma]) cosh (c/[gamma]) -
+      sinh (b/[gamma]) sinh (c/[gamma]) cos A,   (5)
+
+  with two similar equations. The sum of the three angles of a triangle
+  is always less than two right angles. The area of the triangle ABC is
+  [lambda]^2([pi] - A - B - C). If the base BC of a triangle is kept
+  fixed and the vertex A moves in the fixed plane ABC so that the area
+  ABC is constant, then the locus of A is a line of equal distance from
+  BC. This locus is not a straight line. The whole theory of similarity
+  is inapplicable; two triangles are either congruent, or their angles
+  are not equal two by two. Thus the elements of a triangle are
+  determined when its three angles are given. By keeping A and B and the
+  line BC fixed, but by making C move off to infinity along BC, the
+  lines BC and AC become parallel, and the sides a and b become
+  infinite. Hence from equation (5) above, it follows that two parallel
+  lines (cf. Section VII. _Axioms of Geometry_) must be considered as
+  making a zero angle with each other. Also if B be a right angle, from
+  the equation (5), remembering that, in the limit,
+
+    cosh (a/[gamma])/cosh (b/[gamma]) = cosh (a/[gamma])/sinh (b/[gamma]) = 1,
+
+  we have cos A = tanh (c/2[gamma]) .... (6).
+
+  The angle A is called by N.I. Lobatchewsky the "angle of parallelism."
+
+  The whole theory of lines and planes at right angles to each other is
+  simply the theory of conjugate elements with respect to the absolute,
+  where ideal lines and planes are introduced.
+
+  Thus if l and l' be any two conjugate lines with respect to the
+  absolute (of which one of the two must be improper, say l'), then any
+  plane through l' and containing proper points is perpendicular to l.
+  Also if p is any plane containing proper points, and P is its pole,
+  which is necessarily improper, then the lines through P are the
+  normals to P. The equation of the sphere, centre (x1, y1, z1, w1) and
+  radius [rho], is
+
+    (w1^2- x1^2- y1^2- z1^2)(w^2 - x^2 - y^2 - z^2) cosh^2([rho]/[gamma]) = (w1w -
+      x1x - y1y -z1z)^2   (7).
+
+  The equation of the surface of equal distance ([sigma]) from the plane
+  lx + my + nz + rw = 0 is
+
+    (l^2 + m^2 + n^2 - r^2)(w^2 - x^2 - y^2 - z^2) sinh^2([sigma]/[gamma]) = (rw +
+      lx + my + nz)^2    (8).
+
+  A surface of equal distance is a sphere whose centre is improper; and
+  both types of surface are included in the family
+
+    k^2(w^2 - x^2 - y^2 - z^2) = (ax + by + cz + dw)^2   (9).
+
+  But this family also includes a third type of surfaces, which can be
+  looked on either as the limits of spheres whose centres have
+  approached the absolute, or as the limits of surfaces of equal
+  distance whose central planes have approached a position tangential to
+  the absolute. These surfaces are called limit-surfaces. Thus (9)
+  denotes a limit-surface, if d^2 - a^2 - b^2 - c^2 = 0. Two
+  limit-surfaces only differ in position. Thus the two limit-surfaces
+  which touch the plane YOZ at O, but have their concavities turned in
+  opposite directions, have as their equations
+
+    w^2 - x^2 - y^2 - z^2 = (w [+-] x)^2.
+
+  The geodesic geometry of a sphere is elliptic, that of a surface of
+  equal distance is hyperbolic, and that of a limit-surface is parabolic
+  (i.e. _Euclidean_). The equation of the surface (cylinder) of equal
+  distance ([delta]) from the line OX is
+
+    (w^2 - x^2) tanh^2([delta]/[gamma]) - y^2 - z^2 = 0.
+
+  This is not a ruled surface. Hence in this geometry it is not possible
+  for two straight lines to be at a constant distance from each other.
+
+  Secondly, let the equation of the absolute be x^2 + y^2 + z^2 + w^2 =
+  0. The absolute is now imaginary and the geometry is elliptic.
+
+  The distance (d12) between the two points (x1, y1, z1, w1) and (x2,
+  y2, z2, w2) is given by
+
+    cos (d12/[gamma]) = [+-](x1x2 + y1y2 + z1z2 + w1w2)
+      / {(x1^2 + y1^2 + z1^2 + w1^2) {(x2^2 + y2^2 + z2^2 + w2^2)}^1/2   (10).
+
+  Thus there are two distances between the points, and if one is d12,
+  the other is [pi][gamma]-d12. Every straight line returns into itself,
+  forming a closed series. Thus there are two segments between any two
+  points, together forming the whole line which contains them; one
+  distance is associated with one segment, and the other distance with
+  the other segment. The complete length of every straight line is
+  [pi][gamma].
+
+  The angle between the two planes l1x + m1y + n1z + r + 1w = 0 and l2x
+  + m2y + n2z + r2w = 0 is
+
+    cos [theta]12 = (l1l2 + m1m2 + n1n2 + r1r2)/ {(l1^2 + m1^2 + n1^2 +r1^2)
+      (l2^2 + m2^2 + n2^2 + r2^2)}^1/2 (11).
+
+  The polar plane with respect to the absolute of the point (x1, y1, z1,
+  w1) is the real plane x1x + y1y + z1z + w1w = 0, and the pole of the
+  plane l1x + m1y + n1z + r1w = 0 is the point (l1, m1, n1, r1). Thus
+  (from equations 10 and 11) it follows that the angle between the polar
+  planes of the points (x1, ...) and (x2, ...) is d12/[gamma], and that
+  the distance between the poles of the planes (l1, ...) and (l2, ...)
+  is [gamma][theta]12. Thus there is complete reciprocity between points
+  and planes in respect to all properties. This complete reign of the
+  principle of duality is one of the great beauties of this geometry.
+  The theory of lines and planes at right angles is simply the theory of
+  conjugate elements with respect to the absolute. A tetrahedron
+  self-conjugate with respect to the absolute has all its intersecting
+  elements (edges and planes) at right angles. If l and l' are two
+  conjugate lines, the planes through one are the planes perpendicular
+  to the other. If P is the pole of the plane p, the lines through P are
+  the normals to the plane p. The distance from P to p is 1/2[pi][gamma].
+  Thus every sphere is also a surface of equal distance from the polar
+  of its centre, and conversely. A plane does not divide space; for the
+  line joining any two points P and Q only cuts the plane once, in L
+  say, then it is always possible to go from P to Q by the segment of
+  the line PQ which does not contain L. But P and Q may be said to be
+  separated by a plane p, if the point in which PQ cuts p lies on the
+  shortest segment between P and Q. With this sense of "separation," it
+  is possible[2] to find three points P, Q, R such that P and Q are
+  separated by the plane p, but P and R are not separated by p, nor are
+  Q and R.
+
+  Let A, B, C be any three non-collinear points, then four triangles are
+  defined by these points. Thus if a, b, c and A, B, C are the elements
+  of any one triangle, then the four triangles have as their elements:
+
+    (1)      a,                  b,                  c,             A,           B,           C.
+
+    (2)      a,            [pi][gamma] - b,   [pi][gamma] - c,      A,       [pi] - B,    [pi] - C.
+
+    (3) [pi][gamma] - a,          b,          [pi][gamma] - c,   [pi] - A,       B,       [pi] - C.
+
+    (4) [pi][gamma] - a,   [pi][gamma] - b,          c,          [pi] - A,   [pi] - B,        C.
+
+  The formulae connecting the elements are
+
+    sin A/sin (a/[gamma]) = sin B/sin (b/[gamma]) = sin C/sin (c/[gamma]),
+         (12)
+
+  and
+
+    cos (a/[gamma]) = cos (b/[gamma]) cos (c/[gamma]) + sin (b/[gamma])
+      sin (c/[gamma]) cos A,   (13)
+
+  with two similar equations.
+
+  Two cases arise, namely (I.) according as one of the four triangles
+  has as its sides the shortest segments between the angular points, or
+  (II.) according as this is not the case. When case I. holds there is
+  said to be a "principal triangle."[3] If all the figures considered
+  lie within a sphere of radius 1/4[pi][gamma] only case I. can hold, and
+  the principal triangle is the triangle wholly within this sphere, also
+  the peculiarities in respect to the separation of points by a plane
+  cannot then arise. The sum of the three angles of a triangle ABC is
+  always greater than two right angles, and the area of the triangle is
+  [gamma]^2(A + B + C--[pi]). Thus as in hyperbolic geometry the theory
+  of similarity does not hold, and the elements of a triangle are
+  determined when its three angles are given. The coordinates of a point
+  can be written in the form
+
+    sin ([rho]/[gamma]) sin [Phi] cos [phi], sin ([rho]/[gamma]) sin [Phi]
+      sin [phi], sin ([rho]/[gamma]) cos [Phi], cos ([rho]/[gamma]),
+
+  where [rho], [Phi] and [phi] have the same meanings as in the
+  corresponding formulae in hyperbolic geometry. Again, suppose a watch
+  is laid on the plane OXY, face upwards with its centre at O, and the
+  line 12 to 6 (as marked on dial) along the line YOY. Let the watch be
+  continually pushed along the plane along the line OX, that is, in the
+  direction 9 to 3. Then the line XOX being of finite length, the watch
+  will return to O, but at its first return it will be found to be face
+  downwards on the other side of the plane, with the line 12 to 6
+  reversed in direction along the line YOY. This peculiarity was first
+  pointed out by Felix Klein. The theory of parallels as it exists in
+  hyperbolic space has no application in elliptic geometry. But another
+  property of Euclidean parallel lines holds in elliptic geometry, and
+  by the use of it parallel lines are defined. For the equation of the
+  surface (cylinder) of equal distance ([delta]) from the line XOX is
+
+    (x^2 + w^2) tan^2([delta]/[gamma]) - (y^2 + z^2) = 0.
+
+  This is also the surface of equal distance, 1/2[pi][gamma]-[delta],
+  from the line conjugate to XOX. Now from the form of the above
+  equation this is a ruled surface, and through every point of it two
+  generators pass. But these generators are lines of equal distance from
+  XOX. Thus throughout every point of space two lines can be drawn which
+  are lines of equal distance from a given line l. This property was
+  discovered by W.K. Clifford. The two lines are called Clifford's right
+  and left parallels to l through the point. This property of
+  parallelism is reciprocal, so that if m is a left parallel to l, then
+  l is a left parallel to m. Note also that two parallel lines l and m
+  are not coplanar. Many of those properties of Euclidean parallels,
+  which do not hold for Lobatchewsky's parallels in hyperbolic geometry,
+  do hold for Clifford's parallels in elliptic geometry. The geodesic
+  geometry of spheres is elliptic, the geodesic geometry of surfaces of
+  equal distance from lines (cylinders) is Euclidean, and surfaces of
+  revolution can be found[4] of which the geodesic geometry is
+  hyperbolic. But it is to be noticed that the connectivity of these
+  surfaces is different to that of a Euclidean plane. For instance there
+  are only [&infin]^2 congruence transformations of the cylindrical
+  surfaces of equal distance into themselves, instead of the [&infin]^3
+  for the ordinary plane. It would obviously be possible to state
+  "axioms" which these geodesics satisfy, and thus to define
+  independently, and not as loci, quasi-spaces of these peculiar types.
+  The existence of such Euclidean quasi-geometries was first pointed out
+  by Clifford.[5]
+
+In both elliptic and hyperbolic geometry the spherical geometry, i.e.
+the relations between the angles formed by lines and planes passing
+through the same point, is the same as the "spherical trigonometry" in
+Euclidean geometry. The constant [gamma], which appears in the formulae
+both of hyperbolic and elliptic geometry, does not by its variation
+produce different types of geometry. There is only one type of elliptic
+geometry and one type of hyperbolic geometry; and the magnitude of the
+constant [gamma] in each case simply depends upon the magnitude of the
+arbitrary unit of length in comparison with the natural unit of length
+which each particular instance of either geometry presents. The
+existence of a natural unit of length is a peculiarity common both to
+hyperbolic and elliptic geometries, and differentiates them from
+Euclidean geometry. It is the reason for the failure of the theory of
+similarity in them. If [gamma] is very large, that is, if the natural
+unit is very large compared to the arbitrary unit, and if the lengths
+involved in the figures considered are not large compared to the
+arbitrary unit, then both the elliptic and hyperbolic geometries
+approximate to the Euclidean. For from formulae (4) and (5) and also
+from (12) and (13) we find, after retaining only the lowest powers of
+small quantities, as the formulae for any triangle ABC,
+
+  a/ sin A = b/ sin B = c/ sin C,
+
+and
+
+  a^2 = b^2 + c^2 - 2bc cos A,
+
+with two similar equations. Thus the geometries of small figures are in
+both types Euclidean.
+
+
+  Theory of parallels before Gauss.
+
+_History._--"In pulcherrimo Geometriae corpore," wrote Sir Henry Savile
+in 1621, "duo sunt naevi, duae labes nec quod sciam plures, in quibus
+eluendis et emaculendis cum veterum tum recentiorum ... vigilavit
+industria." These two blemishes are the theory of parallels and the
+theory of proportion. The "industry of the moderns," in both respects,
+has given rise to important branches of mathematics, while at the same
+time showing that Euclid is in these respects more free from blemish
+than had been previously credible. It was from endeavours to improve the
+theory of parallels that non-Euclidean geometry arose; and though it has
+now acquired a far wider scope, its historical origin remains
+instructive and interesting. Euclid's "axiom of parallels" appears as
+Postulate V. to the first book of his _Elements_, and is stated thus,
+"And that, if a straight line falling on two straight lines make the
+angles, internal and on the same side, less than two right angles, the
+two straight lines, being produced indefinitely, meet on the side on
+which are the angles less than two right angles." The original Greek is
+[Greek: kai ean eis duo eutheias eutheia empiptousa tas entos kai epi ta
+auta mere gonias duo orthon elassonas poie, ekballomenas tas duo
+eutheias ep' apeiron sympiptein, eph' ha mere eisin hai ton duo orthon
+elassones].
+
+To Euclid's successors this axiom had signally failed to appear
+self-evident, and had failed equally to appear indemonstrable. Without
+the use of the postulate its converse is proved in Euclid's 28th
+proposition, and it was hoped that by further efforts the postulate
+itself could be also proved. The first step consisted in the discovery
+of equivalent axioms. Christoph Clavius in 1574 deduced the axiom from
+the assumption that a line whose points are all equidistant from a
+straight line is itself straight. John Wallis in 1663 showed that the
+postulate follows from the possibility of similar triangles on different
+scales. Girolamo Saccheri (1733) showed that it is sufficient to have a
+single triangle, the sum of whose angles is two right angles. Other
+equivalent forms may be obtained, but none shows any essential
+superiority to Euclid's. Indeed plausibility, which is chiefly aimed at,
+becomes a positive demerit where it conceals a real assumption.
+
+
+  Saccheri.
+
+A new method, which, though it failed to lead to the desired goal,
+proved in the end immensely fruitful, was invented by Saccheri, in a
+work entitled _Euclides ab omni naevo vindicatus_ (Milan, 1733). If the
+postulate of parallels is involved in Euclid's other assumptions,
+contradictions must emerge when it is denied while the others are
+maintained. This led Saccheri to attempt a _reductio ad absurdum_, in
+which he mistakenly believed himself to have succeeded. What is
+interesting, however, is not his fallacious conclusion, but the
+non-Euclidean results which he obtains in the process. Saccheri
+distinguishes three hypotheses (corresponding to what are now known as
+Euclidean or parabolic, elliptic and hyperbolic geometry), and proves
+that some one of the three must be universally true. His three
+hypotheses are thus obtained: equal perpendiculars AC, BD are drawn from
+a straight line AB, and CD are joined. It is shown that the angles ACD,
+BDC are equal. The first hypothesis is that these are both right angles;
+the second, that they are both obtuse; and the third, that they are both
+acute. Many of the results afterwards obtained by Lobatchewsky and
+Bolyai are here developed. Saccheri fails to be the founder of
+non-Euclidean geometry only because he does not perceive the possible
+truth of his non-Euclidean hypotheses.
+
+
+  Lambert.
+
+Some advance is made by Johann Heinrich Lambert in his _Theorie der
+Parallellinien_ (written 1766; posthumously published 1786). Though he
+still believed in the necessary truth of Euclidean geometry, he
+confessed that, in all his attempted proofs, something remained
+undemonstrated. He deals with the same three hypotheses as Saccheri,
+showing that the second holds on a sphere, while the third would hold on
+a sphere of purely imaginary radius. The second hypothesis he succeeds
+in condemning, since, like all who preceded Bernhard Riemann, he is
+unable to conceive of the straight line as finite and closed. But the
+third hypothesis, which is the same as Lobatchewsky's, is not even
+professedly refuted.[6]
+
+
+  Three periods of non-Euclidean geometry.
+
+Non-Euclidean geometry proper begins with Karl Friedrich Gauss. The
+advance which he made was rather philosophical than mathematical: it was
+he (probably) who first recognized that the postulate of parallels is
+possibly false, and should be empirically tested by measuring the angles
+of large triangles. The history of non-Euclidean geometry has been aptly
+divided by Felix Klein into three very distinct periods. The
+first--which contains only Gauss, Lobatchewsky and Bolyai--is
+characterized by its synthetic method and by its close relation to
+Euclid. The attempt at indirect proof of the disputed postulate would
+seem to have been the source of these three men's discoveries; but when
+the postulate had been denied, they found that the results, so far from
+showing contradictions, were just as self-consistent as Euclid. They
+inferred that the postulate, if true at all, can only be proved by
+observations and measurements. Only one kind of non-Euclidean space is
+known to them, namely, that which is now called hyperbolic. The second
+period is analytical, and is characterized by a close relation to the
+theory of surfaces. It begins with Riemann's inaugural dissertation,
+which regards space as a particular case of a _manifold_; but the
+characteristic standpoint of the period is chiefly emphasized by Eugenio
+Beltrami. The conception of measure of curvature is extended by Riemann
+from surfaces to spaces, and a new kind of space, finite but unbounded
+(corresponding to the second hypothesis of Saccheri and Lambert), is
+shown to be possible. As opposed to the second period, which is purely
+metrical, the third period is essentially projective in its method. It
+begins with Arthur Cayley, who showed that metrical properties are
+projective properties relative to a certain fundamental quadric, and
+that different geometries arise according as this quadric is real,
+imaginary or degenerate. Klein, to whom the development of Cayley's work
+is due, showed further that there are two forms of Riemann's space,
+called by him the elliptic and the spherical. Finally, it has been shown
+by Sophus Lie, that if figures are to be freely movable throughout all
+space in [oo]^6 ways, no other three-dimensional spaces than the above
+four are possible.
+
+
+  Gauss.
+
+Gauss published nothing on the theory of parallels, and it was not
+generally known until after his death that he had interested himself in
+that theory from a very early date. In 1799 he announces that Euclidean
+geometry would follow from the assumption that a triangle can be drawn
+greater than any given triangle. Though unwilling to assume this, we
+find him in 1804 still hoping to prove the postulate of parallels. In
+1830 he announces his conviction that geometry is not an a priori
+science; in the following year he explains that non-Euclidean geometry
+is free from contradictions, and that, in this system, the angles of a
+triangle diminish without limit when all the sides are increased. He
+also gives for the circumference of a circle of radius r the formula
+[pi]k(e^(r/k) - e^(r-/k)), where k is a constant depending upon the
+nature of the space. In 1832, in reply to the receipt of Bolyai's
+_Appendix_, he gives an elegant proof that the amount by which the sum
+of the angles of a triangle falls short of two right angles is
+proportional to the area of the triangle. From these and a few other
+remarks it appears that Gauss possessed the foundations of hyperbolic
+geometry, which he was probably the first to regard as perhaps true. It
+is not known with certainty whether he influenced Lobatchewsky and
+Bolyai, but the evidence we possess is against such a view.[7]
+
+
+  Lobatchewsky.
+
+The first to publish a non-Euclidean geometry was Nicholas Lobatchewsky,
+professor of mathematics in the new university of Kazan.[8] In the place
+of the disputed postulate he puts the following: "All straight lines
+which, in a plane, radiate from a given point, can, with respect to any
+other straight line in the same plane, be divided into two classes, the
+_intersecting_ and the _non-intersecting_. The _boundary line_ of the
+one and the other class is called _parallel to the given line_." It
+follows that there are two parallels to the given line through any
+point, each meeting the line at infinity, like a Euclidean parallel.
+(Hence a line has two distinct points at infinity, and not one only as
+in ordinary geometry.) The two parallels to a line through a point make
+equal acute angles with the perpendicular to the line through the point.
+If p be the length of the perpendicular, either of these angles is
+denoted by [Pi](p). The determination of [Pi](p) is the chief problem
+(cf. equation (6) above); it appears finally that, with a suitable
+choice of the unit of length,
+
+  tan 1/2 [Pi](p) = e^(-p).
+
+Before obtaining this result it is shown that spherical trigonometry is
+unchanged, and that the normals to a circle or a sphere still pass
+through its centre. When the radius of the circle or sphere becomes
+infinite all these normals become parallel, but the circle or sphere
+does not become a straight line or plane. It becomes what Lobatchewsky
+calls a limit-line or limit-surface. The geometry on such a surface is
+shown to be Euclidean, limit-lines replacing Euclidean straight lines.
+(It is, in fact, a surface of zero measure of curvature.) By the help of
+these propositions Lobatchewsky obtains the above value of [Pi](p), and
+thence the solution of triangles. He points out that his formulae result
+from those of spherical trigonometry by substituting ia, ib, ic, for the
+sides a, b, c.
+
+
+  Bolyai.
+
+John Bolyai, a Hungarian, obtained results closely corresponding to
+those of Lobatchewsky. These he published in an appendix to a work by
+his father, entitled _Appendix Scientiam spatii absolute veram exhibens:
+a veritate aut falsitate Axiomatis XI. Euclidei (a priori haud unquam
+decidenda) independentem: adjecta ad casum falsitatis, quadratura
+circuli geometrica_.[9] This work was published in 1831, but its
+conception dates from 1823. It reveals a profounder appreciation of the
+importance of the new ideas, but otherwise differs little from
+Lobatchewsky's. Both men point out that Euclidean geometry as a limiting
+case of their own more general system, that the geometry of very small
+spaces is always approximately Euclidean, that no a priori grounds exist
+for a decision, and that observation can only give an approximate
+answer. Bolyai gives also, as his title indicates, a geometrical
+construction, in hyperbolic space, for the quadrature of the circle, and
+shows that the area of the greatest possible triangle, which has all its
+sides parallel and all its angles zero, is [pi][iota]^2, where i is what
+we should now call the space-constant.
+
+
+  Riemann.
+
+The works of Lobatchewsky and Bolyai, though known and valued by Gauss,
+remained obscure and ineffective until, in 1866, they were translated
+into French by J. Houel. But at this time Riemann's dissertation, _Uber
+die Hypothesen, welche der Geometrie zu Grunde liegen_,[10] was already
+about to be published. In this work Riemann, without any knowledge of
+his predecessors in the same field, inaugurated a far more profound
+discussion, based on a far more general standpoint; and by its
+publication in 1867 the attention of mathematicians and philosophers was
+at last secured. (The dissertation dates from 1854, but owing to changes
+which Riemann wished to make in it, it remained unpublished until after
+his death.)
+
+
+  Definition of a manifold.
+
+Riemann's work contains two fundamental conceptions, that of a manifold
+and that of the _measure of curvature_ of a continuous manifold
+possessed of what he calls flatness in the smallest parts. By means of
+these conceptions space is made to appear at the end of a gradual series
+of more and more specialized conceptions. Conceptions of magnitude, he
+explains, are only possible where we have a general conception capable
+of determination in various ways. The manifold consists of all these
+various determinations, each of which is an element of the manifold. The
+passage from one element to another may be discrete or continuous; the
+manifold is called discrete or continuous accordingly. Where it is
+discrete two portions of it can be compared, as to magnitude, by
+counting; where continuous, by measurement. But measurement demands
+superposition, and consequently some magnitude independent of its place
+in the manifold. In passing, in a continuous manifold, from one element
+to another in a determinate way, we pass through a series of
+intermediate terms, which form a one-dimensional manifold. If this whole
+manifold be similarly caused to pass over into another, each of its
+elements passes through a one-dimensional manifold, and thus on the
+whole a two-dimensional manifold is generated. In this way we can
+proceed to n dimensions. Conversely, a manifold of n dimensions can be
+analysed into one of one dimension and one of (n - 1) dimensions. By
+repetitions of this process the position of an element may be at last
+determined by n magnitudes. We may here stop to observe that the above
+conception of a manifold is akin to that due to Hermann Grassmann in the
+first edition (1847) of his _Ausdehnungslehre_.[11]
+
+
+  Measure of curvature.
+
+Both concepts have been elaborated and superseded by the modern
+procedure in respect to the axioms of geometry, and by the conception of
+abstract geometry involved therein. Riemann proceeds to specialize the
+manifold by considerations as to measurement. If measurement is to be
+possible, some magnitude, we saw, must be independent of position; let
+us consider manifolds in which lengths of lines are such magnitudes, so
+that every line is measurable by every other. The coordinates of a point
+being x1, x2, ... x_n, let us confine ourselves to lines along which the
+ratios dx1 : dx2 : ... : dx_n alter continuously. Let us also assume
+that the element of length, ds, is unchanged (to the first order) when
+all its points undergo the same infinitesimal motion. Then if all the
+increments dx be altered in the same ratio, ds is also altered in this
+ratio. Hence ds is a homogeneous function of the first degree of the
+increments dx. Moreover, ds must be unchanged when all the dx change
+sign. The simplest possible case is, therefore, that in which ds is the
+square root of a quadratic function of the dx. This case includes space,
+and is alone considered in what follows. It is called the case of
+flatness in the smallest parts. Its further discussion depends upon the
+measure of curvature, the second of Riemann's fundamental conceptions.
+This conception, derived from the theory of surfaces, is applied as
+follows. Any one of the shortest lines which issue from a given point
+(say the origin) is completely determined by the initial ratios of the
+dx. Two such lines, defined by dx and [delta]x say, determine a pencil,
+or one-dimensional series, of shortest lines, any one of which is
+defined by [lambda]dx + [mu][delta]x, where the parameter [lambda] :
+[mu] may have any value. This pencil generates a two-dimensional series
+of points, which may be regarded as a surface, and for which we may
+apply Gauss's formula for the measure of curvature at any point. Thus at
+every point of our manifold there is a measure of curvature
+corresponding to every such pencil; but all these can be found when
+n.[/(n-1)]/2 of them are known. If figures are to be freely movable, it
+is necessary and sufficient that the measure of curvature should be the
+same for all points and all directions at each point. Where this is the
+case, if [alpha] be the measure of curvature, the linear element can be
+put into the form
+
+  ds = [root]([Sigma]dx^2)/(1 + 1/4[alpha][Sigma]x^2).
+
+If [alpha] be positive, space is finite, though still unbounded, and
+every straight line is closed--a possibility first recognized by
+Riemann. It is pointed out that, since the possible values of a form a
+continuous series, observations cannot prove that our space is strictly
+Euclidean. It is also regarded as possible that, in the infinitesimal,
+the measure of curvature of our space should be variable.
+
+There are four points in which this profound and epoch-making work is
+open to criticism or development--(1) the idea of a manifold requires
+more precise determination; (2) the introduction of coordinates is
+entirely unexplained and the requisite presuppositions are unanalysed;
+(3) the assumption that ds is the square root of a quadratic function of
+dx1, dx2, ... is arbitrary; (4) the idea of superposition, or
+congruence, is not adequately analysed. The modern solution of these
+difficulties is properly considered in connexion with the general
+subject of the axioms of geometry.
+
+
+  Helmholtz.
+
+The publication of Riemann's dissertation was closely followed by two
+works of Hermann von Helmholtz,[12] again undertaken in ignorance of the
+work of predecessors. In these a proof is attempted that ds must be a
+rational integral quadratic function of the increments of the
+coordinates. This proof has since been shown by Lie to stand in need of
+correction (see VII. _Axioms of Geometry_). Helmholtz's remaining works
+on the subject[13] are of almost exclusively philosophical interest. We
+shall return to them later.
+
+
+  Beltrami.
+
+The only other writer of importance in the second period is Eugenio
+Beltrami, by whom Riemann's work was brought into connexion with that of
+Lobatchewsky and Bolyai. As he gave, by an elegant method, a convenient
+Euclidean interpretation of hyperbolic plane geometry, his results will
+be stated at some length[14]. The _Saggio_ shows that Lobatchewsky's
+plane geometry holds in Euclidean geometry on surfaces of constant
+negative curvature, straight lines being replaced by geodesics. Such
+surfaces are capable of a conformal representation on a plane, by which
+geodesics are represented by straight lines. Hence if we take, as
+coordinates on the surface, the Cartesian coordinates of corresponding
+points on the plane, the geodesics must have linear equations.
+
+  Hence it follows that
+
+    ds^2 = R^2w^(-4){([alpha]^2 - v^2)du^2 + 2uvdudv + ([alpha]^2 - u^2)dv^2}
+
+  where w^2 = [alpha]^2 - u^2 - v^2, and (-1)/R^2 is the measure of
+  curvature of our surface (note that k = [gamma] as used above). The
+  angle between two geodesics u = const., v = const. is [theta], where
+
+    cos [theta] = uv/[root]{([alpha]^2 - u^2)([alpha]^2 - v^2)}, sin [theta] =
+      aw/[root]{(a^2 - u^2)(a^2 - v^2)}.
+
+  Thus u = 0 is orthogonal to all geodesies v = const., and vice versa.
+  In order that sin [theta] may be real, w^2 must be positive; thus
+  geodesics have no real intersection when the corresponding straight
+  lines intersect outside the circle u^2 + v^2 = [alpha]^2. When they
+  intersect on this circle, [theta] = 0. Thus Lobatchewsky's parallels
+  are represented by straight lines intersecting on the circle. Again,
+  transforming to polar coordinates u = r cos [mu], v = r sin [mu], and
+  calling [rho] the geodesic distance of u, v from the origin, we have,
+  for a geodesic through the origin,
+
+    d[rho] = Radr/(a^2 - r^2), [rho] = 1/2R log(a + r)/(a - r), r = a tan h
+      ([rho]/R).
+
+  Thus points on the surface corresponding to points in the plane on the
+  limiting circle r = a, are all at an infinite distance from the
+  origin. Again, considering r constant, the arc of a geodesic circle
+  subtending an angle [mu] at the origin is
+
+    [sigma] = Rr[mu]/[root](a^2 - r^2) = [mu]R sin h ([rho]/R),
+
+  whence the circumference of a circle of radius [rho] is 2[pi]R sin h
+  ([rho]/R). Again, if [alpha] be the angle between any two geodesics
+
+    V - v = m(U - u), V - v = n(U - u),
+
+  then tan [alpha] = a(n - m)w/{(1 + mn)a^2 - (v - mu) (v - nu)}.
+
+  Thus [alpha] is imaginary when u, v is outside the limiting circle,
+  and is zero when, and only when, u, v is on the limiting circle. All
+  these results agree with those of Lobatchewsky and Bolyai. The maximum
+  triangle, whose angles are all zero, is represented in the auxiliary
+  plane by a triangle inscribed in the limiting circle. The angle of
+  parallelism is also easily obtained. The perpendicular to v = 0 at a
+  distance [delta] from the origin is u = a tan h ([delta]/R), and the
+  parallel to this through the origin is u = v sin h ([delta]/R). Hence
+  [Pi] ([delta]), the angle which this parallel makes with v = 0, is
+  given by
+
+    tan [Pi]([delta]) . sin h ([delta]/R) = 1, or tan 1/2[Pi]([delta]) =
+      e^(-[delta]/R)
+
+  which is Lobatchewsky's formula. We also obtain easily for the area of
+  a triangle the formula R^2([pi] - A - B - C).
+
+  Beltrami's treatment connects two curves which, in the earlier
+  treatment, had no connexion. These are limit-lines and curves of
+  constant distance from a straight line. Both may be regarded as
+  circles, the first having an infinite, the second an imaginary radius.
+  The equation to a circle of radius [rho] and centre u0v0 is
+
+    (a^2 - uu0 - vv0)^2 = cos h^2 ([rho]/R)w0^2w^2 = C^2w^2     (say).
+
+  This equation remains real when [rho] is a pure imaginary, and remains
+  finite when w0 = 0, provided [rho] becomes infinite in such a way that
+  w0 cos h ([rho]/R) remains finite. In the latter case the equation
+  represents a limit-line. In the former case, by giving different
+  values to C, we obtain concentric circles with the imaginary centre
+  u0v0. One of these, obtained by putting C = 0, is the straight line
+  a^2 - uu0 - vv0 = 0. Hence the others are each throughout at a
+  constant distance from this line. (It may be shown that all motions in
+  a hyperbolic plane consist, in a general sense, of rotations; but
+  three types must be distinguished according as the centre is real,
+  imaginary or at infinity. All points describe, accordingly, one of the
+  three types of circles.)
+
+  The above Euclidean interpretation fails for three or more dimensions.
+  In the _Teoria fondamentale_, accordingly, where n dimensions are
+  considered, Beltrami treats hyperbolic space in a purely analytical
+  spirit. The paper shows that Lobatchewsky's space of any number of
+  dimensions has, in Riemann's sense, a constant negative measure of
+  curvature. Beltrami starts with the formula (analogous to that of the
+  _Saggio_)
+
+    ds^2 = R^2x^(-2)(dx^2 + dx1^2 + dx2^2 + ... + dx_n^2)
+
+  where x^2 + x1^2 + x2^2 + ... + x_n^2 = a^2.
+
+  He shows that geodesics are represented by linear equations between
+  x1, x2, ..., x_n, and that the geodesic distance [rho] between two
+  points x and x' is given by
+
+          [rho]                    a^2 - x1x'1 - x2x'2 - ... - x_n x'_n
+     cosh ----- = ---------------------------------------------------------------------------
+            R     {(a^2 - x1^2 - x2^2 - ... - x_n^2)(a^2 - x'1^2 - x'2^2 - ... - x'_n^2)}^1/2
+
+  (a formula practically identical with Cayley's, though obtained by a
+  very different method). In order to show that the measure of curvature
+  is constant, we make the substitutions
+
+    x1 = r[lambda]1, x2 = r[lambda]2 ... x_n = r[lambda]_n, where
+      [Sigma][lambda]^2 = 1.
+
+  Hence
+                  _________
+    ds^2 = (Radr/(a^2 - r^2)])^2 + R^2r^2d[Delta]^2/(a^2 - r^2).
+
+  where
+
+    d[Delta]^2 = [Sigma]d[lambda]^2.
+
+  Also calling [rho] the geodesic distance from the origin, we have
+
+         [rho]           a               [rho]           r
+    cosh ----- = -----------------, sinh ----- = -----------------.
+           R     [root](a^2 - r^2)         R     [root](a^2 - r^2)
+
+  Hence
+
+    ds^2 = d[rho]^2 + (R sin h ([rho]/R))^2d[Delta]^2.
+
+  Putting
+
+    z1 = [rho][lambda]1, z2 = [rho][lambda]2, ... z_n = [rho][lambda]_n,
+
+  we obtain
+                                 _                       _
+                            1   |  /  R        [rho]\^2   |
+    ds^2 = [Sigma]dz^2 + ------ | ( ----- sinh ----- ) - 1| [Sigma](z_i dz_k - z_k dz_i)^2.
+                        [rho]^2 |_ \[rho]        R  /    _|
+
+  Hence when [rho] is small, we have approximately
+
+                          1
+    ds^2 = [Sigma]dz^2 + ----[Sigma](z_i dz_k - z_k dz_i)^2    (1).
+                         3R^2
+
+  Considering a surface element through the origin, we may choose our
+  axes so that, for this element,
+
+    z3 = Z4 = ... = z_n = 0.
+
+  Thus
+
+                            1
+    ds^2 = dz1^2 + dz2^2 + ----(z1dz2 - z2dz1)^2    (2).
+                           3R^2
+
+  Now the area of the triangle whose vertices are (0, 0), (z1, z2),
+  (dz1, dz2) is 1/2(z1, dz2 - z2dz1). Hence the quotient when the terms
+  of the fourth order in (2) are divided by the square of this triangle
+  is 4/3R^2; hence, returning to general axes, the same is the quotient
+  when the terms of the fourth order in (1) are divided by the square of
+  the triangle whose vertices are (0, 0, ... 0), (z1, z2, z3, ... z_n),
+  (dz1, dz2, dz3 ... dz_n). But -3/4 of this quotient is defined by
+  Riemann as the measure of curvature.[15] Hence the measure of
+  curvature is -1/R^2, i.e. is constant and negative. The properties of
+  parallels, triangles, &c., are as in the _Saggio_. It is also shown
+  that the analogues of limit surfaces have zero curvature; and that
+  spheres of radius [rho] have constant positive curvature 1/R^2 sinh^2
+  ([rho]/R), so that spherical geometry may be regarded as contained in
+  the pseudo-spherical (as Beltrami calls Lobatchewsky's system).
+
+
+  Transition to the projective method.
+
+The _Saggio_, as we saw, gives a Euclidean interpretation confined to
+two dimensions. But a consideration of the auxiliary plane suggests a
+different interpretation, which may be extended to any number of
+dimensions. If, instead of referring to the pseudosphere, we merely
+_define_ distance and angle, in the Euclidean plane, as those functions
+of the coordinates which gave us distance and angle on the pseudosphere,
+we find that the geometry of our plane has become Lobatchewsky's. All
+the points of the limiting circle are now at infinity, and points beyond
+it are imaginary. If we give our circle an imaginary radius the geometry
+on the plane becomes elliptic. Replacing the circle by a sphere, we
+obtain an analogous representation for three dimensions. Instead of a
+circle or sphere we may take any conic or quadric. With this definition,
+if the fundamental quadric be [Sigma]_(xx) = 0, and if [Sigma]_(xx)' be
+the polar form of [Sigma]_(xx), the distance [rho] between x and x' is
+given by the projective formula
+
+  cos([rho]/k) = [Sigma]_xx'/{[Sigma]_(xx).[Sigma]_x'x'}^1/2.
+
+That this formula is projective is rendered evident by observing that
+e^(-2i[rho]/k) is the anharmonic ratio of the range consisting of the
+two points and the intersections of the line joining them with the
+fundamental quadric. With this we are brought to the third or projective
+period. The method of this period is due to Cayley; its application to
+previous non-Euclidean geometry is due to Klein. The projective method
+contains a generalization of discoveries already made by Laguerre[16] in
+1853 as regards Euclidean geometry. The arbitrariness of this procedure
+of deriving metrical geometry from the properties of conics is removed
+by Lie's theory of congruence. We then arrive at the stage of thought
+which finds its expression in the modern treatment of the axioms of
+geometry.
+
+
+  The two kinds of elliptic space.
+
+The projective method leads to a discrimination, first made by
+Klein,[17] of two varieties of Riemann's space; Klein calls these
+elliptic and spherical. They are also called the polar and antipodal
+forms of elliptic space. The latter names will here be used. The
+difference is strictly analogous to that between the diameters and the
+points of a sphere. In the polar form two straight lines in a plane
+always intersect in one and only one point; in the antipodal form they
+intersect always in two points, which are antipodes. According to the
+definition of geometry adopted in section VII. (_Axioms of Geometry_),
+the antipodal form is not to be termed "geometry," since any pair of
+coplanar straight lines intersect each other in two points. It may be
+called a "quasi-geometry." Similarly in the antipodal form two diameters
+always determine a plane, but two points on a sphere do not determine a
+great circle when they are antipodes, and two great circles always
+intersect in two points. Again, a plane does not form a boundary among
+lines through a point: we can pass from any one such line to any other
+without passing through the plane. But a great circle does divide the
+surface of a sphere. So, in the polar form, a complete straight line
+does not divide a plane, and a plane does not divide space, and does
+not, like a Euclidean plane, have two sides.[18] But, in the antipodal
+form, a plane is, in these respects, like a Euclidean plane.
+
+It is explained in section VII. in what sense the metrical geometry of
+the material world can be considered to be determinate and not a matter
+of arbitrary choice. The scientific question as to the best available
+evidence concerning the nature of this geometry is one beset with
+difficulties of a peculiar kind. We are obstructed by the fact that all
+existing physical science assumes the Euclidean hypothesis. This
+hypothesis has been involved in all actual measurements of large
+distances, and in all the laws of astronomy and physics. The principle
+of simplicity would therefore lead us, in general, where an observation
+conflicted with one or more of those laws, to ascribe this anomaly, not
+to the falsity of Euclidean geometry, but to the falsity of the laws in
+question. This applies especially to astronomy. On the earth our means
+of measurement are many and direct, and so long as no great accuracy is
+sought they involve few scientific laws. Thus we acquire, from such
+direct measurements, a very high degree of probability that the
+space-constant, if not infinite, is yet large as compared with
+terrestrial distances. But astronomical distances and triangles can only
+be measured by means of the received laws of astronomy and optics, all
+of which have been established by assuming the truth of the Euclidean
+hypothesis. It therefore remains possible (until a detailed proof of the
+contrary is forthcoming) that a large but finite space-constant, with
+different laws of astronomy and optics, would have equally explained the
+phenomena. We cannot, therefore, accept the measurements of stellar
+parallaxes, &c., as conclusive evidence that the space-constant is large
+as compared with stellar distances. For the present, on grounds of
+simplicity, we may rightly adopt this view; but it must remain possible
+that, in view of some hitherto undiscovered discrepancy, a slight
+correction of the sort suggested might prove the simplest alternative.
+But conversely, a finite parallax for very distant stars, or a negative
+parallax for any star, could not be accepted as conclusive evidence that
+our geometry is non-Euclidean, unless it were shown--and this seems
+scarcely possible--that no modification of astronomy or optics could
+account for the phenomenon. Thus although we may admit a probability
+that the space-constant is large in comparison with stellar distances, a
+conclusive proof or disproof seems scarcely possible.
+
+Finally, it is of interest to note that, though it is theoretically
+possible to prove, by scientific methods, that our geometry is
+non-Euclidean, it is wholly impossible to prove by such methods that it
+is accurately Euclidean. For the unavoidable errors of observation must
+always leave a slight margin in our measurements. A triangle might be
+found whose angles were certainly greater, or certainly less, than two
+right angles; but to prove them _exactly_ equal to two right angles must
+always be beyond our powers. If, therefore, any man cherishes a hope of
+proving the exact truth of Euclid, such a hope must be based, not upon
+scientific, but upon philosophical considerations.
+
+  BIBLIOGRAPHY.--The bibliography appended to section VII. should be
+  consulted in this connexion. Also, in addition to the citations
+  already made, the following works may be mentioned.
+
+  For Lobatchewsky's writings, cf. _Urkunden zur Geschichte der
+  nichteuklidischen Geometrie_, i., _Nikolaj Iwanowitsch Lobatschefsky_,
+  by F. Engel and P. Stackel (Leipzig, 1898). For John Bolyai's
+  _Appendix_, cf. _Absolute Geometrie nach Johann Bolyai_, by J.
+  Frischauf (Leipzig, 1872), and also the new edition of his father's
+  large work, _Tentamen_ ..., published by the Mathematical Society of
+  Budapest; the second volume contains the appendix. Cf. also J.
+  Frischauf, _Elemente der absoluten Geometrie_ (Leipzig, 1876); M.L.
+  Gerard, _Sur la geometrie non-Euclidienne_ (thesis for doctorate)
+  (Paris, 1892); de Tilly, _Essai sur les principes fondamentales de la
+  geometrie et de la mecanique_ (Bordeaux, 1879); Sir R.S. Ball, "On the
+  Theory of Content," _Trans. Roy. Irish Acad._ vol. xxix. (1889); F.
+  Lindemann, "Mechanik bei projectiver Maasbestimmung," _Math. Annal._
+  vol. vii.; W.K. Clifford, "Preliminary Sketch of Biquaternions,"
+  _Proc. of Lond. Math. Soc._ (1873), and _Coll. Works_; A. Buchheim,
+  "On the Theory of Screws in Elliptic Space," _Proc. Lond. Math. Soc._
+  vols. xv., xvi., xvii.; H. Cox, "On the Application of Quaternions and
+  Grassmann's Algebra to different Kinds of Uniform Space," _Trans.
+  Camb. Phil. Soc._ (1882); M. Dehn, "Die Legendarischen Satze uber die
+  Winkelsumme im Dreieck," Math. Ann. vol. 53 (1900), and "Uber den
+  Rauminhalt," _Math. Annal._ vol. 55 (1902).
+
+  For expositions of the whole subject, cf. F. Klein, _Nicht-Euklidische
+  Geometrie_ (Gottingen, 1893); R. Bonola, _La Geometria non-Euclidea_
+  (Bologna, 1906); P. Barbarin, _La Geometrie non-Euclidienne_ (Paris,
+  1902); W. Killing, _Die nicht-Euklidischen Raumformen in analytischer
+  Behandlung_ (Leipzig, 1885). The last-named work also deals with
+  geometry of more than three dimensions; in this connexion cf. also G.
+  Veronese, _Fondamenti di geometria a piu dimensioni ed a piu specie_
+  _di unita rettilinee_ ... (Padua, 1891, German translation, Leipzig,
+  1894); G. Fontene, _L'Hyperespace a (n-1) dimensions_ (Paris, 1892);
+  and A.N. Whitehead, _loc. cit._ Cf. also E. Study, "Uber
+  nicht-Euklidische und Liniengeometrie," _Jahr. d. Deutsch. Math. Ver._
+  vol. xv. (1906); W. Burnside, "On the Kinematics of non-Euclidean
+  Space," _Proc. Lond. Math. Soc._ vol. xxvi. (1894). A bibliography on
+  the subject up to 1878 has been published by G.B. Halsted, _Amer.
+  Journ. of Math._ vols. i. and ii.; and one up to 1900 by R. Bonola,
+  _Index operum ad geometriam absolutam spectantium_ ... (1902, and
+  Leipzig, 1903).     (B. A. W. R.; A. N. W.)
+
+
+VII. AXIOMS OF GEOMETRY
+
+  Theories of space.
+
+Until the discovery of the non-Euclidean geometries (Lobatchewsky, 1826
+and 1829; J. Bolyai, 1832; B. Riemann, 1854), geometry was universally
+considered as being exclusively the science of existent space. (See
+section VI. _Non-Euclidean Geometry_.) In respect to the science, as
+thus conceived, two controversies may be noticed. First, there is the
+controversy respecting the absolute and relational theories of space.
+According to the absolute theory, which is the traditional view (held
+explicitly by Newton), space has an existence, in some sense whatever it
+may be, independent of the bodies which it contains. The bodies occupy
+space, and it is not intrinsically unmeaning to say that any definite
+body occupies _this_ part of space, and not _that_ part of space,
+without reference to other bodies occupying space. According to the
+relational theory of space, of which the chief exponent was
+Leibnitz,[19] space is nothing but a certain assemblage of the relations
+between the various particular bodies in space. The idea of space with
+no bodies in it is absurd. Accordingly there can be no meaning in saying
+that a body is _here_ and not _there_, apart from a reference to the
+other bodies in the universe. Thus, on this theory, absolute motion is
+intrinsically unmeaning. It is admitted on all hands that in practice
+only relative motion is directly measurable. Newton, however, maintains
+in the _Principia_ (scholium to the 8th definition) that it is
+indirectly measurable by means of the effects of "centrifugal force" as
+it occurs in the phenomena of rotation. This irrelevance of absolute
+motion (if there be such a thing) to science has led to the general
+adoption of the relational theory by modern men of science. But no
+decisive argument for either view has at present been elaborated.[20]
+Kant's view of space as being a form of perception at first sight
+appears to cut across this controversy. But he, saturated as he was with
+the spirit of the Newtonian physics, must (at least in both editions of
+the _Critique_) be classed with the upholders of the absolute theory.
+The form of perception has a type of existence proper to itself
+independently of the particular bodies which it contains. For example he
+writes:[21] "Space does not represent any quality of objects by
+themselves, or objects in their relation to one another, i.e. space does
+not represent any determination which is inherent in the objects
+themselves, and would remain, even if all subjective conditions of
+intuition were removed."
+
+
+  Axioms.
+
+The second controversy is that between the view that the axioms
+applicable to space are known only from experience, and the view that in
+some sense these axioms are given _a priori_. Both these views, thus
+broadly stated, are capable of various subtle modifications, and a
+discussion of them would merge into a general treatise on epistemology.
+The cruder forms of the _a priori_ view have been made quite untenable
+by the modern mathematical discoveries. Geometers now profess ignorance
+in many respects of the exact axioms which apply to existent space, and
+it seems unlikely that a profound study of the question should thus
+obliterate _a priori_ intuitions.
+
+Another question irrelevant to this article, but with some relevance to
+the above controversy, is that of the derivation of our perception of
+existent space from our various types of sensation. This is a question
+for psychology.[22]
+
+_Definition of Abstract Geometry._--Existent space is the subject matter
+of only one of the applications of the modern science of abstract
+geometry, viewed as a branch of pure mathematics. Geometry has been
+defined[23] as "the study of series of two or more dimensions." It has
+also been defined[24] as "the science of cross classification." These
+definitions are founded upon the actual practice of mathematicians in
+respect to their use of the term "Geometry." Either of them brings out
+the fact that geometry is not a science with a determinate subject
+matter. It is concerned with any subject matter to which the formal
+axioms may apply. Geometry is not peculiar in this respect. All branches
+of pure mathematics deal merely with types of relations. Thus the
+fundamental ideas of geometry (e.g. those of _points_ and of _straight
+lines_) are not ideas of determinate entities, but of any entities for
+which the axioms are true. And a set of formal geometrical axioms cannot
+in themselves be true or false, since they are not determinate
+propositions, in that they do not refer to a determinate subject matter.
+The axioms are propositional functions.[25] When a set of axioms is
+given, we can ask (1) whether they are consistent, (2) whether their
+"existence theorem" is proved, (3) whether they are independent. Axioms
+are consistent when the contradictory of any axiom cannot be deduced
+from the remaining axioms. Their existence theorem is the proof that
+they are true when the fundamental ideas are considered as denoting some
+determinate subject matter, so that the axioms are developed into
+determinate propositions. It follows from the logical law of
+contradiction that the proof of the existence theorem proves also the
+consistency of the axioms. This is the only method of proof of
+consistency. The axioms of a set are independent of each other when no
+axiom can be deduced from the remaining axioms of the set. The
+independence of a given axiom is proved by establishing the consistency
+of the remaining axioms of the set, together with the contradictory of
+the given axiom. The enumeration of the axioms is simply the enumeration
+of the hypotheses[26] (with respect to the undetermined subject matter)
+of which some at least occur in each of the subsequent propositions.
+
+Any science is called a "geometry" if it investigates the theory of the
+classification of a set of entities (the points) into classes (the
+straight lines), such that (1) there is one and only one class which
+contains any given pair of the entities, and (2) every such class
+contains more than two members. In the two geometries, important from
+their relevance to existent space, axioms which secure an order of the
+points on any line also occur. These geometries will be called
+"Projective Geometry" and "Descriptive Geometry." In projective geometry
+any two straight lines in a plane intersect, and the straight lines are
+closed series which return into themselves, like the circumference of a
+circle. In descriptive geometry two straight lines in a plane do not
+necessarily intersect, and a straight line is an open series without
+beginning or end. Ordinary Euclidean geometry is a descriptive geometry;
+it becomes a projective geometry when the so-called "points at infinity"
+are added.
+
+
+_Projective Geometry._
+
+Projective geometry may be developed from two undefined fundamental
+ideas, namely, that of a "point" and that of a "straight line." These
+undetermined ideas take different specific meanings for the various
+specific subject matters to which projective geometry can be applied.
+The number of the axioms is always to some extent arbitrary, being
+dependent upon the verbal forms of statement which are adopted. They
+will be presented[27] here as twelve in number, eight being "axioms of
+classification," and four being "axioms of order."
+
+_Axioms of Classification._--The eight axioms of classification are as
+follows:
+
+1. Points form a class of entities with at least two members.
+
+2. Any straight line is a class of points containing at least three
+members.
+
+3. Any two distinct points lie in one and only one straight line.
+
+4. There is at least one straight line which does not contain all the
+points.
+
+5. If A, B, C are non-collinear points, and A' is on the straight line
+BC, and B' is on the straight line CA, then the straight lines AA' and
+BB' possess a point in common.
+
+  _Definition._--If A, B, C are any three non-collinear points, the
+  _plane_ ABC is the class of points lying on the straight lines joining
+  A with the various points on the straight line BC.
+
+6. There is at least one plane which does not contain all the points.
+
+7. There exists a plane [alpha], and a point A not incident in [alpha],
+such that any point lies in some straight line which contains both A and
+a point in [alpha].
+
+  _Definition._--Harm. (ABCD) symbolizes the following conjoint
+  statements: (1) that the points A, B, C, D are collinear, and (2) that
+  a quadrilateral can be found with one pair of opposite sides
+  intersecting at A, with the other pair intersecting at C, and with its
+  diagonals passing through B and D respectively. Then B and D are said
+  to be "harmonic conjugates" with respect to A and C.
+
+8. Harm. (ABCD) implies that B and D are distinct points.
+
+In the above axioms 4 secures at least two dimensions, axiom 5 is the
+fundamental axiom of the plane, axiom 6 secures at least three
+dimensions, and axiom 7 secures at most three dimensions. From axioms
+1-5 it can be proved that any two distinct points in a straight line
+determine that line, that any three non-collinear points in a plane
+determine that plane, that the straight line containing any two points
+in a plane lies wholly in that plane, and that any two straight lines in
+a plane intersect. From axioms 1-6 Desargue's well-known theorem on
+triangles in perspective can be proved.
+
+  The enunciation of this theorem is as follows: If ABC and A'B'C' are
+  two coplanar triangles such that the lines AA', BB', CC' are
+  concurrent, then the three points of intersection of BC and B'C' of CA
+  and C'A', and of AB and A'B' are collinear; and conversely if the
+  three points of intersection are collinear, the three lines are
+  concurrent. The proof which can be applied is the usual projective
+  proof by which a third triangle A"B"C" is constructed not coplanar
+  with the other two, but in perspective with each of them.
+
+  It has been proved[28] that Desargues's theorem cannot be deduced from
+  axioms 1-5, that is, if the geometry be confined to two dimensions.
+  All the proofs proceed by the method of producing a specification of
+  "points" and "straight lines" which satisfies axioms 1-5, and such
+  that Desargues's theorem does not hold.
+
+  It follows from axioms 1-5 that Harm. (ABCD) implies Harm. (ADCB) and
+  Harm. (CBAD), and that, if A, B, C be any three distinct collinear
+  points, there exists at least one point D such that Harm. (ABCD). But
+  it requires Desargues's theorem, and hence axiom 6, to prove that
+  Harm. (ABCD) and Harm. (ABCD') imply the identity of D and D'.
+
+The necessity for axiom 8 has been proved by G. Fano,[29] who has
+produced a three dimensional geometry of fifteen points, i.e. a method
+of cross classification of fifteen entities, in which each straight line
+contains three points, and each plane contains seven straight lines. In
+this geometry axiom 8 does not hold. Also from axioms 1-6 and 8 it
+follows that Harm. (ABCD) implies Harm. (BCDA).
+
+  _Definitions._--When two plane figures can be derived from one another
+  by a single projection, they are said to be in _perspective_. When two
+  plane figures can be derived one from the other by a finite series of
+  perspective relations between intermediate figures, they are said to
+  be _projectively_ related. Any property of a plane figure which
+  necessarily also belongs to any projectively related figure, is called
+  a _projective_ property.
+
+  The following theorem, known from its importance as "the fundamental
+  theorem of projective geometry," cannot be proved[30] from axioms 1-8.
+  The enunciation is: "A projective correspondence between the points on
+  two straight lines is completely determined when the correspondents of
+  three distinct points on one line are determined on the other." This
+  theorem is equivalent[31] (assuming axioms 1-8) to another theorem,
+  known as Pappus's Theorem, namely: "If l and l' are two distinct
+  coplanar lines, and A, B, C are three distinct points on l, and A',
+  B', C' are three distinct points on l', then the three points of
+  intersection of AA' and B'C, of A'B and CC', of BB' and C'A, are
+  collinear." This theorem is obviously Pascal's well-known theorem
+  respecting a hexagon inscribed in a conic, for the special case when
+  the conic has degenerated into the two lines l and l'. Another theorem
+  also equivalent (assuming axioms 1-8) to the fundamental theorem is
+  the following:[32] If the three collinear pairs of points, A and A', B
+  and B', C and C', are such that the three pairs of opposite sides of a
+  complete quadrangle pass respectively through them, i.e. one pair
+  through A and A' respectively, and so on, and if also the three sides
+  of the quadrangle which pass through A, B, and C, are concurrent in
+  one of the corners of the quadrangle, then another quadrangle can be
+  found with the same relation to the three pairs of points, except that
+  its three sides which pass through A, B, and C, are not concurrent.
+
+  Thus, if we choose to take any one of these three theorems as an
+  axiom, all the theorems of projective geometry which do not require
+  ordinal or metrical ideas for their enunciation can be proved. Also a
+  conic can be defined as the locus of the points found by the usual
+  construction, based upon Pascal's theorem, for points on the conic
+  through five given points. But it is unnecessary to assume here any
+  one of the suggested axioms; for the fundamental theorem can be
+  deduced from the axioms of order together with axioms 1-8.
+
+_Axioms of Order._--It is possible to define (cf. Pieri, _loc. cit._)
+the property upon which the order of points on a straight line depends.
+But to secure that this property does in fact range the points in a
+serial order, some axioms are required. A straight line is to be a
+closed series; thus, when the points are in order, it requires two
+points on the line to divide it into two distinct complementary
+segments, which do not overlap, and together form the whole line.
+Accordingly the problem of the definition of order reduces itself to the
+definition of these two segments formed by any two points on the line;
+and the axioms are stated relatively to these segments.
+
+  _Definition._--If A, B, C are three collinear points, the points on
+  the _segment_ ABC are defined to be those points such as X, for which
+  there exist two points Y and Y' with the property that Harm. (AYCY')
+  and Harm. (BYXY') both hold. The _supplementary segment_ ABC is
+  defined to be the rest of the points on the line. This definition is
+  elucidated by noticing that with our ordinary geometrical ideas, if B
+  and X are any two points between A and C, then the two pairs of
+  points, A and C, B and X, define an involution with real double
+  points, namely, the Y and Y' of the above definition. The property of
+  belonging to a segment ABC is projective, since the harmonic relation
+  is projective.
+
+The first three axioms of order (cf. Pieri, _loc. cit._) are:
+
+9. If A, B, C are three distinct collinear points, the supplementary
+segment ABC is contained within the segment BCA.
+
+10. If A, B, C are three distinct collinear points, the common part of
+the segments BCA and CAB is contained in the supplementary segment ABC.
+
+11. If A, B, C are three distinct collinear points, and D lies In the
+segment ABC, then the segment ADC is contained within the segment ABC.
+
+From these axioms all the usual properties of a closed order follow. It
+will be noticed that, if A, B, C are any three collinear points, C is
+necessarily traversed in passing from A to B by one route along the
+line, and is not traversed in passing from A to B along the other route.
+Thus there is no meaning, as referred to closed straight lines, in the
+simple statement that C lies between A and B. But there may be a
+relation of separation between two pairs of collinear points, such as A
+and C, and B and D. The couple B and D is said to separate A and C, if
+the four points are collinear and D lies in the segment complementary to
+the segment ABC. The property of the separation of pairs of points by
+pairs of points is projective. Also it can be proved that Harm. (ABCD)
+implies that B and D separate A and C.
+
+  _Definitions._--A series of entities arranged in a serial order, open
+  or closed, is said to be _compact_, if the series contains no
+  immediately consecutive entities, so that in traversing the series
+  from any one entity to any other entity it is necessary to pass
+  through entities distinct from either. It was the merit of R. Dedekind
+  and of G. Cantor explicitly to formulate another fundamental property
+  of series. The Dedekind property[33] as applied to an open series can
+  be defined thus: An open series possesses the Dedekind property, if,
+  however, it be divided into two mutually exclusive classes u and v,
+  which (1) contain between them the whole series, and (2) are such that
+  every member of u precedes in the serial order every member of v,
+  there is always a member of the series, belonging to one of the two, u
+  or v, which precedes every member of v (other than itself if it belong
+  to v), and also succeeds every member of u (other than itself if it
+  belong to u). Accordingly in an open series with the Dedekind property
+  there is always a member of the series marking the junction of two
+  classes such as u and v. An open series is _continuous_ if it is
+  compact and possesses the Dedekind property. A closed series can
+  always be transformed into an open series by taking any arbitrary
+  member as the first term and by taking one of the two ways round as
+  the ascending order of the series. Thus the definitions of compactness
+  and of the Dedekind property can be at once transferred to a closed
+  series.
+
+12. The last axiom of order is that there exists at least one straight
+line for which the point order possesses the Dedekind property.
+
+It follows from axioms 1-12 by projection that the Dedekind property is
+true for all lines. Again the _harmonic system_ ABC, where A, B, C are
+collinear points, is defined[34] thus: take the harmonic conjugates A',
+B', C' of each point with respect to the other two, again take the
+harmonic conjugates of each of the six points A, B, C, A', B', C' with
+respect to each pair of the remaining five, and proceed in this way by
+an unending series of steps. The set of points thus obtained is called
+the harmonic system ABC. It can be proved that a harmonic system is
+compact, and that every segment of the line containing it possesses
+members of it. Furthermore, it is easy to prove that the fundamental
+theorem holds for harmonic systems, in the sense that, if A, B, C are
+three points on a line l, and A', B', C' are three points on a line l',
+and if by any two distinct series of projections A, B, C are projected
+into A', B', C', then any point of the harmonic system ABC corresponds
+to the same point of the harmonic system A'B'C' according to both the
+projective relations which are thus established between l and l'. It now
+follows immediately that the fundamental theorem must hold for all the
+points on the lines l and l', since (as has been pointed out) harmonic
+systems are "everywhere dense" on their containing lines. Thus the
+fundamental theorem follows from the axioms of order.
+
+A system of numerical coordinates can now be introduced, possessing the
+property that linear equations represent planes and straight lines. The
+outline of the argument by which this remarkable problem (in that
+"distance" is as yet undefined) is solved, will now be given. It is
+first proved that the points on any line can in a certain way be
+definitely associated with all the positive and negative real numbers,
+so as to form with them a one-one correspondence. The arbitrary elements
+in the establishment of this relation are the points on the line
+associated with 0, 1 and [oo].
+
+This association[35] is most easily effected by considering a class of
+projective relations of the line with itself, called by F. Schur (_loc.
+cit._) _prospectivities_.
+
+  Let l (fig. 69) be the given line, m and n any two lines intersecting
+  at U on l, S and S' two points on n. Then a projective relation
+  between l and itself is formed by projecting l from S on to m, and
+  then by projecting m from S' back on to l. All such projective
+  relations, however m, n, S and S' be varied, are called
+  "prospectivities," and U is the double point of the prospectivity. If
+  a point O on l is related to A by a prospectivity, then all
+  prospectivities, which (1) have the same double point U, and (2)
+  relate O to A, give the same correspondent (Q, in figure) to any point
+  P on the line l; in fact they are all the same prospectivity, however
+  m, n, S, and S' may have been varied subject to these conditions. Such
+  a prospectivity will be denoted by (OAU^2).
+
+  [Illustration: FIG. 69.]
+
+  The sum of two prospectivities, written (OAU^2) + (OBU^2), is defined
+  to be that transformation of the line l into itself which is obtained
+  by first applying the prospectivity (OAU^2) and then applying the
+  prospectivity (OBU^2). Such a transformation, when the two summands
+  have the same double point, is itself a prospectivity with that double
+  point.
+
+  [Illustration: FIG. 70]
+
+  With this definition of addition it can be proved that prospectivities
+  with the same double point satisfy all the axioms of magnitude.
+  Accordingly they can be associated in a one-one correspondence with
+  the positive and negative real numbers. Let E (fig. 70) be any point
+  on l, distinct from O and U. Then the prospectivity (OEU^2) is
+  associated with unity, the prospectivity (OOU^2) is associated with
+  zero, and (OUU^2) with [infinity]. The prospectivities of the type
+  (OPU^2), where P is any point on the segment OEU, correspond to the
+  positive numbers; also if P' is the harmonic conjugate of P with
+  respect to O and U, the prospectivity (OP'U^2) is associated with the
+  corresponding negative number. (The subjoined figure explains this
+  relation of the positive and negative prospectivities.) Then any point
+  P on l is associated with the same number as is the prospectivity
+  (OPU^2).
+
+  [Illustration: FIG. 71.]
+
+  It can be proved that the order of the numbers in algebraic order of
+  magnitude agrees with the order on the line of the associated points.
+  Let the numbers, assigned according to the preceding specification, be
+  said to be associated with the points according to the
+  "numeration-system (OEU)." The introduction of a coordinate system for
+  a plane is now managed as follows: Take any triangle OUV in the plane,
+  and on the lines OU and OV establish the numeration systems (OE1U) and
+  (OE2V), where E1 and E2 are arbitrarily chosen. Then (cf. fig. 71) if
+  M and N are associated with the numbers x and y according to these
+  systems, the coordinates of P are x and y. It then follows that the
+  equation of a straight line is of the form ax + by + c = 0. Both
+  coordinates of any point on the line UV are infinite. This can be
+  avoided by introducing homogeneous coordinates X, Y, Z, where x = X/Z,
+  and y = Y/Z, and Z = 0 is the equation of UV.
+
+  [Illustration: FIG. 72.]
+
+  The procedure for three dimensions is similar. Let OUVW (fig. 72) be
+  any tetrahedron, and associate points on OU, OV, OW with numbers
+  according to the numeration systems (OE1U), (OE2V), and (OE3W). Let
+  the planes VWP, WUP, UVP cut OU, OV, OW in L, M, N respectively; and
+  let x, y, z be the numbers associated with L, M, N respectively. Then
+  P is the point (x, y, z). Also homogeneous coordinates can be
+  introduced as before, thus avoiding the infinities on the plane UVW.
+
+  The cross ratio of a range of four collinear points can now be defined
+  as a number characteristic of that range. Let the coordinates of any
+  point P_r of the range P1 P2 P3 P4 be
+
+    [lambda]_r a + [mu]_r + a'  [lambda]_r b + [mu]_r b'
+    -------------------------,  ------------------------,
+       [lambda]_r + [mu]_r         [lambda]_r + [mu]_r
+
+      [lambda]_r c + [mu]_r c'
+      ------------------------, (r = 1, 2, 3, 4)
+        [lambda]_r + [mu]_r
+
+  and let ([lambda]_r [mu]_s) be written for [lambda]_r [mu]_s
+  -[lambda]_s [mu]_r. Then the cross ratio {P1 P2 P3 P4} is defined to
+  be the number ([lambda]1[mu]2) ([lambda]3[mu]4) / ([lambda]1[mu]4)
+  ([lambda]3[mu]2). The equality of the cross ratios of the ranges (P1
+  P2 P3 P4) and (Q1 Q2 Q3 Q4) is proved to be the necessary and
+  sufficient condition for their mutual projectivity. The cross ratios
+  of all harmonic ranges are then easily seen to be all equal to -1, by
+  comparing with the range (OE1UE'1) on the axis of x.
+
+  Thus all the ordinary propositions of geometry in which distance and
+  angular measure do not enter otherwise than in cross ratios can now be
+  enunciated and proved. Accordingly the greater part of the analytical
+  theory of conics and quadrics belongs to geometry at this stage The
+  theory of distance will be considered after the principles of
+  descriptive geometry have been developed.
+
+
+_Descriptive Geometry._
+
+Descriptive geometry is essentially the science of multiple order for
+open series. The first satisfactory system of axioms was given by M.
+Pasch.[36] An improved version is due to G. Peano.[37] Both these
+authors treat the idea of the class of points constituting the segment
+lying _between_ two points as an undefined fundamental idea. Thus in
+fact there are in this system two fundamental ideas, namely, of points
+and of segments. It is then easy enough to define the prolongations of
+the segments, so as to form the complete straight lines. D.
+Hilbert's[38] formulation of the axioms is in this respect practically
+based on the same fundamental ideas. His work is justly famous for some
+of the mathematical investigations contained in it, but his exposition
+of the axioms is distinctly inferior to that of Peano. Descriptive
+geometry can also be considered[39] as the science of a class of
+relations, each relation being a two-termed serial relation, as
+considered in the logic of relations, ranging the points between which
+it holds into a linear open order. Thus the relations are the straight
+lines, and the terms between which they hold are the points. But a
+combination of these two points of view yields[40] the simplest
+statement of all. Descriptive geometry is then conceived as the
+investigation of an undefined fundamental relation between three terms
+(points); and when the relation holds between three points A, B, C, the
+points are said to be "in the [linear] order ABC."
+
+O. Veblen's axioms and definitions, slightly modified, are as follows:--
+
+1. If the points A, B, C are in the order ABC, they are in the order
+CBA.
+
+2. If the points A, B, C are in the order ABC, they are not in the order
+BCA.
+
+3. If the points A, B, C are in the order ABC, A is distinct from C.
+
+4. If A and B are any two distinct points, there exists a point C such
+that A, B, C are in the order ABC.
+
+  _Definition._--The _line_ AB (A =| B) consists of A and B, and of all
+  points X in one of the possible orders, ABX, AXB, XAB. The points X in
+  the order AXB constitute the _segment_ AB.
+
+5. If points C and D (C =| D) lie on the line AB, then A lies on the
+line CD.
+
+6. There exist three distinct points A, B, C not in any of the orders
+ABC, BCA, CAB.
+
+7. If three distinct points A, B, C (fig. 73) do not lie on the same
+line, and D and E are two distinct points in the orders BCD and CEA,
+then a point F exists in the order AFB, and such that D, E, F are
+collinear.
+
+[Illustration: FIG. 73.]
+
+  _Definition._--If A, B, C are three non-collinear points, the _plane_
+  ABC is the class of points which lie on any one of the lines joining
+  any two of the points belonging to the _boundary_ of the triangle ABC,
+  the boundary being formed by the segments BC, CA and AB. The
+  _interior_ of the triangle ABC is formed by the points in segments
+  such as PQ, where P and Q are points respectively on two of the
+  segments BC, CA, AB.
+
+8. There exists a plane ABC, which does not contain all the points.
+
+  _Definition._--If A, B, C, D are four non-coplanar points, the space
+  ABCD is the class of points which lie on any of the lines containing
+  two points on the surface of the tetrahedron ABCD, the _surface_ being
+  formed by the interiors of the triangles ABC, BCD, DCA, DAB.
+
+9. There exists a space ABCD which contains all the points.
+
+10. The Dedekind property holds for the order of the points on any
+straight line.
+
+It follows from axioms 1-9 that the points on any straight line are
+arranged in an open serial order. Also all the ordinary theorems
+respecting a point dividing a straight line into two parts, a straight
+line dividing a plane into two parts, and a plane dividing space into
+two parts, follow.
+
+  Again, in any plane [alpha] consider a line l and a point A (fig. 74).
+
+  [Illustration: FIG. 74.]
+
+  Let any point B divide l into two half-lines l1 and l2. Then it can be
+  proved that the set of half-lines, emanating from A and intersecting
+  l1 (such as m), are bounded by two half-lines, of which ABC is one.
+  Let r be the other. Then it can be proved that r does not intersect
+  l1. Similarly for the half-line, such as n, intersecting l2. Let s be
+  its bounding half-line. Then two cases are possible. (1) The
+  half-lines r and s are collinear, and together form one complete line.
+  In this case, there is one and only one line (viz. r + s) through A
+  and lying in [alpha] which does not intersect l. This is the Euclidean
+  case, and the assumption that this case holds is the _Euclidean
+  parallel axiom_. But (2) the half-lines r and s may not be collinear.
+  In this case there will be an infinite number of lines, such as k for
+  instance, containing A and lying in [alpha], which do not intersect l.
+  Then the lines through A in [alpha] are divided into two classes by
+  reference to l, namely, the _secant_ lines which intersect l, and the
+  _non-secant_ lines which do not intersect l. The two boundary
+  non-secant lines, of which r and s are respectively halves, may be
+  called the two parallels to l through A.
+
+  The perception of the possibility of case 2 constituted the
+  starting-point from which Lobatchewsky constructed the first explicit
+  coherent theory of non-Euclidean geometry, and thus created a
+  revolution in the philosophy of the subject. For many centuries the
+  speculations of mathematicians on the foundations of geometry were
+  almost confined to hopeless attempts to prove the "parallel axiom"
+  without the introduction of some equivalent axiom.[41]
+
+_Associated Projective and Descriptive Spaces._--A region of a
+projective space, such that one, and only one, of the two supplementary
+segments between any pair of points within it lies entirely within it,
+satisfies the above axioms (1-10) of descriptive geometry, where the
+points of the region are the descriptive points, and the portions of
+straight lines within the region are the descriptive lines. If the
+excluded part of the original projective space is a single plane, the
+Euclidean parallel axiom also holds, otherwise it does not hold for the
+descriptive space of the limited region. Again, conversely, starting
+from an original descriptive space an associated projective space can be
+constructed by means of the concept of _ideal points_.[42] These are
+also called _projective points_, where it is understood that the simple
+points are the points of the original descriptive space. An _ideal
+point_ is the class of straight lines which is composed of two coplanar
+lines a and b, together with the lines of intersection of all pairs of
+intersecting planes which respectively contain a and b, together with
+the lines of intersection with the plane ab of all planes containing any
+one of the lines (other than a or b) already specified as belonging to
+the ideal point. It is evident that, if the two original lines a and b
+intersect, the corresponding ideal point is nothing else than the whole
+class of lines which are concurrent at the point ab. But the essence of
+the definition is that an ideal point has an existence when the lines a
+and b do not intersect, so long as they are coplanar. An ideal point is
+termed _proper_, if the lines composing it intersect; otherwise it is
+_improper_.
+
+A theorem essential to the whole theory is the following: if any two of
+the three lines a, b, c are coplanar, but the three lines are not all
+coplanar, and similarly for the lines a, b, d, then c and d are
+coplanar. It follows that any two lines belonging to an ideal point can
+be used as the pair of guiding lines in the definition. An ideal point
+is said to be _coherent_ with a plane, if any of the lines composing it
+lie in the plane. An _ideal line_ is the class of ideal points each of
+which is coherent with two given planes. If the planes intersect, the
+ideal line is termed _proper_, otherwise it is _improper_. It can be
+proved that any two planes, with which any two of the ideal points are
+both coherent, will serve as the guiding planes used in the definition.
+The ideal planes are defined as in projective geometry, and all the
+other definitions (for segments, order, &c.) of projective geometry are
+applied to the ideal elements. If an ideal plane contains some proper
+ideal points, it is called _proper_, otherwise it is _improper_. Every
+ideal plane contains some improper ideal points.
+
+It can now be proved that all the axioms of projective geometry hold of
+the ideal elements as thus obtained; and also that the order of the
+ideal points as obtained by the projective method agrees with the order
+of the proper ideal points as obtained from that of the associated
+points of the descriptive geometry. Thus a projective space has been
+constructed out of the ideal elements, and the proper ideal elements
+correspond element by element with the associated descriptive elements.
+Thus the proper ideal elements form a region in the projective space
+within which the descriptive axioms hold. Accordingly, by substituting
+ideal elements, a descriptive space can always be considered as a region
+within a projective space. This is the justification for the ordinary
+use of the "points at infinity" in the ordinary Euclidean geometry; the
+reasoning has been transferred from the original descriptive space to
+the associated projective space of ideal elements; and with the
+Euclidean parallel axiom the improper ideal elements reduce to the ideal
+points on a single improper ideal plane, namely, the plane at
+infinity.[43]
+
+_Congruence and Measurement._--The property of physical space which is
+expressed by the term "measurability" has now to be considered. This
+property has often been considered as essential to the very idea of
+space. For example, Kant writes,[44] "Space is represented as an
+infinite given _quantity_." This quantitative aspect of space arises
+from the measurability of distances, of angles, of surfaces and of
+volumes. These four types of quantity depend upon the two first among
+them as fundamental. The measurability of space is essentially connected
+with the idea of _congruence_, of which the simplest examples are to be
+found in the proofs of equality by the method of superposition, as used
+in elementary plane geometry. The mere concepts of "part" and of "whole"
+must of necessity be inadequate as the foundation of measurement, since
+we require the comparison as to quantity of regions of space which have
+no portions in common. The idea of congruence, as exemplified by the
+method of superposition in geometrical reasoning, appears to be founded
+upon that of the "rigid body," which moves from one position to another
+with its internal spatial relations unchanged. But unless there is a
+previous concept of the metrical relations between the parts of the
+body, there can be no basis from which to deduce that they are
+unchanged.
+
+It would therefore appear as if the idea of the congruence, or metrical
+equality, of two portions of space (as empirically suggested by the
+motion of rigid bodies) must be considered as a fundamental idea
+incapable of definition in terms of those geometrical concepts which
+have already been enumerated. This was in effect the point of view of
+Pasch.[45] It has, however, been proved by Sophus Lie[46] that
+congruence is capable of definition without recourse to a new
+fundamental idea. This he does by means of his theory of finite
+continuous groups (see GROUPS, THEORY OF), of which the definition is
+possible in terms of our established geometrical ideas, remembering that
+coordinates have already been introduced. The displacement of a rigid
+body is simply a mode of defining to the senses a one-one transformation
+of all space into itself. For at any point of space a particle may be
+conceived to be placed, and to be rigidly connected with the rigid body;
+and thus there is a definite correspondence of any point of space with
+the new point occupied by the associated particle after displacement.
+Again two successive displacements of a rigid body from position A to
+position B, and from position B to position C, are the same in effect as
+one displacement from A to C. But this is the characteristic "group"
+property. Thus the transformations of space into itself defined by
+displacements of rigid bodies form a group.
+
+Call this group of transformations a congruence-group. Now according to
+Lie a congruence-group is defined by the following characteristics:--
+
+1. A congruence-group is a finite continuous group of one-one
+transformations, containing the identical transformation.
+
+2. It is a sub-group of the general projective group, i.e. of the group
+of which any transformation converts planes into planes, and straight
+lines into straight lines.
+
+3. An infinitesimal transformation can always be found satisfying the
+condition that, at least throughout a certain enclosed region, any
+definite line and any definite point on the line are latent, i.e.
+correspond to themselves.
+
+4. No infinitesimal transformation of the group exists, such that, at
+least in the region for which (3) holds, a straight line, a point on it,
+and a plane through it, shall all be latent.
+
+The property enunciated by conditions (3) and (4), taken together, is
+named by Lie "Free mobility in the infinitesimal." Lie proves the
+following theorems for a projective space:--
+
+  1. If the above four conditions are only satisfied by a group
+  throughout part of projective space, this part either ([alpha]) must
+  be the region enclosed by a real closed quadric, or ([beta]) must be
+  the whole of the projective space with the exception of a single
+  plane. In case ([alpha]) the corresponding congruence group is the
+  continuous group for which the enclosing quadric is latent; and in
+  case ([beta]) an imaginary conic (with a real equation) lying in the
+  latent plane is also latent, and the congruence group is the
+  continuous group for which the plane and conic are latent.
+
+  2. If the above four conditions are satisfied by a group throughout
+  the whole of projective space, the congruence group is the continuous
+  group for which some imaginary quadric (with a real equation) is
+  latent.
+
+  By a proper choice of non-homogeneous co-ordinates the equation of any
+  quadrics of the types considered, either in theorem 1 ([alpha]), or in
+  theorem 2, can be written in the form 1 +c(x^2 + y^2 + z^2) = 0, where
+  c is negative for a real closed quadric, and positive for an imaginary
+  quadric. Then the general infinitesimal transformation is defined by
+  the three equations:
+
+    dx/dt = u - [omega]3y + [omega]2z + cx(ux + vy + wz), \
+    dy/dt = v - [omega]1z + [omega]3x + cy(ux + vy + wz),  > (A)
+    dz/dt = w - [omega]2x + [omega]1y + cz(ux + vy + wz). /
+
+  In the ease considered in theorem 1 ([beta]), with the proper choice
+  of co-ordinates the three equations defining the general infinitesimal
+  transformation are:
+
+    dx/dt = u - [omega]3y + [omega]2z, \
+    dy/dt = v - [omega]1z + [omega]3x,  > (B)
+    dz/dt = w - [omega]2x + [omega]1y. /
+
+  In this case the latent plane is the plane for which at least one of
+  x, y, z are infinite, that is, the plane 0.x + 0.y + 0.z + a = 0; and
+  the latent conic is the conic in which the cone x^2 + y^2 + z^2 = 0
+  intersects the latent plane.
+
+It follows from theorems 1 and 2 that there is not one unique
+congruence-group, but an indefinite number of them. There is one
+congruence-group corresponding to each closed real quadric, one to each
+imaginary quadric with a real equation, and one to each imaginary conic
+in a real plane and with a real equation. The quadric thus associated
+with each congruence-group is called the _absolute_ for that group, and
+in the degenerate case of 1 ([beta]) the absolute is the latent plane
+together with the latent imaginary conic. If the absolute is real, the
+congruence-group is _hyperbolic_; if imaginary, it is _elliptic_; if the
+absolute is a plane and imaginary conic, the group is parabolic.
+Metrical geometry is simply the theory of the properties of some
+particular congruence-group selected for study.
+
+  The definition of distance is connected with the corresponding
+  congruence-group by two considerations in respect to a range of five
+  points (A1, A2, P1, P2, P3), of which A1 and A2 are on the absolute.
+
+  Let {A1P1A2P2} stand for the cross ratio (as defined above) of the
+  range (A1P1A2P2), with a similar notation for the other ranges. Then
+
+  (1)   log{A1P1A2P2} + log{A1P2A2P3} = log{A1P1A2P3},
+
+  and
+
+  (2), if the points A1, A2, P1, P2 are transformed into A'1, A'2, P'1,
+  P'2 by any transformation of the congruence-group, ([alpha])
+  {A1P(1}A2P2 = {A'1P'1A'2P'2}, since the transformation is projective,
+  and ([beta]) A'1, A'2 are on the absolute since A1 and A2 are on it.
+  Thus if we define the distance P1P2 to be 1/2k log {A1P1A2P2}, where
+  A1 and A2 are the points in which the line P1P2 cuts the absolute, and
+  k is some constant, the two characteristic properties of distance,
+  namely, (1) the addition of consecutive lengths on a straight line,
+  and (2) the invariability of distances during a transformation of the
+  congruence-group, are satisfied. This is the well-known Cayley-Klein
+  projective definition[47] of distance, which was elaborated in view of
+  the addition property alone, previously to Lie's discovery of the
+  theory of congruence-groups. For a hyperbolic group when P1 and P2 are
+  in the region enclosed by the absolute, log {A1P1A2P2} is real, and
+  therefore k must be real. For an elliptic group A1 and A2 are
+  conjugate imaginaries, and log {A1P1A2P2} is a pure imaginary, and k
+  is chosen to be [kappa]/[iota], where [kappa] is real and [iota] =
+  [root]-.
+
+  Similarly the angle between two planes, p1 and p2, is defined to be
+  (1/2[iota]) log (t1p1t2p2), where t1 and t2 are tangent planes to the
+  absolute through the line p1p2. The planes t1 and t2 are imaginary for
+  an elliptic group, and also for an hyperbolic group when the planes p1
+  and p2 intersect at points within the region enclosed by the absolute.
+  The development of the consequences of these metrical definitions is
+  the subject of non-Euclidean geometry.
+
+  The definitions for the parabolic case can be arrived at as limits of
+  those obtained in either of the other two cases by making k ultimately
+  to vanish. It is also obvious that, if P1 and P2 be the points (x1,
+  y1, z1) and (x2, y2, z2), it follows from equations (B) above that
+  {(x1 - x2)^2 + (y1 - y2)^2 + (z1 - z2)^2}^1/2 is unaltered by a
+  congruence transformation and also satisfies the addition property for
+  collinear distances. Also the previous definition of an angle can be
+  adapted to this case, by making t1 and t2 to be the tangent planes
+  through the line p1p2 to the imaginary conic. Similarly if p1 and p2
+  are intersecting lines, the same definition of an angle holds, where
+  t1 and t2 are now the lines from the point p1p2 to the two points
+  where the plane p1p2 cuts the imaginary conic. These points are in
+  fact the "circular points at infinity" on the plane. The development
+  of the consequences of these definitions for the parabolic case gives
+  the ordinary Euclidean metrical geometry.
+
+Thus the only metrical geometry for the whole of projective space is of
+the elliptic type. But the actual measure-relations (though not their
+general properties) differ according to the elliptic congruence-group
+selected for study. In a descriptive space a congruence-group should
+possess the four characteristics of such a group throughout the whole of
+the space. Then form the associated ideal projective space. The
+associated congruence-group for this ideal space must satisfy the four
+conditions throughout the region of the proper ideal points. Thus the
+boundary of this region is the absolute. Accordingly there can be no
+metrical geometry for the whole of a descriptive space unless its
+boundary (in the associated ideal space) is a closed quadric or a plane.
+If the boundary is a closed quadric, there is one possible
+congruence-group of the hyperbolic type. If the boundary is a plane (the
+plane at infinity), the possible congruence-groups are parabolic; and
+there is a congruence-group corresponding to each imaginary conic in
+this plane, together with a Euclidean metrical geometry corresponding to
+each such group. Owing to these alternative possibilities, it would
+appear to be more accurate to say that systems of quantities can be
+found in a space, rather than that space is a quantity.
+
+Lie has also deduced[48] the same results with respect to
+congruence-groups from another set of defining properties, which
+explicitly assume the existence of a quantitative relation (the
+distance) between any two points, which is invariant for any
+transformation of the congruence-group.[49]
+
+The above results, in respect to congruence and metrical geometry,
+considered in relation to existent space, have led to the doctrine[50]
+that it is intrinsically unmeaning to ask which system of metrical
+geometry is true of the physical world. Any one of these systems can be
+applied, and in an indefinite number of ways. The only question before
+us is one of convenience in respect to simplicity of statement of the
+physical laws. This point of view seems to neglect the consideration
+that science is to be relevant to the definite perceiving minds of men;
+and that (neglecting the ambiguity introduced by the invariable slight
+inexactness of observation which is not relevant to this special
+doctrine) we have, in fact, presented to our senses a definite set of
+transformations forming a congruence-group, resulting in a set of
+measure relations which are in no respect arbitrary. Accordingly our
+scientific laws are to be stated relevantly to that particular
+congruence-group. Thus the investigation of the type (elliptic,
+hyperbolic or parabolic) of this special congruence-group is a perfectly
+definite problem, to be decided by experiment. The consideration of
+experiments adapted to this object requires some development of
+non-Euclidean geometry (see section VI., _Non-Euclidean Geometry_). But
+if the doctrine means that, assuming some sort of objective reality for
+the material universe, beings can be imagined, to whom _either_ all
+congruence-groups are equally important, _or_ some other
+congruence-group is specially important, the doctrine appears to be an
+immediate deduction from the mathematical facts. Assuming a definite
+congruence-group, the investigation of surfaces (or three-dimensional
+loci in space of four dimensions) with geodesic geometries of the form
+of metrical geometries of other types of congruence-groups forms an
+important chapter of non-Euclidean geometry. Arising from this
+investigation there is a widely-spread fallacy, which has found its way
+into many philosophic writings, namely, that the possibility of the
+geometry of existent three-dimensional space being other than Euclidean
+depends on the physical existence of Euclidean space of four or more
+dimensions. The foregoing exposition shows the baselessness of this
+idea.
+
+  BIBLIOGRAPHY.--For an account of the investigations on the axioms of
+  geometry during the Greek period, see M. Cantor, _Vorlesungen uber die
+  Geschichte der Mathematik_, Bd. i. and iii.; T.L. Heath, _The Thirteen
+  Books of Euclid's Elements, a New Translation from the Greek, with
+  Introductory Essays and Commentary, Historical, Critical, and
+  Explanatory_ (Cambridge, 1908)--this work is the standard source of
+  information; W.B. Frankland, _Euclid, Book I., with a Commentary_
+  (Cambridge, 1905)--the commentary contains copious extracts from the
+  ancient commentators. The next period of really substantive importance
+  is that of the 18th century. The leading authors are: G. Saccheri,
+  S.J., _Euclides ab omni naevo vindicatus_ (Milan, 1733). Saccheri was
+  an Italian Jesuit who unconsciously discovered non-Euclidean geometry
+  in the course of his efforts to prove its impossibility. J.H. Lambert,
+  _Theorie der Parallellinien_ (1766); A.M. Legendre, _Elements de
+  geometrie_ (1794). An adequate account of the above authors is given
+  by P. Stackel and F. Engel, _Die Theorie der Parallellinien von Euklid
+  bis auf Gauss_ (Leipzig, 1895). The next period of time (roughly from
+  1800 to 1870) contains two streams of thought, both of which are
+  essential to the modern analysis of the subject. The first stream is
+  that which produced the discovery and investigation of non-Euclidean
+  geometries, the second stream is that which has produced the geometry
+  of position, comprising both projective and descriptive geometry not
+  very accurately discriminated. The leading authors on non-Euclidean
+  geometry are K.F. Gauss, in private letters to Schumacher, cf. Stackel
+  and Engel, _loc. cit._; N. Lobatchewsky, rector of the university of
+  Kazan, to whom the honour of the effective discovery of non-Euclidean
+  geometry must be assigned. His first publication was at Kazan in 1826.
+  His various memoirs have been re-edited by Engel; cf. _Urkunden zur
+  Geschichte der nichteuklidischen Geometrie_ by Stackel and Engel, vol.
+  i. "Lobatchewsky." J. Bolyai discovered non-Euclidean geometry
+  apparently in independence of Lobatchewsky. His memoir was published
+  in 1831 as an appendix to a work by his father W. Bolyai, _Tentamen
+  juventutem...._ This memoir has been separately edited by J.
+  Frischauf, _Absolute Geometrie nach J. Bolyai_ (Leipzig, 1872); B.
+  Riemann, _Uber die Hypothesen, welche der Geometrie zu Grunde liegen_
+  (1854); cf. _Gesamte Werke_, a translation in The Collected Papers of
+  W.K. Clifford. This is a fundamental memoir on the subject and must
+  rank with the work of Lobatchewsky. Riemann discovered elliptic
+  metrical geometry, and Lobatchewsky hyperbolic geometry. A full
+  account of Riemann's ideas, with the subsequent developments due to
+  Clifford, F. Klein and W. Killing, will be found in _The Boston
+  Colloquium for 1903_ (New York, 1905), article "Forms of Non-Euclidean
+  Space," by F.S. Woods. A. Cayley, _loc. cit._ (1859), and F. Klein,
+  "Uber die sogenannte nichteuklidische Geometrie," _Math. Annal._ vols.
+  iv. and vi. (1871 and 1872), between them elaborated the projective
+  theory of distance; H. Helmholtz, "Uber die thatsachlichen Grundlagen
+  der Geometrie" (1866), and "Uber die Thatsachen, die der Geometrie zu
+  Grunde liegen" (1868), both in his _Wissenschaftliche Abhandlungen_,
+  vol. ii., and S. Lie, _loc. cit._ (1890 and 1893), between them
+  elaborated the group theory of congruence.
+
+  The numberless works which have been written to suggest equivalent
+  alternatives to Euclid's parallel axioms may be neglected as being of
+  trivial importance, though many of them are marvels of geometric
+  ingenuity.
+
+  The second stream of thought confined itself within the circle of
+  ideas of Euclidean geometry. Its origin was mainly due to a succession
+  of great French mathematicians, for example, G. Monge, _Geometrie
+  descriptive_ (1800); J.V. Poncelet, _Traite des proprietes projectives
+  des figures_ (1822); M. Chasles, _Apercu historique sur l'origine et
+  le developpement des methodes en geometrie_ (Bruxelles, 1837), and
+  _Traite de geometrie superieure_ (Paris, 1852); and many others. But
+  the works which have been, and are still, of decisive influence on
+  thought as a store-house of ideas relevant to the foundations of
+  geometry are K.G.C. von Staudt's two works, _Geometrie der Lage_
+  (Nurnberg, 1847); and _Beitrage zur Geometrie der Lage_ (Nurnberg,
+  1856, 3rd ed. 1860).
+
+  The final period is characterized by the successful production of
+  exact systems of axioms, and by the final solution of problems which
+  have occupied mathematicians for two thousand years. The successful
+  analysis of the ideas involved in serial continuity is due to R.
+  Dedekind, _Stetigkeit und irrationale Zahlen_ (1872), and to G.
+  Cantor, _Grundlagen einer allgemeinen Mannigfaltigkeitslehre_
+  (Leipzig, 1883), and _Acta math._ vol. 2.
+
+  Complete systems of axioms have been stated by M. Pasch, _loc. cit._;
+  G. Peano, _loc. cit._; M. Pieri, _loc. cit._; B. Russell, _Principles
+  of Mathematics_; O. Veblen, _loc. cit._; and by G. Veronese in his
+  treatise, _Fondamenti di geometria_ (Padua, 1891; German transl. by A.
+  Schepp, _Grundzuge der Geometrie_, Leipzig, 1894). Most of the leading
+  memoirs on special questions involved have been cited in the text; in
+  addition there may be mentioned M. Pieri, "Nuovi principii di
+  geometria projettiva complessa," _Trans. Accad. R. d. Sci._ (Turin,
+  1905); E.H. Moore, "On the Projective Axioms of Geometry," _Trans.
+  Amer. Math. Soc._, 1902; O. Veblen and W.H. Bussey, "Finite Projective
+  Geometries," _Trans. Amer. Math. Soc._, 1905; A.B. Kempe, "On the
+  Relation between the Logical Theory of Classes and the Geometrical
+  Theory of Points," _Proc. Lond. Math. Soc._, 1890; J. Royce, "The
+  Relation of the Principles of Logic to the Foundations of Geometry,"
+  _Trans. of Amer. Math. Soc._, 1905; A. Schoenflies, "Uber die
+  Moglichkeit einer projectiven Geometrie bei transfiniter
+  (nichtarchimedischer) Massbestimmung," _Deutsch. M.-V. Jahresb._,
+  1906.
+
+  For general expositions of the bearings of the above investigations,
+  cf. Hon. Bertrand Russell, _loc. cit._; L. Couturat, _Les Principes
+  des mathematiques_ (Paris, 1905); H. Poincare, _loc. cit._; Russell
+  and Whitehead, _Principia mathematica_ (Cambridge, Univ. Press). The
+  philosophers whose views on space and geometric truth deserve especial
+  study are Descartes, Leibnitz, Hume, Kant and J.S. Mill.     (A. N. W.)
+
+
+FOOTNOTES:
+
+  [1] For Egyptian geometry see EGYPT, S _Science and Mathematics_.
+
+  [2] Cf. A.N. Whitehead, _Universal Algebra_, Bk. vi. (Cambridge,
+    1898).
+
+  [3] Cf. A.N. Whitehead, _loc. cit._
+
+  [4] Cf. A.N. Whitehead, "The Geodesic Geometry of Surfaces in
+    non-Euclidean Space," _Proc. Lond. Math. Soc._ vol. xxix.
+
+  [5] Cf. Klein, "Zur nicht-Euklidischen Geometrie," _Math. Annal._
+    vol. xxxvii.
+
+  [6] On the theory of parallels before Lobatchewsky, see Stackel und
+    Engel, _Theorie der Parallellinien von Euklid bis auf Gauss_
+    (Leipzig, 1895). The foregoing remarks are based upon the materials
+    collected in this work.
+
+  [7] See Stackel und Engel, _op. cit._, and "Gauss, die beiden Bolyai,
+    und die nicht-Euklidische Geometrie," _Math. Annalen_, Bd. xlix.;
+    also Engel's translation of Lobatchewsky (Leipzig, 1898), pp. 378 ff.
+
+  [8] Lobatchewsky's works on the subject are the following:--"On the
+    Foundations of Geometry," _Kazan Messenger_, 1829-1830; "New
+    Foundations of Geometry, with a complete Theory of Parallels,"
+    _Proceedings of the University of Kazan_, 1835 (both in Russian, but
+    translated into German by Engel, Leipzig, 1898); "Geometrie
+    imaginaire," Crelle's Journal, 1837; _Theorie der Parallellinien_
+    (Berlin, 1840; 2nd ed., 1887; translated by Halsted, Austin, Texas,
+    1891). His results appear to have been set forth in a paper (now
+    lost) which he read at Kazan in 1826.
+
+  [9] Translated by Halsted (Austin, Texas, 4th ed., 1896.)
+
+  [10] _Abhandlungen d. Konigl. Ges. d. Wiss. zu Gottingen_, Bd. xiii.;
+    _Ges. math. Werke_, pp. 254-269; translated by Clifford, _Collected
+    Mathematical Papers_.
+
+  [11] Cf. _Gesamm. math. und phys. Werke_, vol. i. (Leipzig, 1894).
+
+  [12] _Wiss. Abh._ vol. ii. pp. 610, 618 (1866, 1868).
+
+  [13] _Mind_, O.S., vols. i. and iii.; _Vortrage und Reden_, vol. ii.
+    pp. 1, 256.
+
+  [14] His papers are "Saggio di interpretazione della geometria
+    non-Euclidea," _Giornale di matematiche_, vol. vi. (1868); "Teoria
+    fondamentale degli spazii di curvatura costante," _Annali di
+    matematica_, vol. ii. (1868-1869). Both were translated into French
+    by J. Houel, _Annales scientifiques de l'Ecole Normale superieure_,
+    vol. vi. (1869).
+
+  [15] Beltrami shows also that this definition agrees with that of
+    Gauss.
+
+  [16] "Sur la theorie des foyers," _Nouv. Ann._ vol. xii.
+
+  [17] _Math. Annalen_, iv. vi., 1871-1872.
+
+  [18] For an investigation of these and similar properties, see
+    Whitehead, _Universal Algebra_ (Cambridge, 1898), bk. vi. ch. ii. The
+    polar form was independently discovered by Simon Newcomb in 1877.
+
+  [19] For an analysis of Leibnitz's ideas on space, cf. B. Russell,
+    _The Philosophy of Leibnitz_, chs. viii.-x.
+
+  [20] Cf. Hon. Bertrand Russell, "Is Position in Time and Space
+    Absolute or Relative?" _Mind_, n.s. vol. 10 (1901), and A.N.
+    Whitehead, "Mathematical Concepts of the Material World," _Phil.
+    Trans._ (1906), p. 205.
+
+  [21] Cf. _Critique of Pure Reason_, 1st section: "Of Space,"
+    conclusion A, Max Muller's translation.
+
+  [22] Cf. Ernst Mach, _Erkenntniss und Irrtum_ (Leipzig); the relevant
+    chapters are translated by T.J. McCormack, _Space and Geometry_
+    (London, 1906); also A. Meinong, _Uber die Stellung der
+    Gegenstandstheorie im System der Wissenschaften_ (Leipzig, 1907).
+
+  [23] Cf. Russell, _Principles of Mathematics_, S 352 (Cambridge,
+    1903).
+
+  [24] Cf. A.N. Whitehead, _The Axioms of Projective Geometry_, S 3
+    (Cambridge, 1906).
+
+  [25] Cf. Russell, _Princ. of Math._, ch. i.
+
+  [26] Cf. Russell, _loc. cit._, and G. Frege, "Uber die Grundlagen der
+    Geometrie," _Jahresber. der Deutsch. Math. Ver._ (1906).
+
+  [27] This formulation--though not in respect to number--is in all
+    essentials that of M. Pieri, cf. "I principii della Geometria di
+    Posizione," _Accad. R. di Torino_ (1898); also cf. Whitehead, _loc.
+    cit._
+
+  [28] Cf. G. Peano, "Sui fondamenti della Geometria," p. 73, _Rivista
+    di matematica_, vol. iv. (1894), and D. Hilbert, _Grundlagen der
+    Geometrie_ (Leipzig, 1899); and R.F. Moulton, "A Simple
+    non-Desarguesian Plane Geometry," _Trans. Amer. Math. Soc._, vol.
+    iii. (1902).
+
+  [29] Cf. "Sui postulati fondamentali della geometria projettiva,"
+    _Giorn. di matematica_, vol. xxx. (1891); also of Pieri, _loc. cit._,
+    and Whitehead, _loc. cit._
+
+  [30] Cf. Hilbert, _loc. cit._; for a fuller exposition of Hilbert's
+    proof cf. K.T. Vahlen, _Abstrakte Geometrie_ (Leipzig, 1905), also
+    Whitehead, _loc. cit._
+
+  [31] Cf. H. Wiener, _Jahresber. der Deutsch. Math. Ver._ vol. i.
+    (1890); and F. Schur, "Uber den Fundamentalsatz der projectiven
+    Geometrie," _Math. Ann._ vol. li. (1899).
+
+  [32] Cf. Hilbert, _loc. cit._, and Whitehead, _loc. cit._
+
+  [33] Cf. Dedekind, _Stetigkeit und irrationale Zahlen_ (1872).
+
+  [34] Cf. v. Staudt, _Geometrie der Lage_ (1847).
+
+  [35] Cf. Pasch, _Vorlesungen uber neuere Geometrie_ (Leipzig, 1882),
+    a classic work; also Fiedler, _Die darstellende Geometrie_ (1st ed.,
+    1871, 3rd ed., 1888); Clebsch, _Vorlesungen uber Geometrie_, vol.
+    iii.; Hilbert, _loc. cit._; F. Schur, _Math. Ann. Bd._ lv. (1902);
+    Vahlen, _loc. cit._; Whitehead, _loc. cit._
+
+  [36] Cf. _loc. cit._
+
+  [37] Cf. _I Principii di geometria_ (Turin, 1889) and "Sui fondamenti
+    della geometria," _Rivista di mat._ vol. iv. (1894).
+
+  [38] Cf. _loc. cit._
+
+  [39] Cf. Vailati, _Rivista di mat._ vol. iv. and Russell, _loc. cit._
+    S 376.
+
+  [40] Cf. O. Veblen, "On the Projective Axioms of Geometry," _Trans.
+    Amer. Math. Soc._ vol. iii. (1902).
+
+  [41] Cf. P. Stackel and F. Engel, _Die Theorie der Parallellinien von
+    Euklid bis auf Gauss_ (Leipzig, 1895).
+
+  [42] Cf. Pasch, _loc. cit._, and R. Bonola, "Sulla introduzione degli
+    enti improprii in geometria projettive," _Giorn. di mat._ vol.
+    xxxviii. (1900); and Whitehead, _Axioms of Descriptive Geometry_
+    (Cambridge, 1907).
+
+  [43] The original idea (confined to this particular case) of ideal
+    points is due to von Staudt (_loc. cit._).
+
+  [44] Cf. _Critique_, "Trans. Aesth." Sect. I.
+
+  [45] Cf. _loc. cit._
+
+  [46] Cf. _Uber die Grundlagen der Geometrie_ (Leipzig, Ber., 1890);
+    and _Theorie der Transformationsgruppen_ (Leipzig, 1893), vol. iii.
+
+  [47] Cf. A. Cayley, "A Sixth Memoir on Quantics," _Trans. Roy. Soc._,
+    1859, and _Coll. Papers_, vol. ii.; and F. Klein, _Math. Ann._ vol.
+    iv., 1871.
+
+  [48] Cf. _loc. cit._
+
+  [49] For similar deductions from a third set of axioms, suggested in
+    essence by Peano, Riv. mat. vol. iv. _loc. cit._ cf. Whitehead, _Desc.
+    Geom. loc. cit._
+
+  [50] Cf. H. Poincare, _La Science et l'hypothese_, ch. iii.
+
+
+
+
+
+
+
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