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diff --git a/37461.txt b/37461.txt new file mode 100644 index 0000000..d5f857a --- /dev/null +++ b/37461.txt @@ -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. + + + + + + + +End of the Project Gutenberg EBook of Encyclopaedia Britannica, 11th +Edition, Volume 11, Slice 6, by Various + +*** END OF THIS PROJECT GUTENBERG EBOOK ENCYCLOPAEDIA BRITANNICA *** + +***** This file should be named 37461.txt or 37461.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/7/4/6/37461/ + +Produced by Marius Masi, Don Kretz and the Online +Distributed Proofreading Team at http://www.pgdp.net + + +Updated editions will replace the previous one--the old editions +will be renamed. + +Creating the works from public domain print editions means that no +one owns a United States copyright in these works, so the Foundation +(and you!) can copy and distribute it in the United States without +permission and without paying copyright royalties. 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